GCSE Maths Linear Equations (Foundation): worked examples and questions with answers
By the end of this chapter you will be able to
- Explain what an equation is and use the balance idea: do the same to both sides
- Solve one-step and two-step equations, including negative and fractional solutions
- Solve equations with the unknown on both sides and equations with brackets
- Solve equations that contain fractions (Higher: including algebraic fractions)
- Form an equation from words, perimeters, angles, ages and costs, solve it and interpret the answer
- Check a solution by substitution, and find approximate solutions using a graph
You should already know
- Negative numbers and the four operations (Chapter 1)
- Collecting like terms and expanding brackets (Chapter 7)
- Fractions and common denominators (Chapter 3)
Where this leads
- Rearranging formulae (Chapter 8)
- Inequalities (Chapter 10)
- Straight-line graphs (Chapter 12)
- Simultaneous equations (Chapter 14)
Edexcel GCSE (9–1) Mathematics 1MA1, Foundation tier. Content references: A17, A21.
- 9.1 Equations and the balance idea: Worksheet (PDF) · Answers (PDF) · lesson page
- 9.2 Two-step equations: Worksheet (PDF) · Answers (PDF) · lesson page
- 9.3 Negative and fractional solutions, and checking: Worksheet (PDF) · Answers (PDF) · lesson page
- 9.4 The unknown on both sides: Worksheet (PDF) · Answers (PDF) · lesson page
- 9.5 Equations with brackets: Worksheet (PDF) · Answers (PDF) · lesson page
- 9.6 Equations with fractions: Worksheet (PDF) · Answers (PDF) · lesson page
- 9.7 Forming equations from words: Worksheet (PDF) · Answers (PDF) · lesson page
- 9.8 Equations from geometry: perimeters and angles: Worksheet (PDF) · Answers (PDF) · lesson page
- 9.9 Solving equations using graphs, and interpreting solutions: Worksheet (PDF) · Answers (PDF) · lesson page
- Check-up and exam-style questions: Worksheet (PDF) · Answers (PDF) · lesson page
Exercise P Do you remember?
Non-calculatorMethod. Use a number line: subtracting a bigger number from a smaller one takes you below zero.
- Start at 4 and count back 9. Count back 4 to reach 0, then 5 more.4 − 4 = 0, then 0 − 5 = −5
- Write the answer. The answer is below zero, so it is negative.4 − 9 = −5
Check. −5 + 9 = 4. ✓
Watch out. 9 − 4 = 5, but 4 − 9 is not the same: the order matters in subtraction.
Method. Collect like terms: add the x terms together and leave the plain number.
- Find the like terms. 3x and 5x are both x terms. The −2 is a plain number, so it is a different kind of term.3x and 5x go together
- Add them. 3 + 5 = 8, so 3x + 5x = 8x.8x − 2
Check. Try x = 1: 3 + 5 − 2 = 6 and 8 − 2 = 6. ✓
Watch out. You cannot add 8x and −2 any further, because they are different types of term.
Method. Multiply every term inside the bracket by the number outside.
- Multiply the first term. 3 times x.3 × x = 3x
- Multiply the second term. 3 times 4.3 × 4 = 12
- Join them with the + sign. 3x + 12
Check. Try x = 2: 3 × 6 = 18 and 3 × 2 + 12 = 18. ✓
Watch out. Multiply both terms, not only the first.
Method. Substitute 4 in place of n, then follow the order of operations (multiply before adding).
- Replace n with 4. 2n means 2 × n.2 × 4 + 3
- Multiply first. Multiplication comes before addition.8 + 3
- Add. = 11
Check. If n = 4 then 2n = 8, and 8 + 3 = 11. ✓
Watch out. Do not work from left to right: 2 + 3 = 5 and then 5 × 4 would be wrong.
Method. List the multiples of each number and find the smallest one they share.
- Multiples of 2. Count in twos.2, 4, 6, 8, 10, …
- Multiples of 3. Count in threes.3, 6, 9, 12, …
- Find the smallest in both lists. 6 is in both lists, so the LCM is 6
Check. 6 ÷ 2 = 3 and 6 ÷ 3 = 2, both exact. ✓
Watch out. The LCM is the smallest shared multiple, not the product of the two numbers (although here it happens to be 6).
Equations and the balance idea
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
In algebra a letter stands for a number that we do not know yet. An expression such as 3x + 5 has no equals sign. It is just a recipe for making a number, so it has no single answer. An equation such as 3x + 5 = 20 does have an equals sign. It says that two things have the same value. To solve an equation means to find the value of the letter that makes the statement true.
The best way to think about an equation is as a pair of balanced scales. Whatever is on the left pan weighs exactly the same as whatever is on the right pan. In the picture, the blue box is an unknown mass x and each orange circle is 1 kg. The scales balance, so x + 3 = 7.
This gives us the one rule of equation solving: whatever you do to one side, do the same to the other side. If you take 3 kg off one pan but not the other, the scales tip and the equation is no longer true. Our aim is always to get the letter on its own, with nothing added to it, taken away from it, multiplying it or dividing it.
To get the letter on its own we undo what has been done to it. We use the inverse operation, which is the operation that does the opposite. Adding and subtracting are inverses of each other. Multiplying and dividing are inverses of each other.
| The equation looks like | It means | Do this to both sides |
|---|---|---|
| x + 5 = 12 | 5 has been added | subtract 5 |
| x − 5 = 12 | 5 has been taken away | add 5 |
| 5x = 20 | x has been multiplied by 5 | divide by 5 |
| x5 = 4 | x has been divided by 5 | multiply by 5 |
Remember that 5x is short for 5 × x: in algebra we do not write the multiplication sign. Also remember that x5 means x ÷ 5, as you saw with fractions in Chapter 3.
Key ideas
- An equation has an equals sign; an expression does not.
- To solve an equation is to find the value of the letter that makes it true.
- Do the same to both sides so that the equation stays balanced.
- Use the inverse operation: the opposite of + is −, and the opposite of × is ÷.
- Finish with the letter on its own: x = a number.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x + 7 = 15
- Subtract 7 from both sides. This undoes the +7, because +7 − 7 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 7 − 7 = 15 − 7
x = 8 - Write the solution. x is now on its own, so the equation tells us its value.x = 8
Check. Substitute x = 8 into the original equation. Left side: 8 + 7 = 15. Right side: 15. Both sides equal 15, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x − 4 = 9
- Add 4 to both sides. This undoes the −4, because −4 + 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x − 4 + 4 = 9 + 4
x = 13 - Write the solution. x is now on its own, so the equation tells us its value.x = 13
Check. Substitute x = 13 into the original equation. Left side: 13 − 4 = 9. Right side: 9. Both sides equal 9, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5x = 35
- Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 355
x = 355 = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
Check. Substitute x = 7 into the original equation. Left side: 5 × 7 = 35. Right side: 35. Both sides equal 35, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x3 = 6
- Multiply both sides by 3. x3 means x divided by 3. The inverse of dividing by 3 is multiplying by 3. We do it to both sides to keep the balance.3 × x3 = 3 × 6
x = 18 - Write the solution. x is now on its own, so the equation tells us its value.x = 18
Check. Substitute x = 18 into the original equation. Left side: 183 = 6. Right side: 6. Both sides equal 6, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Watch out
Do the opposite operation. To solve x + 6 = 15 you subtract 6 from both sides, giving x = 9. Adding 6 would give 21, which is wrong. Always check your answer by putting it back into the equation.Exercise 9A Equations and one-step equations
Non-calculatorMethod. An equation has an equals sign (=) and says two things are equal. An expression has no equals sign.
- Look for an equals sign. This is the quickest way to tell the difference.4x + 1: there is no = sign.
- Decide. An equals sign means the statement can be true or false, so we can solve it. Without one it is only a recipe for working out a number.This is an expression.
Check. There is nothing to solve, because there is no equals sign. ✓
Watch out. Do not call 3x + 5 an equation: it has no equals sign.
Method. An equation has an equals sign (=) and says two things are equal. An expression has no equals sign.
- Look for an equals sign. This is the quickest way to tell the difference.4x + 1 = 13: there is an = sign.
- Decide. An equals sign means the statement can be true or false, so we can solve it. Without one it is only a recipe for working out a number.This is an equation.
Check. An equation can be solved to find a value of x; an expression cannot, so it is not solved. ✓
Watch out. Do not call 3x + 5 an equation: it has no equals sign.
Method. An equation has an equals sign (=) and says two things are equal. An expression has no equals sign.
- Look for an equals sign. This is the quickest way to tell the difference.7 − x: there is no = sign.
- Decide. An equals sign means the statement can be true or false, so we can solve it. Without one it is only a recipe for working out a number.This is an expression.
Check. There is nothing to solve, because there is no equals sign. ✓
Watch out. Do not call 3x + 5 an equation: it has no equals sign.
Method. An equation has an equals sign (=) and says two things are equal. An expression has no equals sign.
- Look for an equals sign. This is the quickest way to tell the difference.x5 = 2: there is an = sign.
- Decide. An equals sign means the statement can be true or false, so we can solve it. Without one it is only a recipe for working out a number.This is an equation.
Check. An equation can be solved to find a value of x; an expression cannot, so it is not solved. ✓
Watch out. Do not call 3x + 5 an equation: it has no equals sign.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x + 3 = 10
- Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 3 − 3 = 10 − 3
x = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
Check. Substitute x = 7 into the original equation. Left side: 7 + 3 = 10. Right side: 10. Both sides equal 10, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x + 8 = 15
- Subtract 8 from both sides. This undoes the +8, because +8 − 8 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 8 − 8 = 15 − 8
x = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
Check. Substitute x = 7 into the original equation. Left side: 7 + 8 = 15. Right side: 15. Both sides equal 15, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x + 6 = 9
- Subtract 6 from both sides. This undoes the +6, because +6 − 6 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 6 − 6 = 9 − 6
x = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
Check. Substitute x = 3 into the original equation. Left side: 3 + 6 = 9. Right side: 9. Both sides equal 9, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x + 15 = 40
- Subtract 15 from both sides. This undoes the +15, because +15 − 15 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 15 − 15 = 40 − 15
x = 25 - Write the solution. x is now on its own, so the equation tells us its value.x = 25
Check. Substitute x = 25 into the original equation. Left side: 25 + 15 = 40. Right side: 40. Both sides equal 40, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.7 + x = 12
- Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: write the x term first (adding can be done in any order).
The equation is now x + 7 = 12 - Subtract 7 from both sides. This undoes the +7, because +7 − 7 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 7 − 7 = 12 − 7
x = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 7 + 5 = 12. Right side: 12. Both sides equal 12, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x + 25 = 61
- Subtract 25 from both sides. This undoes the +25, because +25 − 25 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 25 − 25 = 61 − 25
x = 36 - Write the solution. x is now on its own, so the equation tells us its value.x = 36
Check. Substitute x = 36 into the original equation. Left side: 36 + 25 = 61. Right side: 61. Both sides equal 61, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x − 2 = 7
- Add 2 to both sides. This undoes the −2, because −2 + 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x − 2 + 2 = 7 + 2
x = 9 - Write the solution. x is now on its own, so the equation tells us its value.x = 9
Check. Substitute x = 9 into the original equation. Left side: 9 − 2 = 7. Right side: 7. Both sides equal 7, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x − 5 = 9
- Add 5 to both sides. This undoes the −5, because −5 + 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x − 5 + 5 = 9 + 5
x = 14 - Write the solution. x is now on its own, so the equation tells us its value.x = 14
Check. Substitute x = 14 into the original equation. Left side: 14 − 5 = 9. Right side: 9. Both sides equal 9, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x − 10 = 4
- Add 10 to both sides. This undoes the −10, because −10 + 10 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x − 10 + 10 = 4 + 10
x = 14 - Write the solution. x is now on its own, so the equation tells us its value.x = 14
Check. Substitute x = 14 into the original equation. Left side: 14 − 10 = 4. Right side: 4. Both sides equal 4, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x − 1 = 20
- Add 1 to both sides. This undoes the −1, because −1 + 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x − 1 + 1 = 20 + 1
x = 21 - Write the solution. x is now on its own, so the equation tells us its value.x = 21
Check. Substitute x = 21 into the original equation. Left side: 21 − 1 = 20. Right side: 20. Both sides equal 20, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x − 8 = 8
- Add 8 to both sides. This undoes the −8, because −8 + 8 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x − 8 + 8 = 8 + 8
x = 16 - Write the solution. x is now on its own, so the equation tells us its value.x = 16
Check. Substitute x = 16 into the original equation. Left side: 16 − 8 = 8. Right side: 8. Both sides equal 8, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x − 3 = 0
- Add 3 to both sides. This undoes the −3, because −3 + 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x − 3 + 3 = 0 + 3
x = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
Check. Substitute x = 3 into the original equation. Left side: 3 − 3 = 0. Right side: 0. Both sides equal 0, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x = 14
- Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 142
x = 142 = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
Check. Substitute x = 7 into the original equation. Left side: 2 × 7 = 14. Right side: 14. Both sides equal 14, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5x = 20
- Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 205
x = 205 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 5 × 4 = 20. Right side: 20. Both sides equal 20, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3x = 27
- Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 273
x = 273 = 9 - Write the solution. x is now on its own, so the equation tells us its value.x = 9
Check. Substitute x = 9 into the original equation. Left side: 3 × 9 = 27. Right side: 27. Both sides equal 27, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.10x = 90
- Divide both sides by 10. 10x means 10 multiplied by x. The inverse of multiplying by 10 is dividing by 10, and we do it to both sides to keep the balance. This leaves x on its own.10x10 = 9010
x = 9010 = 9 - Write the solution. x is now on its own, so the equation tells us its value.x = 9
Check. Substitute x = 9 into the original equation. Left side: 10 × 9 = 90. Right side: 90. Both sides equal 90, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4x = 36
- Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 364
x = 364 = 9 - Write the solution. x is now on its own, so the equation tells us its value.x = 9
Check. Substitute x = 9 into the original equation. Left side: 4 × 9 = 36. Right side: 36. Both sides equal 36, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.7x = 56
- Divide both sides by 7. 7x means 7 multiplied by x. The inverse of multiplying by 7 is dividing by 7, and we do it to both sides to keep the balance. This leaves x on its own.7x7 = 567
x = 567 = 8 - Write the solution. x is now on its own, so the equation tells us its value.x = 8
Check. Substitute x = 8 into the original equation. Left side: 7 × 8 = 56. Right side: 56. Both sides equal 56, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x2 = 5
- Multiply both sides by 2. x2 means x divided by 2. The inverse of dividing by 2 is multiplying by 2. We do it to both sides to keep the balance.2 × x2 = 2 × 5
x = 10 - Write the solution. x is now on its own, so the equation tells us its value.x = 10
Check. Substitute x = 10 into the original equation. Left side: 102 = 5. Right side: 5. Both sides equal 5, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x3 = 4
- Multiply both sides by 3. x3 means x divided by 3. The inverse of dividing by 3 is multiplying by 3. We do it to both sides to keep the balance.3 × x3 = 3 × 4
x = 12 - Write the solution. x is now on its own, so the equation tells us its value.x = 12
Check. Substitute x = 12 into the original equation. Left side: 123 = 4. Right side: 4. Both sides equal 4, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x5 = 6
- Multiply both sides by 5. x5 means x divided by 5. The inverse of dividing by 5 is multiplying by 5. We do it to both sides to keep the balance.5 × x5 = 5 × 6
x = 30 - Write the solution. x is now on its own, so the equation tells us its value.x = 30
Check. Substitute x = 30 into the original equation. Left side: 305 = 6. Right side: 6. Both sides equal 6, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x10 = 3
- Multiply both sides by 10. x10 means x divided by 10. The inverse of dividing by 10 is multiplying by 10. We do it to both sides to keep the balance.10 × x10 = 10 × 3
x = 30 - Write the solution. x is now on its own, so the equation tells us its value.x = 30
Check. Substitute x = 30 into the original equation. Left side: 3010 = 3. Right side: 3. Both sides equal 3, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x4 = 8
- Multiply both sides by 4. x4 means x divided by 4. The inverse of dividing by 4 is multiplying by 4. We do it to both sides to keep the balance.4 × x4 = 4 × 8
x = 32 - Write the solution. x is now on its own, so the equation tells us its value.x = 32
Check. Substitute x = 32 into the original equation. Left side: 324 = 8. Right side: 8. Both sides equal 8, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x7 = 7
- Multiply both sides by 7. x7 means x divided by 7. The inverse of dividing by 7 is multiplying by 7. We do it to both sides to keep the balance.7 × x7 = 7 × 7
x = 49 - Write the solution. x is now on its own, so the equation tells us its value.x = 49
Check. Substitute x = 49 into the original equation. Left side: 497 = 7. Right side: 7. Both sides equal 7, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Method. To undo what has been done to x, use the inverse (opposite) operation on both sides.
- Look at what has been done to x. Say it in words.9 has been added to x.
- Choose the opposite operation. Adding and subtracting undo each other. Multiplying and dividing undo each other.The inverse is: subtract 9.
- Do it to both sides. Both sides must change in the same way, so the equation stays balanced.x + 9 = 14 → subtract 9 on both sides
Check. Doing this leaves x on its own. ✓
Watch out. The inverse of + is − and the inverse of × is ÷. Never use the same operation again.
Method. To undo what has been done to x, use the inverse (opposite) operation on both sides.
- Look at what has been done to x. Say it in words.6 has been taken away from x.
- Choose the opposite operation. Adding and subtracting undo each other. Multiplying and dividing undo each other.The inverse is: add 6.
- Do it to both sides. Both sides must change in the same way, so the equation stays balanced.x − 6 = 10 → add 6 on both sides
Check. Doing this leaves x on its own. ✓
Watch out. The inverse of + is − and the inverse of × is ÷. Never use the same operation again.
Method. To undo what has been done to x, use the inverse (opposite) operation on both sides.
- Look at what has been done to x. Say it in words.x has been multiplied by 4.
- Choose the opposite operation. Adding and subtracting undo each other. Multiplying and dividing undo each other.The inverse is: divide by 4.
- Do it to both sides. Both sides must change in the same way, so the equation stays balanced.4x = 20 → divide by 4 on both sides
Check. Doing this leaves x on its own. ✓
Watch out. The inverse of + is − and the inverse of × is ÷. Never use the same operation again.
Method. To undo what has been done to x, use the inverse (opposite) operation on both sides.
- Look at what has been done to x. Say it in words.x has been divided by 5.
- Choose the opposite operation. Adding and subtracting undo each other. Multiplying and dividing undo each other.The inverse is: multiply by 5.
- Do it to both sides. Both sides must change in the same way, so the equation stays balanced.x5 = 3 → multiply by 5 on both sides
Check. Doing this leaves x on its own. ✓
Watch out. The inverse of + is − and the inverse of × is ÷. Never use the same operation again.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x + 47 = 91
- Subtract 47 from both sides. This undoes the +47, because +47 − 47 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 47 − 47 = 91 − 47
x = 44 - Write the solution. x is now on its own, so the equation tells us its value.x = 44
Check. Substitute x = 44 into the original equation. Left side: 44 + 47 = 91. Right side: 91. Both sides equal 91, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.8x = 96
- Divide both sides by 8. 8x means 8 multiplied by x. The inverse of multiplying by 8 is dividing by 8, and we do it to both sides to keep the balance. This leaves x on its own.8x8 = 968
x = 968 = 12 - Write the solution. x is now on its own, so the equation tells us its value.x = 12
Check. Substitute x = 12 into the original equation. Left side: 8 × 12 = 96. Right side: 96. Both sides equal 96, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x − 29 = 31
- Add 29 to both sides. This undoes the −29, because −29 + 29 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x − 29 + 29 = 31 + 29
x = 60 - Write the solution. x is now on its own, so the equation tells us its value.x = 60
Check. Substitute x = 60 into the original equation. Left side: 60 − 29 = 31. Right side: 31. Both sides equal 31, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x6 = 12
- Multiply both sides by 6. x6 means x divided by 6. The inverse of dividing by 6 is multiplying by 6. We do it to both sides to keep the balance.6 × x6 = 6 × 12
x = 72 - Write the solution. x is now on its own, so the equation tells us its value.x = 72
Check. Substitute x = 72 into the original equation. Left side: 726 = 12. Right side: 12. Both sides equal 12, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x + 148 = 372
- Subtract 148 from both sides. This undoes the +148, because +148 − 148 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 148 − 148 = 372 − 148
x = 224 - Write the solution. x is now on its own, so the equation tells us its value.x = 224
Check. Substitute x = 224 into the original equation. Left side: 224 + 148 = 372. Right side: 372. Both sides equal 372, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.12x = 180
- Divide both sides by 12. 12x means 12 multiplied by x. The inverse of multiplying by 12 is dividing by 12, and we do it to both sides to keep the balance. This leaves x on its own.12x12 = 18012
x = 18012 = 15 - Write the solution. x is now on its own, so the equation tells us its value.x = 15
Check. Substitute x = 15 into the original equation. Left side: 12 × 15 = 180. Right side: 180. Both sides equal 180, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number Ravi thought of.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Ravi thought of x, added 9 (x + 9) and got 25.
Equation: x + 9 = 25 - Subtract 9 from both sides. This undoes the +9, because +9 − 9 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 9 − 9 = 25 − 9
x = 16 - Write the solution. x is now on its own, so the equation tells us its value.x = 16
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.Ravi thought of the number 16.
Check. 16 + 9 = 25, which is what Ravi got ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the cost of one chocolate in pence.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.A box of 8 chocolates costs £6 = 600p. So 8 chocolates cost 8x pence.
Equation: 8x = 600 - Divide both sides by 8. 8x means 8 multiplied by x. The inverse of multiplying by 8 is dividing by 8, and we do it to both sides to keep the balance. This leaves x on its own.8x8 = 6008
x = 6008 = 75 - Write the solution. x is now on its own, so the equation tells us its value.x = 75
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.One chocolate costs 75p.
Check. 8 × 75p = 600p = £6, which is the price of the box ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the mass of the box in kg.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.The left pan holds the box (x kg) and a 5 kg weight, so its mass is x + 5. The right pan holds 12 kg. The scales balance, so the two masses are equal.
Equation: x + 5 = 12 - Subtract 5 from both sides. This undoes the +5, because +5 − 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 5 − 5 = 12 − 5
x = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The box has a mass of 7 kg.
Check. 7 kg + 5 kg = 12 kg, which balances 12 kg ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Substitute the value into both sides of the equation, work each side out, and compare.
- Write the equation and the value to test. A number is a solution of an equation if it makes both sides equal. We test it by substituting it (putting it in place of the letter).x + 5 = 9, x = 4
- Substitute into the left-hand side. Replace every x with the number. Use brackets around a negative number so that the signs stay clear.Left side = 4 + 5 = 9
- Substitute into the right-hand side. Do exactly the same to the other side of the equation.Right side = 9
- Compare the two sides. If the two sides are equal the number is a solution; if not, it is not.9 = 9, so yes, x = 4 is a solution.
Check. The two sides match, which is exactly what we wanted to see. ✓
Watch out. Work out the left side and the right side separately. Do not use an = sign between the two until you have shown they are equal.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Substitute the value into both sides of the equation, work each side out, and compare.
- Write the equation and the value to test. A number is a solution of an equation if it makes both sides equal. We test it by substituting it (putting it in place of the letter).3x = 15, x = 4
- Substitute into the left-hand side. Replace every x with the number. Use brackets around a negative number so that the signs stay clear.Left side = 3 × 4 = 12
- Substitute into the right-hand side. Do exactly the same to the other side of the equation.Right side = 15
- Compare the two sides. If the two sides are equal the number is a solution; if not, it is not.12 ≠ 15, so no, x = 4 is not a solution.
Check. The correct solution is x = 5: substituting it gives 15 on both sides. ✓
Watch out. Work out the left side and the right side separately. Do not use an = sign between the two until you have shown they are equal.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Substitute the value into both sides of the equation, work each side out, and compare.
- Write the equation and the value to test. A number is a solution of an equation if it makes both sides equal. We test it by substituting it (putting it in place of the letter).x − 2 = 3, x = 4
- Substitute into the left-hand side. Replace every x with the number. Use brackets around a negative number so that the signs stay clear.Left side = 4 − 2 = 2
- Substitute into the right-hand side. Do exactly the same to the other side of the equation.Right side = 3
- Compare the two sides. If the two sides are equal the number is a solution; if not, it is not.2 ≠ 3, so no, x = 4 is not a solution.
Check. The correct solution is x = 5: substituting it gives 3 on both sides. ✓
Watch out. Work out the left side and the right side separately. Do not use an = sign between the two until you have shown they are equal.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Substitute the value into both sides of the equation, work each side out, and compare.
- Write the equation and the value to test. A number is a solution of an equation if it makes both sides equal. We test it by substituting it (putting it in place of the letter).x2 = 2, x = 4
- Substitute into the left-hand side. Replace every x with the number. Use brackets around a negative number so that the signs stay clear.Left side = 42 = 2
- Substitute into the right-hand side. Do exactly the same to the other side of the equation.Right side = 2
- Compare the two sides. If the two sides are equal the number is a solution; if not, it is not.2 = 2, so yes, x = 4 is a solution.
Check. The two sides match, which is exactly what we wanted to see. ✓
Watch out. Work out the left side and the right side separately. Do not use an = sign between the two until you have shown they are equal.
Method. Check Sam's answer by substituting it, then find the correct step.
- Test Sam's answer. Put x = 21 back into the equation.21 + 6 = 27, but the right-hand side is 15. 27 ≠ 15, so Sam is wrong.
- Find his mistake. x + 6 means 6 has been added to x. The inverse is to subtract 6, but Sam added 6 (15 + 6 = 21).He used the same operation instead of the opposite one.
- Do the correct step. Subtract 6 from both sides.x + 6 − 6 = 15 − 6
x = 9
Check. 9 + 6 = 15. ✓
Watch out. Always use the opposite operation.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.“a number plus 7” is x + 7, and “is 19” means = 19.
Equation: x + 7 = 19 - Subtract 7 from both sides. This undoes the +7, because +7 − 7 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 7 − 7 = 19 − 7
x = 12 - Write the solution. x is now on its own, so the equation tells us its value.x = 12
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The number is 12.
Check. Substitute x = 12 into the original equation. Left side: 12 + 7 = 19. Right side: 19. Both sides equal 19, so the solution is correct. ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.“a number minus 6” is x − 6, and “is 11” means = 11.
Equation: x − 6 = 11 - Add 6 to both sides. This undoes the −6, because −6 + 6 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x − 6 + 6 = 11 + 6
x = 17 - Write the solution. x is now on its own, so the equation tells us its value.x = 17
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The number is 17.
Check. Substitute x = 17 into the original equation. Left side: 17 − 6 = 11. Right side: 11. Both sides equal 11, so the solution is correct. ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.“5 times a number” is 5x, and “is 40” means = 40.
Equation: 5x = 40 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 405
x = 405 = 8 - Write the solution. x is now on its own, so the equation tells us its value.x = 8
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The number is 8.
Check. Substitute x = 8 into the original equation. Left side: 5 × 8 = 40. Right side: 40. Both sides equal 40, so the solution is correct. ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.“a number divided by 4” is x4, and “is 9” means = 9.
Equation: x4 = 9 - Multiply both sides by 4. x4 means x divided by 4. The inverse of dividing by 4 is multiplying by 4. We do it to both sides to keep the balance.4 × x4 = 4 × 9
x = 36 - Write the solution. x is now on its own, so the equation tells us its value.x = 36
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The number is 36.
Check. Substitute x = 36 into the original equation. Left side: 364 = 9. Right side: 9. Both sides equal 9, so the solution is correct. ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Method. Solve the equation that has no unknown number in it, then use that value in the other equation.
- Solve 3x = 18. Divide both sides by 3.x = 18 ÷ 3 = 6
- Use x = 6 in the other equation. The two equations have the same solution, so x is also 6 in x + a = 12.6 + a = 12
- Solve for a. Subtract 6 from both sides.a = 12 − 6 = 6
Check. 6 + 6 = 12 and 3 × 6 = 18. ✓
Watch out. Both equations are true for the same value of x, which is 6.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x + 1 = 4
- Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 1 − 1 = 4 − 1
x = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
Check. Substitute x = 3 into the original equation. Left side: 3 + 1 = 4. Right side: 4. Both sides equal 4, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x + 2 = 5
- Subtract 2 from both sides. This undoes the +2, because +2 − 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 2 − 2 = 5 − 2
x = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
Check. Substitute x = 3 into the original equation. Left side: 3 + 2 = 5. Right side: 5. Both sides equal 5, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x = 6
- Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 62
x = 62 = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
Check. Substitute x = 3 into the original equation. Left side: 2 × 3 = 6. Right side: 6. Both sides equal 6, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x − 1 = 3
- Add 1 to both sides. This undoes the −1, because −1 + 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x − 1 + 1 = 3 + 1
x = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 4 − 1 = 3. Right side: 3. Both sides equal 3, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number of sweets in one bag.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.5 bags hold 5x sweets. The total is 40.
Equation: 5x = 40 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 405
x = 405 = 8 - Write the solution. x is now on its own, so the equation tells us its value.x = 8
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.Each bag holds 8 sweets.
Check. 5 × 8 = 40 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Two-step equations
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
Many equations need two steps. Look at 3x + 5 = 20. Two things have been done to x: it was first multiplied by 3, and then 5 was added. A function machine shows this.
To get x back we undo the steps in reverse order, just as you take your shoes off before your socks if you put your socks on first. We undo the adding first (subtract 5), then the multiplying (divide by 3):
3x + 5 = 20 → subtract 5 from both sides: 3x = 15 → divide both sides by 3: x = 5.
Why this order? The “+ 5” is stuck to the whole of 3x. If you divide by 3 first, you have to divide everything (the 5 and the 20 too), which gives fractions and is much harder. Take away the number added or subtracted first, then deal with the multiplication or division.
The same method works when the equation is written the other way round, such as 17 = 2x + 5. The unknown is on the right-hand side, but you do exactly the same steps. When you reach 6 = x you can write it as x = 6, because an equals sign works both ways.
When x is divided by a number, as in x4 − 2 = 3, we still add 2 first, and then multiply by 4 to undo the division.
Key ideas
- Undo the adding or subtracting first, then undo the multiplying or dividing.
- Do the same to both sides each time.
- Write every step on a new line, one operation per line, and keep the equals signs lined up.
- Always check by substituting your answer back into the original equation.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3x + 5 = 20
- Subtract 5 from both sides. This undoes the +5, because +5 − 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 5 − 5 = 20 − 5
3x = 15 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 153
x = 153 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 3 × 5 + 5 = 15 + 5 = 20. Right side: 20. Both sides equal 20, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4x − 7 = 21
- Add 7 to both sides. This undoes the −7, because −7 + 7 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x − 7 + 7 = 21 + 7
4x = 28 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 284
x = 284 = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
Check. Substitute x = 7 into the original equation. Left side: 4 × 7 − 7 = 28 − 7 = 21. Right side: 21. Both sides equal 21, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x5 + 2 = 6
- Subtract 2 from both sides. This undoes the +2, because +2 − 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x5 + 2 − 2 = 6 − 2
x5 = 4 - Multiply both sides by 5. x5 means x divided by 5. The inverse of dividing by 5 is multiplying by 5. We do it to both sides to keep the balance.5 × x5 = 5 × 4
x = 20 - Write the solution. x is now on its own, so the equation tells us its value.x = 20
Check. Substitute x = 20 into the original equation. Left side: 205 + 2 = 4 + 2 = 6. Right side: 6. Both sides equal 6, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.17 = 2x + 5
- Subtract 5 from both sides. This undoes the +5, because +5 − 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.17 − 5 = 2x + 5 − 5
12 = 2x - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.122 = 2x2
x = 122 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
Check. Substitute x = 6 into the original equation. Left side: 17. Right side: 2 × 6 + 5 = 12 + 5 = 17. Both sides equal 17, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Watch out
Do not divide only one term. In 3x + 5 = 20, dividing just the 3x by 3 gives x + 5 = 20, which is wrong. Subtract 5 first: 3x = 15, then divide both sides by 3.Exercise 9B Two-step equations
Non-calculatorKey words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x + 3 = 11
- Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 3 − 3 = 11 − 3
2x = 8 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 82
x = 82 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 2 × 4 + 3 = 8 + 3 = 11. Right side: 11. Both sides equal 11, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3x + 1 = 16
- Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 1 − 1 = 16 − 1
3x = 15 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 153
x = 153 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 3 × 5 + 1 = 15 + 1 = 16. Right side: 16. Both sides equal 16, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5x + 4 = 24
- Subtract 4 from both sides. This undoes the +4, because +4 − 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5x + 4 − 4 = 24 − 4
5x = 20 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 205
x = 205 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 5 × 4 + 4 = 20 + 4 = 24. Right side: 24. Both sides equal 24, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4x + 7 = 31
- Subtract 7 from both sides. This undoes the +7, because +7 − 7 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x + 7 − 7 = 31 − 7
4x = 24 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 244
x = 244 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
Check. Substitute x = 6 into the original equation. Left side: 4 × 6 + 7 = 24 + 7 = 31. Right side: 31. Both sides equal 31, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x + 9 = 21
- Subtract 9 from both sides. This undoes the +9, because +9 − 9 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 9 − 9 = 21 − 9
2x = 12 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 122
x = 122 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
Check. Substitute x = 6 into the original equation. Left side: 2 × 6 + 9 = 12 + 9 = 21. Right side: 21. Both sides equal 21, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.6x + 2 = 20
- Subtract 2 from both sides. This undoes the +2, because +2 − 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.6x + 2 − 2 = 20 − 2
6x = 18 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 186
x = 186 = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
Check. Substitute x = 3 into the original equation. Left side: 6 × 3 + 2 = 18 + 2 = 20. Right side: 20. Both sides equal 20, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x − 3 = 9
- Add 3 to both sides. This undoes the −3, because −3 + 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x − 3 + 3 = 9 + 3
2x = 12 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 122
x = 122 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
Check. Substitute x = 6 into the original equation. Left side: 2 × 6 − 3 = 12 − 3 = 9. Right side: 9. Both sides equal 9, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3x − 5 = 13
- Add 5 to both sides. This undoes the −5, because −5 + 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x − 5 + 5 = 13 + 5
3x = 18 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 183
x = 183 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
Check. Substitute x = 6 into the original equation. Left side: 3 × 6 − 5 = 18 − 5 = 13. Right side: 13. Both sides equal 13, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5x − 1 = 24
- Add 1 to both sides. This undoes the −1, because −1 + 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5x − 1 + 1 = 24 + 1
5x = 25 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 255
x = 255 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 5 × 5 − 1 = 25 − 1 = 24. Right side: 24. Both sides equal 24, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4x − 9 = 15
- Add 9 to both sides. This undoes the −9, because −9 + 9 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x − 9 + 9 = 15 + 9
4x = 24 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 244
x = 244 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
Check. Substitute x = 6 into the original equation. Left side: 4 × 6 − 9 = 24 − 9 = 15. Right side: 15. Both sides equal 15, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.7x − 2 = 40
- Add 2 to both sides. This undoes the −2, because −2 + 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.7x − 2 + 2 = 40 + 2
7x = 42 - Divide both sides by 7. 7x means 7 multiplied by x. The inverse of multiplying by 7 is dividing by 7, and we do it to both sides to keep the balance. This leaves x on its own.7x7 = 427
x = 427 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
Check. Substitute x = 6 into the original equation. Left side: 7 × 6 − 2 = 42 − 2 = 40. Right side: 40. Both sides equal 40, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.10x − 6 = 34
- Add 6 to both sides. This undoes the −6, because −6 + 6 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.10x − 6 + 6 = 34 + 6
10x = 40 - Divide both sides by 10. 10x means 10 multiplied by x. The inverse of multiplying by 10 is dividing by 10, and we do it to both sides to keep the balance. This leaves x on its own.10x10 = 4010
x = 4010 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 10 × 4 − 6 = 40 − 6 = 34. Right side: 34. Both sides equal 34, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x2 + 3 = 8
- Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x2 + 3 − 3 = 8 − 3
x2 = 5 - Multiply both sides by 2. x2 means x divided by 2. The inverse of dividing by 2 is multiplying by 2. We do it to both sides to keep the balance.2 × x2 = 2 × 5
x = 10 - Write the solution. x is now on its own, so the equation tells us its value.x = 10
Check. Substitute x = 10 into the original equation. Left side: 102 + 3 = 5 + 3 = 8. Right side: 8. Both sides equal 8, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x3 + 1 = 5
- Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x3 + 1 − 1 = 5 − 1
x3 = 4 - Multiply both sides by 3. x3 means x divided by 3. The inverse of dividing by 3 is multiplying by 3. We do it to both sides to keep the balance.3 × x3 = 3 × 4
x = 12 - Write the solution. x is now on its own, so the equation tells us its value.x = 12
Check. Substitute x = 12 into the original equation. Left side: 123 + 1 = 4 + 1 = 5. Right side: 5. Both sides equal 5, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x4 − 2 = 3
- Add 2 to both sides. This undoes the −2, because −2 + 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x4 − 2 + 2 = 3 + 2
x4 = 5 - Multiply both sides by 4. x4 means x divided by 4. The inverse of dividing by 4 is multiplying by 4. We do it to both sides to keep the balance.4 × x4 = 4 × 5
x = 20 - Write the solution. x is now on its own, so the equation tells us its value.x = 20
Check. Substitute x = 20 into the original equation. Left side: 204 − 2 = 5 − 2 = 3. Right side: 3. Both sides equal 3, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x5 + 4 = 9
- Subtract 4 from both sides. This undoes the +4, because +4 − 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x5 + 4 − 4 = 9 − 4
x5 = 5 - Multiply both sides by 5. x5 means x divided by 5. The inverse of dividing by 5 is multiplying by 5. We do it to both sides to keep the balance.5 × x5 = 5 × 5
x = 25 - Write the solution. x is now on its own, so the equation tells us its value.x = 25
Check. Substitute x = 25 into the original equation. Left side: 255 + 4 = 5 + 4 = 9. Right side: 9. Both sides equal 9, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x6 − 1 = 2
- Add 1 to both sides. This undoes the −1, because −1 + 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x6 − 1 + 1 = 2 + 1
x6 = 3 - Multiply both sides by 6. x6 means x divided by 6. The inverse of dividing by 6 is multiplying by 6. We do it to both sides to keep the balance.6 × x6 = 6 × 3
x = 18 - Write the solution. x is now on its own, so the equation tells us its value.x = 18
Check. Substitute x = 18 into the original equation. Left side: 186 − 1 = 3 − 1 = 2. Right side: 2. Both sides equal 2, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x10 + 7 = 10
- Subtract 7 from both sides. This undoes the +7, because +7 − 7 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x10 + 7 − 7 = 10 − 7
x10 = 3 - Multiply both sides by 10. x10 means x divided by 10. The inverse of dividing by 10 is multiplying by 10. We do it to both sides to keep the balance.10 × x10 = 10 × 3
x = 30 - Write the solution. x is now on its own, so the equation tells us its value.x = 30
Check. Substitute x = 30 into the original equation. Left side: 3010 + 7 = 3 + 7 = 10. Right side: 10. Both sides equal 10, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.13 = 2x + 3
- Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.13 − 3 = 2x + 3 − 3
10 = 2x - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.102 = 2x2
x = 102 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 13. Right side: 2 × 5 + 3 = 10 + 3 = 13. Both sides equal 13, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.22 = 4x + 2
- Subtract 2 from both sides. This undoes the +2, because +2 − 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.22 − 2 = 4x + 2 − 2
20 = 4x - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.204 = 4x4
x = 204 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 22. Right side: 4 × 5 + 2 = 20 + 2 = 22. Both sides equal 22, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.17 = 3x − 4
- Add 4 to both sides. This undoes the −4, because −4 + 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.17 + 4 = 3x − 4 + 4
21 = 3x - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.213 = 3x3
x = 213 = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
Check. Substitute x = 7 into the original equation. Left side: 17. Right side: 3 × 7 − 4 = 21 − 4 = 17. Both sides equal 17, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.30 = 5x + 5
- Subtract 5 from both sides. This undoes the +5, because +5 − 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.30 − 5 = 5x + 5 − 5
25 = 5x - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.255 = 5x5
x = 255 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 30. Right side: 5 × 5 + 5 = 25 + 5 = 30. Both sides equal 30, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Method. Subtract the number that has been added, from both sides.
- Subtract 4 from both sides. This undoes the + 4.6x + 4 − 4 = 40 − 4
- Work out the right side. 40 − 4 = 36
- Write the new equation. 6x = 36
Check. 36 + 4 = 40. ✓
Method. Divide both sides by the number multiplying x.
- Divide both sides by 6. This undoes the multiplication by 6.6x ÷ 6 = 36 ÷ 6
- Work out the right side. 36 ÷ 6 = 6
- Write the answer. x = 6
Check. 6 × 6 + 4 = 40. ✓
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.8x + 5 = 93
- Subtract 5 from both sides. This undoes the +5, because +5 − 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.8x + 5 − 5 = 93 − 5
8x = 88 - Divide both sides by 8. 8x means 8 multiplied by x. The inverse of multiplying by 8 is dividing by 8, and we do it to both sides to keep the balance. This leaves x on its own.8x8 = 888
x = 888 = 11 - Write the solution. x is now on its own, so the equation tells us its value.x = 11
Check. Substitute x = 11 into the original equation. Left side: 8 × 11 + 5 = 88 + 5 = 93. Right side: 93. Both sides equal 93, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x7 − 3 = 4
- Add 3 to both sides. This undoes the −3, because −3 + 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x7 − 3 + 3 = 4 + 3
x7 = 7 - Multiply both sides by 7. x7 means x divided by 7. The inverse of dividing by 7 is multiplying by 7. We do it to both sides to keep the balance.7 × x7 = 7 × 7
x = 49 - Write the solution. x is now on its own, so the equation tells us its value.x = 49
Check. Substitute x = 49 into the original equation. Left side: 497 − 3 = 7 − 3 = 4. Right side: 4. Both sides equal 4, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number Mia thought of.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Mia multiplied her number by 4 (4x) and then added 3 (4x + 3). She got 35.
Equation: 4x + 3 = 35 - Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x + 3 − 3 = 35 − 3
4x = 32 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 324
x = 324 = 8 - Write the solution. x is now on its own, so the equation tells us its value.x = 8
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.Mia's number was 8.
Check. 4 × 8 = 32, and 32 + 3 = 35 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the cost of one ticket in pounds.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.3 tickets cost 3x pounds. A booking fee of £4 is added once, so the total is 3x + 4 = 31.
Equation: 3x + 4 = 31 - Subtract 4 from both sides. This undoes the +4, because +4 − 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 4 − 4 = 31 − 4
3x = 27 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 273
x = 273 = 9 - Write the solution. x is now on its own, so the equation tells us its value.x = 9
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.One ticket costs £9.
Check. 3 × £9 = £27, and £27 + £4 = £31 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number of weeks.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Jade has £12 and saves £5 each week. After x weeks she has 12 + 5x pounds. She wants £62.
Equation: 5x + 12 = 62 - Subtract 12 from both sides. This undoes the +12, because +12 − 12 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5x + 12 − 12 = 62 − 12
5x = 50 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 505
x = 505 = 10 - Write the solution. x is now on its own, so the equation tells us its value.x = 10
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.Jade needs 10 weeks.
Check. 12 + 5 × 10 = 12 + 50 = 62 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Method. Solve both equations, then compare the two answers.
- Solve Ali's equation. Subtract 3, then divide by 4.4x = 19 − 3 = 16, so x = 16 ÷ 4 = 4
- Solve Beth's equation. Add 1, then divide by 3.3x = 11 + 1 = 12, so x = 12 ÷ 3 = 4
- Compare. Both solutions are 4.Neither is bigger: they get the same answer, so the difference is 0.
Check. 4 × 4 + 3 = 19 and 3 × 4 − 1 = 11. ✓
Watch out. Read the question: it asks who is bigger, and the answer can be “neither”.
Method. Try Zoe's method on every term, then compare it with subtracting first.
- Divide every term by 3 (Zoe's method). We must divide the whole of each side, including the 5 and the 23.3x3 + 53 = 233
x + 53 = 233 - Notice the fractions. Neither 5 nor 23 divides exactly by 3, so we have to work with thirds.x = 233 − 53 = 183 = 6
- Subtract 5 first. The numbers stay whole.3x = 23 − 5 = 18, so x = 18 ÷ 3 = 6
Check. Both methods give 6, and 3 × 6 + 5 = 23. ✓
Watch out. Zoe's method is correct if every term is divided, but the other order avoids fractions.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number of pages Omar reads each day.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Omar reads x pages a day for 6 days, so 6x pages, and has read 18 pages before. The book has 90 pages: 6x + 18 = 90.
Equation: 6x + 18 = 90 - Subtract 18 from both sides. This undoes the +18, because +18 − 18 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.6x + 18 − 18 = 90 − 18
6x = 72 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 726
x = 726 = 12 - Write the solution. x is now on its own, so the equation tells us its value.x = 12
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.Omar reads 12 pages each day.
Check. 6 × 12 = 72, and 72 + 18 = 90 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x4 − 2 = 3
- Add 2 to both sides. This undoes the −2, because −2 + 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x4 − 2 + 2 = 3 + 2
x4 = 5 - Multiply both sides by 4. x4 means x divided by 4. The inverse of dividing by 4 is multiplying by 4. We do it to both sides to keep the balance.4 × x4 = 4 × 5
x = 20 - Write the solution. x is now on its own, so the equation tells us its value.x = 20
Check. Substitute x = 20 into the original equation. Left side: 204 − 2 = 5 − 2 = 3. Right side: 3. Both sides equal 3, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x + 1 = 9
- Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 1 − 1 = 9 − 1
2x = 8 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 82
x = 82 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 2 × 4 + 1 = 8 + 1 = 9. Right side: 9. Both sides equal 9, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Negative and fractional solutions, and checking
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
The answer to an equation does not have to be a positive whole number. It can be negative, a fraction or a decimal. That is not a sign that you have made a mistake. Look at 2x + 9 = 3.
Subtract 9 from both sides: 2x = 3 − 9 = −6. Now divide both sides by 2: x = −6 ÷ 2 = −3. Remember from Chapter 1 that a negative number divided by a positive number gives a negative answer, and that 3 − 9 goes below zero.
When the number you divide by does not go into the other number exactly, the answer is a fraction. If 3x = 7 then x = 73. You can leave it as an improper fraction, or write it as a mixed number: 73 = 213. If the question asks for a decimal and the fraction is exact, such as 52 = 2.5, give the decimal. Never round a fraction like 73 unless the question tells you to; it is exact, and 2.33 is not.
Sometimes the x term is negative, as in 3 − 2x = 11. Subtract 3 from both sides: −2x = 8. Now −2x means −2 × x, so divide both sides by −2: x = 8 ÷ (−2) = −4. A positive divided by a negative is negative.
Checking by substitution. You can always test an answer. Put it back into the original equation, in place of the letter, and work out the left side and the right side separately. If they are equal the answer is correct. This is especially helpful when the answer is negative or a fraction, because those are easy places to slip. Use brackets around a negative number when you substitute: 2 × (−3) + 9 = −6 + 9 = 3, which equals the right-hand side. So x = −3 works.
Key ideas
- A solution can be negative, a fraction or a decimal.
- Keep the sign with the number. Positive ÷ negative = negative, negative ÷ negative = positive.
- Leave a fraction exact (in simplest form, or as a mixed number); do not round unless asked.
- To check: substitute the answer into the original equation and show that both sides are equal. Use brackets for negative values.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x + 9 = 3
- Subtract 9 from both sides. This undoes the +9, because +9 − 9 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 9 − 9 = 3 − 9
2x = −6 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = −62
x = −62 = −3 - Write the solution. x is now on its own, so the equation tells us its value.x = −3
Check. Substitute x = −3 into the original equation. Left side: 2 × (−3) + 9 = −6 + 9 = 3. Right side: 3. Both sides equal 3, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4x = 10
- Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 104
x = 104 = 52 - Write the solution. x is now on its own, so the equation tells us its value.x = 52
- Write the answer as a mixed number. The question asks for a mixed number. Divide the top by the bottom: the whole number is how many times it goes in, the remainder is the numerator of the fraction.5 ÷ 2 = 2 remainder 1
x = 212
Check. Substitute x = 52 into the original equation. Left side: 4 × 52 = 10. Right side: 10. Both sides equal 10, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3 − 2x = 11
- Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: write the x term first (adding can be done in any order).
The equation is now −2x + 3 = 11 - Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.−2x + 3 − 3 = 11 − 3
−2x = 8 - Divide both sides by −2. −2x means −2 multiplied by x. The inverse of multiplying by −2 is dividing by −2, and we do it to both sides to keep the balance. This leaves x on its own. Careful with signs: a positive divided by a negative gives a negative answer, and a negative divided by a negative gives a positive.−2x−2 = 8−2
x = 8−2 = −4 - Write the solution. x is now on its own, so the equation tells us its value.x = −4
Check. Substitute x = −4 into the original equation. Left side: 3 − 2 × (−4) = 3 + 8 = 11. Right side: 11. Both sides equal 11, so the solution is correct. ✓
Watch out. Dividing by a negative number changes the sign: a positive divided by a negative gives a negative. Check your answer by substitution.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Substitute the value into both sides of the equation, work each side out, and compare.
- Write the equation and the value to test. A number is a solution of an equation if it makes both sides equal. We test it by substituting it (putting it in place of the letter).5x + 7 = 2x + 1, x = −2
- Substitute into the left-hand side. Replace every x with the number. Use brackets around a negative number so that the signs stay clear.Left side = 5 × (−2) + 7 = −10 + 7 = −3
- Substitute into the right-hand side. Do exactly the same to the other side of the equation.Right side = 2 × (−2) + 1 = −4 + 1 = −3
- Compare the two sides. If the two sides are equal the number is a solution; if not, it is not.−3 = −3, so yes, x = −2 is a solution.
Check. The two sides match, which is exactly what we wanted to see. ✓
Watch out. Work out the left side and the right side separately. Do not use an = sign between the two until you have shown they are equal.
Watch out
When you substitute a negative number, put it in brackets. 5 × (−2) = −10. Without brackets, writing 5 −2 looks like a subtraction and the sign gets lost. Also, do not drop the minus sign when you divide: −6 ÷ 2 = −3, not 3.Exercise 9C Negative and fractional solutions
Non-calculatorKey words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x + 9 = 4
- Subtract 9 from both sides. This undoes the +9, because +9 − 9 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 9 − 9 = 4 − 9
x = −5 - Write the solution. x is now on its own, so the equation tells us its value.x = −5
Check. Substitute x = −5 into the original equation. Left side: (−5) + 9 = −5 + 9 = 4. Right side: 4. Both sides equal 4, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x + 12 = 3
- Subtract 12 from both sides. This undoes the +12, because +12 − 12 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 12 − 12 = 3 − 12
x = −9 - Write the solution. x is now on its own, so the equation tells us its value.x = −9
Check. Substitute x = −9 into the original equation. Left side: (−9) + 12 = −9 + 12 = 3. Right side: 3. Both sides equal 3, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x + 10 = 4
- Subtract 10 from both sides. This undoes the +10, because +10 − 10 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 10 − 10 = 4 − 10
2x = −6 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = −62
x = −62 = −3 - Write the solution. x is now on its own, so the equation tells us its value.x = −3
Check. Substitute x = −3 into the original equation. Left side: 2 × (−3) + 10 = −6 + 10 = 4. Right side: 4. Both sides equal 4, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3x + 14 = 5
- Subtract 14 from both sides. This undoes the +14, because +14 − 14 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 14 − 14 = 5 − 14
3x = −9 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = −93
x = −93 = −3 - Write the solution. x is now on its own, so the equation tells us its value.x = −3
Check. Substitute x = −3 into the original equation. Left side: 3 × (−3) + 14 = −9 + 14 = 5. Right side: 5. Both sides equal 5, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x − 3 = −7
- Add 3 to both sides. This undoes the −3, because −3 + 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x − 3 + 3 = −7 + 3
x = −4 - Write the solution. x is now on its own, so the equation tells us its value.x = −4
Check. Substitute x = −4 into the original equation. Left side: (−4) − 3 = −4 − 3 = −7. Right side: −7. Both sides equal −7, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4x − 1 = −13
- Add 1 to both sides. This undoes the −1, because −1 + 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x − 1 + 1 = −13 + 1
4x = −12 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = −124
x = −124 = −3 - Write the solution. x is now on its own, so the equation tells us its value.x = −3
Check. Substitute x = −3 into the original equation. Left side: 4 × (−3) − 1 = −12 − 1 = −13. Right side: −13. Both sides equal −13, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x = 5
- Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 52
x = 52 - Write the solution. x is now on its own, so the equation tells us its value.x = 52
Check. Substitute x = 52 into the original equation. Left side: 2 × 52 = 5. Right side: 5. Both sides equal 5, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3x = 4
- Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 43
x = 43 - Write the solution. x is now on its own, so the equation tells us its value.x = 43
Check. Substitute x = 43 into the original equation. Left side: 3 × 43 = 4. Right side: 4. Both sides equal 4, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4x = 6
- Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 64
x = 64 = 32 - Write the solution. x is now on its own, so the equation tells us its value.x = 32
Check. Substitute x = 32 into the original equation. Left side: 4 × 32 = 6. Right side: 6. Both sides equal 6, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5x = 2
- Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 25
x = 25 - Write the solution. x is now on its own, so the equation tells us its value.x = 25
Check. Substitute x = 25 into the original equation. Left side: 5 × 25 = 2. Right side: 2. Both sides equal 2, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.6x = 9
- Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 96
x = 96 = 32 - Write the solution. x is now on its own, so the equation tells us its value.x = 32
Check. Substitute x = 32 into the original equation. Left side: 6 × 32 = 9. Right side: 9. Both sides equal 9, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x + 1 = 6
- Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 1 − 1 = 6 − 1
2x = 5 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 52
x = 52 - Write the solution. x is now on its own, so the equation tells us its value.x = 52
Check. Substitute x = 52 into the original equation. Left side: 2 × 52 + 1 = 5 + 1 = 6. Right side: 6. Both sides equal 6, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3x = 7
- Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 73
x = 73 - Write the solution. x is now on its own, so the equation tells us its value.x = 73
- Write the answer as a mixed number. The question asks for a mixed number. Divide the top by the bottom: the whole number is how many times it goes in, the remainder is the numerator of the fraction.7 ÷ 3 = 2 remainder 1
x = 213
Check. Substitute x = 73 into the original equation. Left side: 3 × 73 = 7. Right side: 7. Both sides equal 7, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4x + 1 = 10
- Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x + 1 − 1 = 10 − 1
4x = 9 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 94
x = 94 - Write the solution. x is now on its own, so the equation tells us its value.x = 94
- Write the answer as a mixed number. The question asks for a mixed number. Divide the top by the bottom: the whole number is how many times it goes in, the remainder is the numerator of the fraction.9 ÷ 4 = 2 remainder 1
x = 214
Check. Substitute x = 94 into the original equation. Left side: 4 × 94 + 1 = 9 + 1 = 10. Right side: 10. Both sides equal 10, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5x − 2 = 14
- Add 2 to both sides. This undoes the −2, because −2 + 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5x − 2 + 2 = 14 + 2
5x = 16 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 165
x = 165 - Write the solution. x is now on its own, so the equation tells us its value.x = 165
- Write the answer as a mixed number. The question asks for a mixed number. Divide the top by the bottom: the whole number is how many times it goes in, the remainder is the numerator of the fraction.16 ÷ 5 = 3 remainder 1
x = 315
Check. Substitute x = 165 into the original equation. Left side: 5 × 165 − 2 = 16 − 2 = 14. Right side: 14. Both sides equal 14, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x + 3 = 12
- Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 3 − 3 = 12 − 3
2x = 9 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 92
x = 92 - Write the solution. x is now on its own, so the equation tells us its value.x = 92
- Write the answer as a mixed number. The question asks for a mixed number. Divide the top by the bottom: the whole number is how many times it goes in, the remainder is the numerator of the fraction.9 ÷ 2 = 4 remainder 1
x = 412
Check. Substitute x = 92 into the original equation. Left side: 2 × 92 + 3 = 9 + 3 = 12. Right side: 12. Both sides equal 12, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x + 1 = 6
- Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 1 − 1 = 6 − 1
2x = 5 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 52
x = 52 - Write the solution. x is now on its own, so the equation tells us its value.x = 52
- Write the answer as a decimal. The question asks for a decimal. The fraction bar means divide, so divide the top by the bottom.5 ÷ 2 = 2.5
x = 2.5
Check. Substitute x = 52 into the original equation. Left side: 2 × 52 + 1 = 5 + 1 = 6. Right side: 6. Both sides equal 6, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4x = 1
- Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 14
x = 14 - Write the solution. x is now on its own, so the equation tells us its value.x = 14
- Write the answer as a decimal. The question asks for a decimal. The fraction bar means divide, so divide the top by the bottom.1 ÷ 4 = 0.25
x = 0.25
Check. Substitute x = 14 into the original equation. Left side: 4 × 14 = 1. Right side: 1. Both sides equal 1, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5x + 2 = 3
- Subtract 2 from both sides. This undoes the +2, because +2 − 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5x + 2 − 2 = 3 − 2
5x = 1 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 15
x = 15 - Write the solution. x is now on its own, so the equation tells us its value.x = 15
- Write the answer as a decimal. The question asks for a decimal. The fraction bar means divide, so divide the top by the bottom.1 ÷ 5 = 0.2
x = 0.2
Check. Substitute x = 15 into the original equation. Left side: 5 × 15 + 2 = 1 + 2 = 3. Right side: 3. Both sides equal 3, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.8x − 3 = 1
- Add 3 to both sides. This undoes the −3, because −3 + 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.8x − 3 + 3 = 1 + 3
8x = 4 - Divide both sides by 8. 8x means 8 multiplied by x. The inverse of multiplying by 8 is dividing by 8, and we do it to both sides to keep the balance. This leaves x on its own.8x8 = 48
x = 48 = 12 - Write the solution. x is now on its own, so the equation tells us its value.x = 12
- Write the answer as a decimal. The question asks for a decimal. The fraction bar means divide, so divide the top by the bottom.1 ÷ 2 = 0.5
x = 0.5
Check. Substitute x = 12 into the original equation. Left side: 8 × 12 − 3 = 4 − 3 = 1. Right side: 1. Both sides equal 1, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5 − x = 2
- Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: write the x term first (adding can be done in any order).
The equation is now −x + 5 = 2 - Subtract 5 from both sides. This undoes the +5, because +5 − 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.−x + 5 − 5 = 2 − 5
−x = −3 - Divide both sides by −1. −x means −1 multiplied by x. The inverse of multiplying by −1 is dividing by −1, and we do it to both sides to keep the balance. This leaves x on its own. Careful with signs: a positive divided by a negative gives a negative answer, and a negative divided by a negative gives a positive.−x−1 = −3−1
x = −3−1 = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
Check. Substitute x = 3 into the original equation. Left side: 5 − 3 = 2. Right side: 2. Both sides equal 2, so the solution is correct. ✓
Watch out. Dividing by a negative number changes the sign: a positive divided by a negative gives a negative. Check your answer by substitution.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.10 − 2x = 4
- Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: write the x term first (adding can be done in any order).
The equation is now −2x + 10 = 4 - Subtract 10 from both sides. This undoes the +10, because +10 − 10 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.−2x + 10 − 10 = 4 − 10
−2x = −6 - Divide both sides by −2. −2x means −2 multiplied by x. The inverse of multiplying by −2 is dividing by −2, and we do it to both sides to keep the balance. This leaves x on its own. Careful with signs: a positive divided by a negative gives a negative answer, and a negative divided by a negative gives a positive.−2x−2 = −6−2
x = −6−2 = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
Check. Substitute x = 3 into the original equation. Left side: 10 − 2 × 3 = 10 − 6 = 4. Right side: 4. Both sides equal 4, so the solution is correct. ✓
Watch out. Dividing by a negative number changes the sign: a positive divided by a negative gives a negative. Check your answer by substitution.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3 − 2x = 11
- Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: write the x term first (adding can be done in any order).
The equation is now −2x + 3 = 11 - Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.−2x + 3 − 3 = 11 − 3
−2x = 8 - Divide both sides by −2. −2x means −2 multiplied by x. The inverse of multiplying by −2 is dividing by −2, and we do it to both sides to keep the balance. This leaves x on its own. Careful with signs: a positive divided by a negative gives a negative answer, and a negative divided by a negative gives a positive.−2x−2 = 8−2
x = 8−2 = −4 - Write the solution. x is now on its own, so the equation tells us its value.x = −4
Check. Substitute x = −4 into the original equation. Left side: 3 − 2 × (−4) = 3 + 8 = 11. Right side: 11. Both sides equal 11, so the solution is correct. ✓
Watch out. Dividing by a negative number changes the sign: a positive divided by a negative gives a negative. Check your answer by substitution.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.7 − 3x = −8
- Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: write the x term first (adding can be done in any order).
The equation is now −3x + 7 = −8 - Subtract 7 from both sides. This undoes the +7, because +7 − 7 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.−3x + 7 − 7 = −8 − 7
−3x = −15 - Divide both sides by −3. −3x means −3 multiplied by x. The inverse of multiplying by −3 is dividing by −3, and we do it to both sides to keep the balance. This leaves x on its own. Careful with signs: a positive divided by a negative gives a negative answer, and a negative divided by a negative gives a positive.−3x−3 = −15−3
x = −15−3 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 7 − 3 × 5 = 7 − 15 = −8. Right side: −8. Both sides equal −8, so the solution is correct. ✓
Watch out. Dividing by a negative number changes the sign: a positive divided by a negative gives a negative. Check your answer by substitution.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.1 − x = 6
- Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: write the x term first (adding can be done in any order).
The equation is now −x + 1 = 6 - Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.−x + 1 − 1 = 6 − 1
−x = 5 - Divide both sides by −1. −x means −1 multiplied by x. The inverse of multiplying by −1 is dividing by −1, and we do it to both sides to keep the balance. This leaves x on its own. Careful with signs: a positive divided by a negative gives a negative answer, and a negative divided by a negative gives a positive.−x−1 = 5−1
x = 5−1 = −5 - Write the solution. x is now on its own, so the equation tells us its value.x = −5
Check. Substitute x = −5 into the original equation. Left side: 1 − (−5) = 1 + 5 = 6. Right side: 6. Both sides equal 6, so the solution is correct. ✓
Watch out. Dividing by a negative number changes the sign: a positive divided by a negative gives a negative. Check your answer by substitution.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.12 − 5x = 2
- Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: write the x term first (adding can be done in any order).
The equation is now −5x + 12 = 2 - Subtract 12 from both sides. This undoes the +12, because +12 − 12 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.−5x + 12 − 12 = 2 − 12
−5x = −10 - Divide both sides by −5. −5x means −5 multiplied by x. The inverse of multiplying by −5 is dividing by −5, and we do it to both sides to keep the balance. This leaves x on its own. Careful with signs: a positive divided by a negative gives a negative answer, and a negative divided by a negative gives a positive.−5x−5 = −10−5
x = −10−5 = 2 - Write the solution. x is now on its own, so the equation tells us its value.x = 2
Check. Substitute x = 2 into the original equation. Left side: 12 − 5 × 2 = 12 − 10 = 2. Right side: 2. Both sides equal 2, so the solution is correct. ✓
Watch out. Dividing by a negative number changes the sign: a positive divided by a negative gives a negative. Check your answer by substitution.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Substitute the value into both sides of the equation, work each side out, and compare.
- Write the equation and the value to test. A number is a solution of an equation if it makes both sides equal. We test it by substituting it (putting it in place of the letter).3x + 8 = 2, x = −2
- Substitute into the left-hand side. Replace every x with the number. Use brackets around a negative number so that the signs stay clear.Left side = 3 × (−2) + 8 = −6 + 8 = 2
- Substitute into the right-hand side. Do exactly the same to the other side of the equation.Right side = 2
- Compare the two sides. If the two sides are equal the number is a solution; if not, it is not.2 = 2, so yes, x = −2 is a solution.
Check. The two sides match, which is exactly what we wanted to see. ✓
Watch out. Work out the left side and the right side separately. Do not use an = sign between the two until you have shown they are equal.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Substitute the value into both sides of the equation, work each side out, and compare.
- Write the equation and the value to test. A number is a solution of an equation if it makes both sides equal. We test it by substituting it (putting it in place of the letter).2x − 1 = 3, x = −2
- Substitute into the left-hand side. Replace every x with the number. Use brackets around a negative number so that the signs stay clear.Left side = 2 × (−2) − 1 = −4 − 1 = −5
- Substitute into the right-hand side. Do exactly the same to the other side of the equation.Right side = 3
- Compare the two sides. If the two sides are equal the number is a solution; if not, it is not.−5 ≠ 3, so no, x = −2 is not a solution.
Check. The correct solution is x = 2: substituting it gives 3 on both sides. ✓
Watch out. Work out the left side and the right side separately. Do not use an = sign between the two until you have shown they are equal.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Substitute the value into both sides of the equation, work each side out, and compare.
- Write the equation and the value to test. A number is a solution of an equation if it makes both sides equal. We test it by substituting it (putting it in place of the letter).5x + 4 = −6, x = −2
- Substitute into the left-hand side. Replace every x with the number. Use brackets around a negative number so that the signs stay clear.Left side = 5 × (−2) + 4 = −10 + 4 = −6
- Substitute into the right-hand side. Do exactly the same to the other side of the equation.Right side = −6
- Compare the two sides. If the two sides are equal the number is a solution; if not, it is not.−6 = −6, so yes, x = −2 is a solution.
Check. The two sides match, which is exactly what we wanted to see. ✓
Watch out. Work out the left side and the right side separately. Do not use an = sign between the two until you have shown they are equal.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Substitute the value into both sides of the equation, work each side out, and compare.
- Write the equation and the value to test. A number is a solution of an equation if it makes both sides equal. We test it by substituting it (putting it in place of the letter).x2 + 4 = 3, x = −2
- Substitute into the left-hand side. Replace every x with the number. Use brackets around a negative number so that the signs stay clear.Left side = (−2)2 + 4 = −22 + 4 = −1 + 4 = 3
- Substitute into the right-hand side. Do exactly the same to the other side of the equation.Right side = 3
- Compare the two sides. If the two sides are equal the number is a solution; if not, it is not.3 = 3, so yes, x = −2 is a solution.
Check. The two sides match, which is exactly what we wanted to see. ✓
Watch out. Work out the left side and the right side separately. Do not use an = sign between the two until you have shown they are equal.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.6x + 7 = −17
- Subtract 7 from both sides. This undoes the +7, because +7 − 7 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.6x + 7 − 7 = −17 − 7
6x = −24 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = −246
x = −246 = −4 - Write the solution. x is now on its own, so the equation tells us its value.x = −4
Check. Substitute x = −4 into the original equation. Left side: 6 × (−4) + 7 = −24 + 7 = −17. Right side: −17. Both sides equal −17, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.9 = 4x + 21
- Subtract 21 from both sides. This undoes the +21, because +21 − 21 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.9 − 21 = 4x + 21 − 21
−12 = 4x - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.−124 = 4x4
x = −124 = −3 - Write the solution. x is now on its own, so the equation tells us its value.x = −3
Check. Substitute x = −3 into the original equation. Left side: 9. Right side: 4 × (−3) + 21 = −12 + 21 = 9. Both sides equal 9, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.The number is multiplied by 3 (3x) and then 11 is added (3x + 11). The answer is 2.
Equation: 3x + 11 = 2 - Subtract 11 from both sides. This undoes the +11, because +11 − 11 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 11 − 11 = 2 − 11
3x = −9 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = −93
x = −93 = −3 - Write the solution. x is now on its own, so the equation tells us its value.x = −3
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The number is −3.
Check. 3 × (−3) = −9, and −9 + 11 = 2 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number of hours.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.The temperature starts at 5°C and falls by 3°C each hour, so after x hours it is 5 − 3x degrees. It is now −7°C.
Equation: 5 − 3x = −7 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: write the x term first (adding can be done in any order).
The equation is now −3x + 5 = −7 - Subtract 5 from both sides. This undoes the +5, because +5 − 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.−3x + 5 − 5 = −7 − 5
−3x = −12 - Divide both sides by −3. −3x means −3 multiplied by x. The inverse of multiplying by −3 is dividing by −3, and we do it to both sides to keep the balance. This leaves x on its own. Careful with signs: a positive divided by a negative gives a negative answer, and a negative divided by a negative gives a positive.−3x−3 = −12−3
x = −12−3 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.It took 4 hours for the temperature to reach −7°C.
Check. 5 − 3 × 4 = 5 − 12 = −7 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the cost of one book in pounds.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.5 books cost 5x pounds, and the total is £12.
Equation: 5x = 12 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 125
x = 125 - Write the solution. x is now on its own, so the equation tells us its value.x = 125
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.One book costs £2.40.
Check. 5 × £2.40 = £120 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Method. Check Dev's answer by substituting, then solve the equation correctly.
- Substitute x = 9. Work out the left side and compare it with the right side.9 + 7 = 16, but the right-hand side is 2. 16 ≠ 2, so Dev is wrong.
- Find the mistake. 7 has been added to x, so we subtract 7 from both sides. Dev added 7 (2 + 7 = 9) instead.x + 7 − 7 = 2 − 7
- Work out the correct answer. 2 − 7 goes below zero, so the answer is negative.x = −5
Check. (−5) + 7 = 2. ✓
Watch out. Subtracting a bigger number from a smaller one gives a negative answer.
Method. Think about the size of the left-hand side when x is positive, then solve to confirm.
- Reason about positive values. If x is positive then 4x is positive, so 4x + 8 is more than 8.4x + 8 > 8 for every positive x
- Compare with the right-hand side. The right-hand side is 3, which is less than 8, so a positive x can never make the two sides equal.8 > 3, so there is no positive solution.
- Solve the equation. Subtract 8, then divide by 4.4x = 3 − 8 = −5
x = −5 ÷ 4 = −54
Check. 4 × (−54) + 8 = −5 + 8 = 3. ✓
Watch out. Estimating the size of the answer first helps you spot sign mistakes.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the cost of one child ticket in pounds.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.4 child tickets cost 4x pounds, and the total is £30.
Equation: 4x = 30 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 304
x = 304 = 152 - Write the solution. x is now on its own, so the equation tells us its value.x = 152
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.One child ticket costs £7.50.
Check. 4 × £7.50 = £300 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x − 9 = −4
- Add 9 to both sides. This undoes the −9, because −9 + 9 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x − 9 + 9 = −4 + 9
x = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 5 − 9 = −4. Right side: −4. Both sides equal −4, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x = −8
- Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = −82
x = −82 = −4 - Write the solution. x is now on its own, so the equation tells us its value.x = −4
Check. Substitute x = −4 into the original equation. Left side: 2 × (−4) = −8. Right side: −8. Both sides equal −8, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x + 6 = 1
- Subtract 6 from both sides. This undoes the +6, because +6 − 6 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 6 − 6 = 1 − 6
x = −5 - Write the solution. x is now on its own, so the equation tells us its value.x = −5
Check. Substitute x = −5 into the original equation. Left side: (−5) + 6 = −5 + 6 = 1. Right side: 1. Both sides equal 1, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3x = −15
- Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = −153
x = −153 = −5 - Write the solution. x is now on its own, so the equation tells us its value.x = −5
Check. Substitute x = −5 into the original equation. Left side: 3 × (−5) = −15. Right side: −15. Both sides equal −15, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
The unknown on both sides
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
Often the letter appears on both sides of the equals sign, such as 3x + 2 = x + 8. Now we cannot get x on its own just by undoing one side. First we must collect all the x terms on one side.
Think of the scales again. The left pan has three boxes and two circles, the right pan has one box and eight circles. Take one box off both pans and the scales still balance.
Here is the method.
- Choose the smaller x term (the one with the smaller number in front of x) and remove it from both sides. Subtracting the smaller one means the x terms that are left are positive.
- Move the plain numbers to the other side: add or subtract the number that is on the same side as the x term, on both sides.
- Divide both sides by the number in front of x.
For 3x + 2 = x + 8, the smaller x term is x (that is 1x). Subtract x from both sides to get 2x + 2 = 8. Then subtract 2 to get 2x = 6, and divide by 2 to get x = 3.
What if the bigger x term is on the right? In 3x + 10 = 7x − 6, the smaller x term is 3x on the left. Subtract 3x from both sides: 10 = 4x − 6. Add 6: 16 = 4x. Divide by 4: 4 = x, which means x = 4. You could remove the bigger x term instead, but then you get a negative number of x and mistakes with signs become more likely. Both ways give the same answer.
Key ideas
- Remove the smaller x term from both sides, so the x terms combine on one side.
- Then move the plain numbers to the other side, and divide.
- When you subtract a term from both sides, subtract it from every part of both sides, and watch the signs.
- Check by substituting into both sides of the original equation.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5x + 3 = 2x + 12
- Remove the x term from the right side: subtract 2x from both sides. We want all the x terms on one side. The right side has 2x; subtracting 2x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.5x + 3 − 2x = 2x + 12 − 2x
3x + 3 = 12 - Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 3 − 3 = 12 − 3
3x = 9 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 93
x = 93 = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
Check. Substitute x = 3 into the original equation. Left side: 5 × 3 + 3 = 15 + 3 = 18. Right side: 2 × 3 + 12 = 6 + 12 = 18. Both sides equal 18, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.7x − 4 = 3x + 20
- Remove the x term from the right side: subtract 3x from both sides. We want all the x terms on one side. The right side has 3x; subtracting 3x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.7x − 4 − 3x = 3x + 20 − 3x
4x − 4 = 20 - Add 4 to both sides. This undoes the −4, because −4 + 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x − 4 + 4 = 20 + 4
4x = 24 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 244
x = 244 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
Check. Substitute x = 6 into the original equation. Left side: 7 × 6 − 4 = 42 − 4 = 38. Right side: 3 × 6 + 20 = 18 + 20 = 38. Both sides equal 38, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3x + 10 = 7x − 6
- Remove the x term from the left side: subtract 3x from both sides. We want all the x terms on one side. The left side has 3x; subtracting 3x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.3x + 10 − 3x = 7x − 6 − 3x
10 = 4x − 6 - Add 6 to both sides. This undoes the −6, because −6 + 6 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.10 + 6 = 4x − 6 + 6
16 = 4x - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.164 = 4x4
x = 164 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 3 × 4 + 10 = 12 + 10 = 22. Right side: 7 × 4 − 6 = 28 − 6 = 22. Both sides equal 22, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x + 9 = 5x + 15
- Remove the x term from the left side: subtract 2x from both sides. We want all the x terms on one side. The left side has 2x; subtracting 2x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.2x + 9 − 2x = 5x + 15 − 2x
9 = 3x + 15 - Subtract 15 from both sides. This undoes the +15, because +15 − 15 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.9 − 15 = 3x + 15 − 15
−6 = 3x - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.−63 = 3x3
x = −63 = −2 - Write the solution. x is now on its own, so the equation tells us its value.x = −2
Check. Substitute x = −2 into the original equation. Left side: 2 × (−2) + 9 = −4 + 9 = 5. Right side: 5 × (−2) + 15 = −10 + 15 = 5. Both sides equal 5, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Watch out
Take the x term away from both sides, and think about the sign. In 5x − 3 = 2x + 9, subtracting 2x from both sides gives 3x − 3 = 9. If you add 2x to the left (a common mistake) you get 7x, which is wrong.Exercise 9D The unknown on both sides
Non-calculatorKey words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5x = 3x + 8
- Remove the x term from the right side: subtract 3x from both sides. We want all the x terms on one side. The right side has 3x; subtracting 3x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.5x − 3x = 3x + 8 − 3x
2x = 8 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 82
x = 82 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 5 × 4 = 20. Right side: 3 × 4 + 8 = 12 + 8 = 20. Both sides equal 20, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4x = x + 12
- Remove the x term from the right side: subtract x from both sides. We want all the x terms on one side. The right side has x; subtracting x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.4x − x = x + 12 − x
3x = 12 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 123
x = 123 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 4 × 4 = 16. Right side: 4 + 12 = 16. Both sides equal 16, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.7x = 4x + 15
- Remove the x term from the right side: subtract 4x from both sides. We want all the x terms on one side. The right side has 4x; subtracting 4x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.7x − 4x = 4x + 15 − 4x
3x = 15 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 153
x = 153 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 7 × 5 = 35. Right side: 4 × 5 + 15 = 20 + 15 = 35. Both sides equal 35, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.6x = 2x + 20
- Remove the x term from the right side: subtract 2x from both sides. We want all the x terms on one side. The right side has 2x; subtracting 2x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.6x − 2x = 2x + 20 − 2x
4x = 20 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 204
x = 204 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 6 × 5 = 30. Right side: 2 × 5 + 20 = 10 + 20 = 30. Both sides equal 30, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.9x = 5x + 28
- Remove the x term from the right side: subtract 5x from both sides. We want all the x terms on one side. The right side has 5x; subtracting 5x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.9x − 5x = 5x + 28 − 5x
4x = 28 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 284
x = 284 = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
Check. Substitute x = 7 into the original equation. Left side: 9 × 7 = 63. Right side: 5 × 7 + 28 = 35 + 28 = 63. Both sides equal 63, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.8x = 3x + 35
- Remove the x term from the right side: subtract 3x from both sides. We want all the x terms on one side. The right side has 3x; subtracting 3x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.8x − 3x = 3x + 35 − 3x
5x = 35 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 355
x = 355 = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
Check. Substitute x = 7 into the original equation. Left side: 8 × 7 = 56. Right side: 3 × 7 + 35 = 21 + 35 = 56. Both sides equal 56, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3x + 5 = x + 11
- Remove the x term from the right side: subtract x from both sides. We want all the x terms on one side. The right side has x; subtracting x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.3x + 5 − x = x + 11 − x
2x + 5 = 11 - Subtract 5 from both sides. This undoes the +5, because +5 − 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 5 − 5 = 11 − 5
2x = 6 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 62
x = 62 = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
Check. Substitute x = 3 into the original equation. Left side: 3 × 3 + 5 = 9 + 5 = 14. Right side: 3 + 11 = 14. Both sides equal 14, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5x + 2 = 2x + 14
- Remove the x term from the right side: subtract 2x from both sides. We want all the x terms on one side. The right side has 2x; subtracting 2x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.5x + 2 − 2x = 2x + 14 − 2x
3x + 2 = 14 - Subtract 2 from both sides. This undoes the +2, because +2 − 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 2 − 2 = 14 − 2
3x = 12 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 123
x = 123 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 5 × 4 + 2 = 20 + 2 = 22. Right side: 2 × 4 + 14 = 8 + 14 = 22. Both sides equal 22, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4x + 1 = x + 16
- Remove the x term from the right side: subtract x from both sides. We want all the x terms on one side. The right side has x; subtracting x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.4x + 1 − x = x + 16 − x
3x + 1 = 16 - Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 1 − 1 = 16 − 1
3x = 15 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 153
x = 153 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 4 × 5 + 1 = 20 + 1 = 21. Right side: 5 + 16 = 21. Both sides equal 21, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.6x + 3 = 4x + 11
- Remove the x term from the right side: subtract 4x from both sides. We want all the x terms on one side. The right side has 4x; subtracting 4x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.6x + 3 − 4x = 4x + 11 − 4x
2x + 3 = 11 - Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 3 − 3 = 11 − 3
2x = 8 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 82
x = 82 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 6 × 4 + 3 = 24 + 3 = 27. Right side: 4 × 4 + 11 = 16 + 11 = 27. Both sides equal 27, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.7x + 4 = 3x + 20
- Remove the x term from the right side: subtract 3x from both sides. We want all the x terms on one side. The right side has 3x; subtracting 3x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.7x + 4 − 3x = 3x + 20 − 3x
4x + 4 = 20 - Subtract 4 from both sides. This undoes the +4, because +4 − 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x + 4 − 4 = 20 − 4
4x = 16 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 164
x = 164 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 7 × 4 + 4 = 28 + 4 = 32. Right side: 3 × 4 + 20 = 12 + 20 = 32. Both sides equal 32, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.8x + 1 = 5x + 19
- Remove the x term from the right side: subtract 5x from both sides. We want all the x terms on one side. The right side has 5x; subtracting 5x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.8x + 1 − 5x = 5x + 19 − 5x
3x + 1 = 19 - Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 1 − 1 = 19 − 1
3x = 18 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 183
x = 183 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
Check. Substitute x = 6 into the original equation. Left side: 8 × 6 + 1 = 48 + 1 = 49. Right side: 5 × 6 + 19 = 30 + 19 = 49. Both sides equal 49, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5x − 3 = 2x + 9
- Remove the x term from the right side: subtract 2x from both sides. We want all the x terms on one side. The right side has 2x; subtracting 2x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.5x − 3 − 2x = 2x + 9 − 2x
3x − 3 = 9 - Add 3 to both sides. This undoes the −3, because −3 + 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x − 3 + 3 = 9 + 3
3x = 12 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 123
x = 123 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 5 × 4 − 3 = 20 − 3 = 17. Right side: 2 × 4 + 9 = 8 + 9 = 17. Both sides equal 17, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.6x − 4 = 3x + 14
- Remove the x term from the right side: subtract 3x from both sides. We want all the x terms on one side. The right side has 3x; subtracting 3x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.6x − 4 − 3x = 3x + 14 − 3x
3x − 4 = 14 - Add 4 to both sides. This undoes the −4, because −4 + 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x − 4 + 4 = 14 + 4
3x = 18 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 183
x = 183 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
Check. Substitute x = 6 into the original equation. Left side: 6 × 6 − 4 = 36 − 4 = 32. Right side: 3 × 6 + 14 = 18 + 14 = 32. Both sides equal 32, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4x − 7 = x + 8
- Remove the x term from the right side: subtract x from both sides. We want all the x terms on one side. The right side has x; subtracting x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.4x − 7 − x = x + 8 − x
3x − 7 = 8 - Add 7 to both sides. This undoes the −7, because −7 + 7 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x − 7 + 7 = 8 + 7
3x = 15 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 153
x = 153 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 4 × 5 − 7 = 20 − 7 = 13. Right side: 5 + 8 = 13. Both sides equal 13, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.7x − 1 = 5x + 13
- Remove the x term from the right side: subtract 5x from both sides. We want all the x terms on one side. The right side has 5x; subtracting 5x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.7x − 1 − 5x = 5x + 13 − 5x
2x − 1 = 13 - Add 1 to both sides. This undoes the −1, because −1 + 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x − 1 + 1 = 13 + 1
2x = 14 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 142
x = 142 = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
Check. Substitute x = 7 into the original equation. Left side: 7 × 7 − 1 = 49 − 1 = 48. Right side: 5 × 7 + 13 = 35 + 13 = 48. Both sides equal 48, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.9x − 8 = 4x + 27
- Remove the x term from the right side: subtract 4x from both sides. We want all the x terms on one side. The right side has 4x; subtracting 4x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.9x − 8 − 4x = 4x + 27 − 4x
5x − 8 = 27 - Add 8 to both sides. This undoes the −8, because −8 + 8 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5x − 8 + 8 = 27 + 8
5x = 35 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 355
x = 355 = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
Check. Substitute x = 7 into the original equation. Left side: 9 × 7 − 8 = 63 − 8 = 55. Right side: 4 × 7 + 27 = 28 + 27 = 55. Both sides equal 55, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.8x − 5 = 3x + 20
- Remove the x term from the right side: subtract 3x from both sides. We want all the x terms on one side. The right side has 3x; subtracting 3x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.8x − 5 − 3x = 3x + 20 − 3x
5x − 5 = 20 - Add 5 to both sides. This undoes the −5, because −5 + 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5x − 5 + 5 = 20 + 5
5x = 25 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 255
x = 255 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 8 × 5 − 5 = 40 − 5 = 35. Right side: 3 × 5 + 20 = 15 + 20 = 35. Both sides equal 35, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x + 12 = 3x + 4
- Remove the x term from the left side: subtract x from both sides. We want all the x terms on one side. The left side has x; subtracting x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.x + 12 − x = 3x + 4 − x
12 = 2x + 4 - Subtract 4 from both sides. This undoes the +4, because +4 − 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.12 − 4 = 2x + 4 − 4
8 = 2x - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.82 = 2x2
x = 82 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 4 + 12 = 16. Right side: 3 × 4 + 4 = 12 + 4 = 16. Both sides equal 16, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x + 9 = 5x
- Remove the x term from the left side: subtract 2x from both sides. We want all the x terms on one side. The left side has 2x; subtracting 2x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.2x + 9 − 2x = 5x − 2x
9 = 3x - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.93 = 3x3
x = 93 = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
Check. Substitute x = 3 into the original equation. Left side: 2 × 3 + 9 = 6 + 9 = 15. Right side: 5 × 3 = 15. Both sides equal 15, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3x + 8 = 5x − 4
- Remove the x term from the left side: subtract 3x from both sides. We want all the x terms on one side. The left side has 3x; subtracting 3x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.3x + 8 − 3x = 5x − 4 − 3x
8 = 2x − 4 - Add 4 to both sides. This undoes the −4, because −4 + 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.8 + 4 = 2x − 4 + 4
12 = 2x - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.122 = 2x2
x = 122 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
Check. Substitute x = 6 into the original equation. Left side: 3 × 6 + 8 = 18 + 8 = 26. Right side: 5 × 6 − 4 = 30 − 4 = 26. Both sides equal 26, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4x + 3 = 7x − 18
- Remove the x term from the left side: subtract 4x from both sides. We want all the x terms on one side. The left side has 4x; subtracting 4x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.4x + 3 − 4x = 7x − 18 − 4x
3 = 3x − 18 - Add 18 to both sides. This undoes the −18, because −18 + 18 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3 + 18 = 3x − 18 + 18
21 = 3x - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.213 = 3x3
x = 213 = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
Check. Substitute x = 7 into the original equation. Left side: 4 × 7 + 3 = 28 + 3 = 31. Right side: 7 × 7 − 18 = 49 − 18 = 31. Both sides equal 31, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x + 15 = 6x − 13
- Remove the x term from the left side: subtract 2x from both sides. We want all the x terms on one side. The left side has 2x; subtracting 2x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.2x + 15 − 2x = 6x − 13 − 2x
15 = 4x − 13 - Add 13 to both sides. This undoes the −13, because −13 + 13 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.15 + 13 = 4x − 13 + 13
28 = 4x - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.284 = 4x4
x = 284 = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
Check. Substitute x = 7 into the original equation. Left side: 2 × 7 + 15 = 14 + 15 = 29. Right side: 6 × 7 − 13 = 42 − 13 = 29. Both sides equal 29, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3x + 7 = 8x − 23
- Remove the x term from the left side: subtract 3x from both sides. We want all the x terms on one side. The left side has 3x; subtracting 3x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.3x + 7 − 3x = 8x − 23 − 3x
7 = 5x − 23 - Add 23 to both sides. This undoes the −23, because −23 + 23 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.7 + 23 = 5x − 23 + 23
30 = 5x - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.305 = 5x5
x = 305 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
Check. Substitute x = 6 into the original equation. Left side: 3 × 6 + 7 = 18 + 7 = 25. Right side: 8 × 6 − 23 = 48 − 23 = 25. Both sides equal 25, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x + 9 = 5x + 15
- Remove the x term from the left side: subtract 2x from both sides. We want all the x terms on one side. The left side has 2x; subtracting 2x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.2x + 9 − 2x = 5x + 15 − 2x
9 = 3x + 15 - Subtract 15 from both sides. This undoes the +15, because +15 − 15 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.9 − 15 = 3x + 15 − 15
−6 = 3x - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.−63 = 3x3
x = −63 = −2 - Write the solution. x is now on its own, so the equation tells us its value.x = −2
Check. Substitute x = −2 into the original equation. Left side: 2 × (−2) + 9 = −4 + 9 = 5. Right side: 5 × (−2) + 15 = −10 + 15 = 5. Both sides equal 5, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3x + 14 = 7x + 22
- Remove the x term from the left side: subtract 3x from both sides. We want all the x terms on one side. The left side has 3x; subtracting 3x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.3x + 14 − 3x = 7x + 22 − 3x
14 = 4x + 22 - Subtract 22 from both sides. This undoes the +22, because +22 − 22 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.14 − 22 = 4x + 22 − 22
−8 = 4x - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.−84 = 4x4
x = −84 = −2 - Write the solution. x is now on its own, so the equation tells us its value.x = −2
Check. Substitute x = −2 into the original equation. Left side: 3 × (−2) + 14 = −6 + 14 = 8. Right side: 7 × (−2) + 22 = −14 + 22 = 8. Both sides equal 8, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5x + 4 = 8x + 13
- Remove the x term from the left side: subtract 5x from both sides. We want all the x terms on one side. The left side has 5x; subtracting 5x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.5x + 4 − 5x = 8x + 13 − 5x
4 = 3x + 13 - Subtract 13 from both sides. This undoes the +13, because +13 − 13 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4 − 13 = 3x + 13 − 13
−9 = 3x - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.−93 = 3x3
x = −93 = −3 - Write the solution. x is now on its own, so the equation tells us its value.x = −3
Check. Substitute x = −3 into the original equation. Left side: 5 × (−3) + 4 = −15 + 4 = −11. Right side: 8 × (−3) + 13 = −24 + 13 = −11. Both sides equal −11, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x − 3 = 4x + 7
- Remove the x term from the left side: subtract 2x from both sides. We want all the x terms on one side. The left side has 2x; subtracting 2x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.2x − 3 − 2x = 4x + 7 − 2x
−3 = 2x + 7 - Subtract 7 from both sides. This undoes the +7, because +7 − 7 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.−3 − 7 = 2x + 7 − 7
−10 = 2x - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.−102 = 2x2
x = −102 = −5 - Write the solution. x is now on its own, so the equation tells us its value.x = −5
Check. Substitute x = −5 into the original equation. Left side: 2 × (−5) − 3 = −10 − 3 = −13. Right side: 4 × (−5) + 7 = −20 + 7 = −13. Both sides equal −13, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x + 1 = 4x + 10
- Remove the x term from the left side: subtract x from both sides. We want all the x terms on one side. The left side has x; subtracting x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.x + 1 − x = 4x + 10 − x
1 = 3x + 10 - Subtract 10 from both sides. This undoes the +10, because +10 − 10 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.1 − 10 = 3x + 10 − 10
−9 = 3x - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.−93 = 3x3
x = −93 = −3 - Write the solution. x is now on its own, so the equation tells us its value.x = −3
Check. Substitute x = −3 into the original equation. Left side: (−3) + 1 = −3 + 1 = −2. Right side: 4 × (−3) + 10 = −12 + 10 = −2. Both sides equal −2, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.6x + 5 = 9x + 20
- Remove the x term from the left side: subtract 6x from both sides. We want all the x terms on one side. The left side has 6x; subtracting 6x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.6x + 5 − 6x = 9x + 20 − 6x
5 = 3x + 20 - Subtract 20 from both sides. This undoes the +20, because +20 − 20 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5 − 20 = 3x + 20 − 20
−15 = 3x - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.−153 = 3x3
x = −153 = −5 - Write the solution. x is now on its own, so the equation tells us its value.x = −5
Check. Substitute x = −5 into the original equation. Left side: 6 × (−5) + 5 = −30 + 5 = −25. Right side: 9 × (−5) + 20 = −45 + 20 = −25. Both sides equal −25, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number of miles.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Firm A charges £3 plus £2 for each mile: 3 + 2x. Firm B charges £1 plus £3 for each mile: 1 + 3x. We want the journey where the prices are equal.
Equation: 3 + 2x = 1 + 3x - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: write the x term first (adding can be done in any order).
On the right side: write the x term first (adding can be done in any order).
The equation is now 2x + 3 = 3x + 1 - Remove the x term from the left side: subtract 2x from both sides. We want all the x terms on one side. The left side has 2x; subtracting 2x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.2x + 3 − 2x = 3x + 1 − 2x
3 = x + 1 - Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3 − 1 = x + 1 − 1
2 = x - Write the solution. x is now on its own, so the equation tells us its value.x = 2
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The two firms charge the same for a journey of 2 miles.
Check. Firm A: 3 + 2 × 2 = £7. Firm B: 1 + 3 × 2 = £7. The same ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Sam's number multiplied by 5 then 6 subtracted: 5x − 6. The same number multiplied by 3 then 10 added: 3x + 10. The two results are the same.
Equation: 5x − 6 = 3x + 10 - Remove the x term from the right side: subtract 3x from both sides. We want all the x terms on one side. The right side has 3x; subtracting 3x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.5x − 6 − 3x = 3x + 10 − 3x
2x − 6 = 10 - Add 6 to both sides. This undoes the −6, because −6 + 6 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x − 6 + 6 = 10 + 6
2x = 16 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 162
x = 162 = 8 - Write the solution. x is now on its own, so the equation tells us its value.x = 8
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The number is 8.
Check. 5 × 8 − 6 = 34 and 3 × 8 + 10 = 34 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Method. Find what Lily did wrong, then do the step correctly.
- Look at her first step. The 2x is on the right-hand side. To remove it we subtract 2x from both sides. Lily added 2x to the left-hand side only (6x + 2x = 8x).Wrong: 6x + 2x + 5 = 17
She changed one side only, so the equation is no longer balanced. - Do the step correctly. Subtract 2x from both sides.6x + 5 − 2x = 2x + 17 − 2x
4x + 5 = 17 - Finish solving. Subtract 5, then divide by 4.4x = 12
x = 3
Check. 6 × 3 + 5 = 23 and 2 × 3 + 17 = 23. ✓
Watch out. The same operation must be applied to the whole of both sides.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Substitute the value into both sides of the equation, work each side out, and compare.
- Write the equation and the value to test. A number is a solution of an equation if it makes both sides equal. We test it by substituting it (putting it in place of the letter).7x − 9 = 3x + 11, x = 5
- Substitute into the left-hand side. Replace every x with the number. Use brackets around a negative number so that the signs stay clear.Left side = 7 × 5 − 9 = 35 − 9 = 26
- Substitute into the right-hand side. Do exactly the same to the other side of the equation.Right side = 3 × 5 + 11 = 15 + 11 = 26
- Compare the two sides. If the two sides are equal the number is a solution; if not, it is not.26 = 26, so yes, x = 5 is a solution.
Check. The two sides match, which is exactly what we wanted to see. ✓
Watch out. Work out the left side and the right side separately. Do not use an = sign between the two until you have shown they are equal.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number of weeks.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.After x weeks Ali has 5 + 4x pounds and Ben has 21 + 2x pounds. We want them to be equal.
Equation: 5 + 4x = 21 + 2x - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: write the x term first (adding can be done in any order).
On the right side: write the x term first (adding can be done in any order).
The equation is now 4x + 5 = 2x + 21 - Remove the x term from the right side: subtract 2x from both sides. We want all the x terms on one side. The right side has 2x; subtracting 2x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.4x + 5 − 2x = 2x + 21 − 2x
2x + 5 = 21 - Subtract 5 from both sides. This undoes the +5, because +5 − 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 5 − 5 = 21 − 5
2x = 16 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 162
x = 162 = 8 - Write the solution. x is now on its own, so the equation tells us its value.x = 8
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.They have the same amount after 8 weeks.
Check. Ali: 5 + 4 × 8 = £37. Ben: 21 + 2 × 8 = £37. The same ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Method. Finish each method and compare the answers.
- Finish Amy's method. Subtract 8 from both sides, then divide by −4.−4x = −4 − 8 = −12
x = −12 ÷ (−4) = 3 - Finish Ben's method. Add 4 to both sides, then divide by 4.12 = 4x
x = 12 ÷ 4 = 3 - Compare. Both methods give the same value.Both get x = 3. Ben's way avoids dividing by a negative number.
Check. 3 × 3 + 8 = 17 and 7 × 3 − 4 = 17. ✓
Watch out. Moving the smaller x term keeps the number in front of x positive.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x + 4 = x + 9
- Remove the x term from the right side: subtract x from both sides. We want all the x terms on one side. The right side has x; subtracting x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.2x + 4 − x = x + 9 − x
x + 4 = 9 - Subtract 4 from both sides. This undoes the +4, because +4 − 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 4 − 4 = 9 − 4
x = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 2 × 5 + 4 = 10 + 4 = 14. Right side: 5 + 9 = 14. Both sides equal 14, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3x = x + 10
- Remove the x term from the right side: subtract x from both sides. We want all the x terms on one side. The right side has x; subtracting x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.3x − x = x + 10 − x
2x = 10 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 102
x = 102 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 3 × 5 = 15. Right side: 5 + 10 = 15. Both sides equal 15, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4x + 2 = 2x + 12
- Remove the x term from the right side: subtract 2x from both sides. We want all the x terms on one side. The right side has 2x; subtracting 2x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.4x + 2 − 2x = 2x + 12 − 2x
2x + 2 = 12 - Subtract 2 from both sides. This undoes the +2, because +2 − 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 2 − 2 = 12 − 2
2x = 10 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 102
x = 102 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 4 × 5 + 2 = 20 + 2 = 22. Right side: 2 × 5 + 12 = 10 + 12 = 22. Both sides equal 22, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Equations with brackets
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
A bracket means “multiply everything inside by the number outside”. In 3(x + 4) the 3 multiplies both the x and the 4. You met expanding in Chapter 7. A picture helps: the rectangle below has height 3 and width (x + 4), so its area is 3(x + 4). Cut it into two rectangles and the area is 3 × x + 3 × 4 = 3x + 12.
To solve an equation with brackets, expand the brackets first, collect like terms if there are any, and then solve as before.
A shortcut when the bracket stands alone. If the whole of one side is a bracket and the other side is a number, such as 3(x + 4) = 27, you can divide both sides by 3 first: x + 4 = 9, so x = 5. This works because 3 is multiplying the entire bracket. Expanding first also works: 3x + 12 = 27, 3x = 15, x = 5. Use whichever you find easier; expanding always works.
Negative numbers and minus signs. A minus sign in front of a bracket multiplies everything inside by −1, so it changes the sign of every term inside. 10 − (x + 2) = 10 − x − 2 = 8 − x. Likewise −2(x − 3) = −2x + 6, because (−2) × (−3) = +6.
When there are brackets on both sides, expand both, then collect the x terms as in the last section.
Key ideas
- Expand first: multiply every term inside the bracket by the number outside.
- A minus sign in front of a bracket changes the sign of every term inside it.
- Collect like terms, then solve (unknown on both sides: remove the smaller x term).
- Shortcut: if one side is just a bracket and the other is a number, you may divide both sides by the number in front of the bracket first.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3(x + 4) = 27
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 3(x + 4):
3 × x = 3x
3 × 4 = 12
The equation is now 3x + 12 = 27 - Subtract 12 from both sides. This undoes the +12, because +12 − 12 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 12 − 12 = 27 − 12
3x = 15 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 153
x = 153 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 3 × (5 + 4) = 3 × 9 = 27. Right side: 27. Both sides equal 27, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2(3x − 5) = 14
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 2(3x − 5):
2 × 3x = 6x
2 × (−5) = −10
The equation is now 6x − 10 = 14 - Add 10 to both sides. This undoes the −10, because −10 + 10 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.6x − 10 + 10 = 14 + 10
6x = 24 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 246
x = 246 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 2 × (3 × 4 − 5) = 2 × 7 = 14. Right side: 14. Both sides equal 14, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5(x − 2) = 3(x + 4)
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 5(x − 2):
5 × x = 5x
5 × (−2) = −10
Expand 3(x + 4):
3 × x = 3x
3 × 4 = 12
The equation is now 5x − 10 = 3x + 12 - Remove the x term from the right side: subtract 3x from both sides. We want all the x terms on one side. The right side has 3x; subtracting 3x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.5x − 10 − 3x = 3x + 12 − 3x
2x − 10 = 12 - Add 10 to both sides. This undoes the −10, because −10 + 10 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x − 10 + 10 = 12 + 10
2x = 22 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 222
x = 222 = 11 - Write the solution. x is now on its own, so the equation tells us its value.x = 11
Check. Substitute x = 11 into the original equation. Left side: 5 × (11 − 2) = 5 × 9 = 45. Right side: 3 × (11 + 4) = 3 × 15 = 45. Both sides equal 45, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.7 − 2(x − 1) = 3
- Expand the brackets. Multiply each term inside the bracket by the number outside it. A negative number multiplied by a negative number gives a positive answer, and a negative multiplied by a positive gives a negative.Expand −2(x − 1):
(−2) × x = −2x
(−2) × (−1) = 2
The equation is now 7 − 2x + 2 = 3 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the numbers 7 + 2 make 9.
The equation is now −2x + 9 = 3 - Subtract 9 from both sides. This undoes the +9, because +9 − 9 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.−2x + 9 − 9 = 3 − 9
−2x = −6 - Divide both sides by −2. −2x means −2 multiplied by x. The inverse of multiplying by −2 is dividing by −2, and we do it to both sides to keep the balance. This leaves x on its own. Careful with signs: a positive divided by a negative gives a negative answer, and a negative divided by a negative gives a positive.−2x−2 = −6−2
x = −6−2 = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
Check. Substitute x = 3 into the original equation. Left side: 7 − 2 × (3 − 1) = 7 − 2 × 2 = 7 − 4 = 3. Right side: 3. Both sides equal 3, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Watch out
Multiply every term in the bracket. 3(x + 4) is 3x + 12, not 3x + 4. And with a negative in front, change both signs: −2(x − 3) = −2x + 6, not −2x − 6.Exercise 9E Equations with brackets
Non-calculatorKey words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2(x + 3) = 14
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 2(x + 3):
2 × x = 2x
2 × 3 = 6
The equation is now 2x + 6 = 14 - Subtract 6 from both sides. This undoes the +6, because +6 − 6 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 6 − 6 = 14 − 6
2x = 8 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 82
x = 82 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 2 × (4 + 3) = 2 × 7 = 14. Right side: 14. Both sides equal 14, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3(x + 1) = 18
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 3(x + 1):
3 × x = 3x
3 × 1 = 3
The equation is now 3x + 3 = 18 - Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 3 − 3 = 18 − 3
3x = 15 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 153
x = 153 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 3 × (5 + 1) = 3 × 6 = 18. Right side: 18. Both sides equal 18, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5(x + 2) = 35
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 5(x + 2):
5 × x = 5x
5 × 2 = 10
The equation is now 5x + 10 = 35 - Subtract 10 from both sides. This undoes the +10, because +10 − 10 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5x + 10 − 10 = 35 − 10
5x = 25 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 255
x = 255 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 5 × (5 + 2) = 5 × 7 = 35. Right side: 35. Both sides equal 35, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4(x + 5) = 36
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 4(x + 5):
4 × x = 4x
4 × 5 = 20
The equation is now 4x + 20 = 36 - Subtract 20 from both sides. This undoes the +20, because +20 − 20 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x + 20 − 20 = 36 − 20
4x = 16 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 164
x = 164 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 4 × (4 + 5) = 4 × 9 = 36. Right side: 36. Both sides equal 36, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.6(x + 1) = 30
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 6(x + 1):
6 × x = 6x
6 × 1 = 6
The equation is now 6x + 6 = 30 - Subtract 6 from both sides. This undoes the +6, because +6 − 6 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.6x + 6 − 6 = 30 − 6
6x = 24 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 246
x = 246 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 6 × (4 + 1) = 6 × 5 = 30. Right side: 30. Both sides equal 30, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2(x + 9) = 30
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 2(x + 9):
2 × x = 2x
2 × 9 = 18
The equation is now 2x + 18 = 30 - Subtract 18 from both sides. This undoes the +18, because +18 − 18 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 18 − 18 = 30 − 18
2x = 12 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 122
x = 122 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
Check. Substitute x = 6 into the original equation. Left side: 2 × (6 + 9) = 2 × 15 = 30. Right side: 30. Both sides equal 30, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3(x − 2) = 15
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 3(x − 2):
3 × x = 3x
3 × (−2) = −6
The equation is now 3x − 6 = 15 - Add 6 to both sides. This undoes the −6, because −6 + 6 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x − 6 + 6 = 15 + 6
3x = 21 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 213
x = 213 = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
Check. Substitute x = 7 into the original equation. Left side: 3 × (7 − 2) = 3 × 5 = 15. Right side: 15. Both sides equal 15, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2(x − 4) = 10
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 2(x − 4):
2 × x = 2x
2 × (−4) = −8
The equation is now 2x − 8 = 10 - Add 8 to both sides. This undoes the −8, because −8 + 8 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x − 8 + 8 = 10 + 8
2x = 18 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 182
x = 182 = 9 - Write the solution. x is now on its own, so the equation tells us its value.x = 9
Check. Substitute x = 9 into the original equation. Left side: 2 × (9 − 4) = 2 × 5 = 10. Right side: 10. Both sides equal 10, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5(x − 1) = 20
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 5(x − 1):
5 × x = 5x
5 × (−1) = −5
The equation is now 5x − 5 = 20 - Add 5 to both sides. This undoes the −5, because −5 + 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5x − 5 + 5 = 20 + 5
5x = 25 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 255
x = 255 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 5 × (5 − 1) = 5 × 4 = 20. Right side: 20. Both sides equal 20, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4(x − 3) = 8
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 4(x − 3):
4 × x = 4x
4 × (−3) = −12
The equation is now 4x − 12 = 8 - Add 12 to both sides. This undoes the −12, because −12 + 12 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x − 12 + 12 = 8 + 12
4x = 20 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 204
x = 204 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 4 × (5 − 3) = 4 × 2 = 8. Right side: 8. Both sides equal 8, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3(2x + 1) = 21
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 3(2x + 1):
3 × 2x = 6x
3 × 1 = 3
The equation is now 6x + 3 = 21 - Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.6x + 3 − 3 = 21 − 3
6x = 18 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 186
x = 186 = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
Check. Substitute x = 3 into the original equation. Left side: 3 × (2 × 3 + 1) = 3 × 7 = 21. Right side: 21. Both sides equal 21, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2(3x − 4) = 22
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 2(3x − 4):
2 × 3x = 6x
2 × (−4) = −8
The equation is now 6x − 8 = 22 - Add 8 to both sides. This undoes the −8, because −8 + 8 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.6x − 8 + 8 = 22 + 8
6x = 30 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 306
x = 306 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 2 × (3 × 5 − 4) = 2 × 11 = 22. Right side: 22. Both sides equal 22, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3(x + 2) + 4 = 19
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 3(x + 2):
3 × x = 3x
3 × 2 = 6
The equation is now 3x + 6 + 4 = 19 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the numbers 6 + 4 make 10.
The equation is now 3x + 10 = 19 - Subtract 10 from both sides. This undoes the +10, because +10 − 10 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 10 − 10 = 19 − 10
3x = 9 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 93
x = 93 = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
Check. Substitute x = 3 into the original equation. Left side: 3 × (3 + 2) + 4 = 3 × 5 + 4 = 15 + 4 = 19. Right side: 19. Both sides equal 19, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2(x + 5) − 3 = 19
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 2(x + 5):
2 × x = 2x
2 × 5 = 10
The equation is now 2x + 10 − 3 = 19 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the numbers 10 − 3 make 7.
The equation is now 2x + 7 = 19 - Subtract 7 from both sides. This undoes the +7, because +7 − 7 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 7 − 7 = 19 − 7
2x = 12 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 122
x = 122 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
Check. Substitute x = 6 into the original equation. Left side: 2 × (6 + 5) − 3 = 2 × 11 − 3 = 22 − 3 = 19. Right side: 19. Both sides equal 19, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4(x − 1) + 7 = 23
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 4(x − 1):
4 × x = 4x
4 × (−1) = −4
The equation is now 4x − 4 + 7 = 23 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the numbers −4 + 7 make 3.
The equation is now 4x + 3 = 23 - Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x + 3 − 3 = 23 − 3
4x = 20 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 204
x = 204 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 4 × (5 − 1) + 7 = 4 × 4 + 7 = 16 + 7 = 23. Right side: 23. Both sides equal 23, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5(x + 2) − 8 = 27
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 5(x + 2):
5 × x = 5x
5 × 2 = 10
The equation is now 5x + 10 − 8 = 27 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the numbers 10 − 8 make 2.
The equation is now 5x + 2 = 27 - Subtract 2 from both sides. This undoes the +2, because +2 − 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5x + 2 − 2 = 27 − 2
5x = 25 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 255
x = 255 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 5 × (5 + 2) − 8 = 5 × 7 − 8 = 35 − 8 = 27. Right side: 27. Both sides equal 27, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2(2x + 1) + 5 = 19
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 2(2x + 1):
2 × 2x = 4x
2 × 1 = 2
The equation is now 4x + 2 + 5 = 19 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the numbers 2 + 5 make 7.
The equation is now 4x + 7 = 19 - Subtract 7 from both sides. This undoes the +7, because +7 − 7 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x + 7 − 7 = 19 − 7
4x = 12 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 124
x = 124 = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
Check. Substitute x = 3 into the original equation. Left side: 2 × (2 × 3 + 1) + 5 = 2 × 7 + 5 = 14 + 5 = 19. Right side: 19. Both sides equal 19, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3(x − 4) + 2 = 8
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 3(x − 4):
3 × x = 3x
3 × (−4) = −12
The equation is now 3x − 12 + 2 = 8 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the numbers −12 + 2 make −10.
The equation is now 3x − 10 = 8 - Add 10 to both sides. This undoes the −10, because −10 + 10 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x − 10 + 10 = 8 + 10
3x = 18 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 183
x = 183 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
Check. Substitute x = 6 into the original equation. Left side: 3 × (6 − 4) + 2 = 3 × 2 + 2 = 6 + 2 = 8. Right side: 8. Both sides equal 8, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3(x + 2) = 2x + 9
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 3(x + 2):
3 × x = 3x
3 × 2 = 6
The equation is now 3x + 6 = 2x + 9 - Remove the x term from the right side: subtract 2x from both sides. We want all the x terms on one side. The right side has 2x; subtracting 2x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.3x + 6 − 2x = 2x + 9 − 2x
x + 6 = 9 - Subtract 6 from both sides. This undoes the +6, because +6 − 6 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 6 − 6 = 9 − 6
x = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
Check. Substitute x = 3 into the original equation. Left side: 3 × (3 + 2) = 3 × 5 = 15. Right side: 2 × 3 + 9 = 6 + 9 = 15. Both sides equal 15, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4(x − 1) = 2x + 8
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 4(x − 1):
4 × x = 4x
4 × (−1) = −4
The equation is now 4x − 4 = 2x + 8 - Remove the x term from the right side: subtract 2x from both sides. We want all the x terms on one side. The right side has 2x; subtracting 2x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.4x − 4 − 2x = 2x + 8 − 2x
2x − 4 = 8 - Add 4 to both sides. This undoes the −4, because −4 + 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x − 4 + 4 = 8 + 4
2x = 12 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 122
x = 122 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
Check. Substitute x = 6 into the original equation. Left side: 4 × (6 − 1) = 4 × 5 = 20. Right side: 2 × 6 + 8 = 12 + 8 = 20. Both sides equal 20, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2(x + 5) = x + 14
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 2(x + 5):
2 × x = 2x
2 × 5 = 10
The equation is now 2x + 10 = x + 14 - Remove the x term from the right side: subtract x from both sides. We want all the x terms on one side. The right side has x; subtracting x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.2x + 10 − x = x + 14 − x
x + 10 = 14 - Subtract 10 from both sides. This undoes the +10, because +10 − 10 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 10 − 10 = 14 − 10
x = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 2 × (4 + 5) = 2 × 9 = 18. Right side: 4 + 14 = 18. Both sides equal 18, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5(x − 3) = 3x + 1
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 5(x − 3):
5 × x = 5x
5 × (−3) = −15
The equation is now 5x − 15 = 3x + 1 - Remove the x term from the right side: subtract 3x from both sides. We want all the x terms on one side. The right side has 3x; subtracting 3x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.5x − 15 − 3x = 3x + 1 − 3x
2x − 15 = 1 - Add 15 to both sides. This undoes the −15, because −15 + 15 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x − 15 + 15 = 1 + 15
2x = 16 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 162
x = 162 = 8 - Write the solution. x is now on its own, so the equation tells us its value.x = 8
Check. Substitute x = 8 into the original equation. Left side: 5 × (8 − 3) = 5 × 5 = 25. Right side: 3 × 8 + 1 = 24 + 1 = 25. Both sides equal 25, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3(2x + 1) = 4x + 11
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 3(2x + 1):
3 × 2x = 6x
3 × 1 = 3
The equation is now 6x + 3 = 4x + 11 - Remove the x term from the right side: subtract 4x from both sides. We want all the x terms on one side. The right side has 4x; subtracting 4x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.6x + 3 − 4x = 4x + 11 − 4x
2x + 3 = 11 - Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 3 − 3 = 11 − 3
2x = 8 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 82
x = 82 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 3 × (2 × 4 + 1) = 3 × 9 = 27. Right side: 4 × 4 + 11 = 16 + 11 = 27. Both sides equal 27, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2(3x − 2) = 5x + 3
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 2(3x − 2):
2 × 3x = 6x
2 × (−2) = −4
The equation is now 6x − 4 = 5x + 3 - Remove the x term from the right side: subtract 5x from both sides. We want all the x terms on one side. The right side has 5x; subtracting 5x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.6x − 4 − 5x = 5x + 3 − 5x
x − 4 = 3 - Add 4 to both sides. This undoes the −4, because −4 + 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x − 4 + 4 = 3 + 4
x = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
Check. Substitute x = 7 into the original equation. Left side: 2 × (3 × 7 − 2) = 2 × 19 = 38. Right side: 5 × 7 + 3 = 35 + 3 = 38. Both sides equal 38, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5(x − 2) = 3(x + 4)
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 5(x − 2):
5 × x = 5x
5 × (−2) = −10
Expand 3(x + 4):
3 × x = 3x
3 × 4 = 12
The equation is now 5x − 10 = 3x + 12 - Remove the x term from the right side: subtract 3x from both sides. We want all the x terms on one side. The right side has 3x; subtracting 3x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.5x − 10 − 3x = 3x + 12 − 3x
2x − 10 = 12 - Add 10 to both sides. This undoes the −10, because −10 + 10 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x − 10 + 10 = 12 + 10
2x = 22 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 222
x = 222 = 11 - Write the solution. x is now on its own, so the equation tells us its value.x = 11
Check. Substitute x = 11 into the original equation. Left side: 5 × (11 − 2) = 5 × 9 = 45. Right side: 3 × (11 + 4) = 3 × 15 = 45. Both sides equal 45, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2(x + 7) = 4(x + 1)
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 2(x + 7):
2 × x = 2x
2 × 7 = 14
Expand 4(x + 1):
4 × x = 4x
4 × 1 = 4
The equation is now 2x + 14 = 4x + 4 - Remove the x term from the left side: subtract 2x from both sides. We want all the x terms on one side. The left side has 2x; subtracting 2x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.2x + 14 − 2x = 4x + 4 − 2x
14 = 2x + 4 - Subtract 4 from both sides. This undoes the +4, because +4 − 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.14 − 4 = 2x + 4 − 4
10 = 2x - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.102 = 2x2
x = 102 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 2 × (5 + 7) = 2 × 12 = 24. Right side: 4 × (5 + 1) = 4 × 6 = 24. Both sides equal 24, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4(x − 1) = 3(x + 2)
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 4(x − 1):
4 × x = 4x
4 × (−1) = −4
Expand 3(x + 2):
3 × x = 3x
3 × 2 = 6
The equation is now 4x − 4 = 3x + 6 - Remove the x term from the right side: subtract 3x from both sides. We want all the x terms on one side. The right side has 3x; subtracting 3x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.4x − 4 − 3x = 3x + 6 − 3x
x − 4 = 6 - Add 4 to both sides. This undoes the −4, because −4 + 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x − 4 + 4 = 6 + 4
x = 10 - Write the solution. x is now on its own, so the equation tells us its value.x = 10
Check. Substitute x = 10 into the original equation. Left side: 4 × (10 − 1) = 4 × 9 = 36. Right side: 3 × (10 + 2) = 3 × 12 = 36. Both sides equal 36, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4(x + 3) = 3(x + 6)
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 4(x + 3):
4 × x = 4x
4 × 3 = 12
Expand 3(x + 6):
3 × x = 3x
3 × 6 = 18
The equation is now 4x + 12 = 3x + 18 - Remove the x term from the right side: subtract 3x from both sides. We want all the x terms on one side. The right side has 3x; subtracting 3x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.4x + 12 − 3x = 3x + 18 − 3x
x + 12 = 18 - Subtract 12 from both sides. This undoes the +12, because +12 − 12 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 12 − 12 = 18 − 12
x = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
Check. Substitute x = 6 into the original equation. Left side: 4 × (6 + 3) = 4 × 9 = 36. Right side: 3 × (6 + 6) = 3 × 12 = 36. Both sides equal 36, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.6(x − 2) = 2(x + 8)
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 6(x − 2):
6 × x = 6x
6 × (−2) = −12
Expand 2(x + 8):
2 × x = 2x
2 × 8 = 16
The equation is now 6x − 12 = 2x + 16 - Remove the x term from the right side: subtract 2x from both sides. We want all the x terms on one side. The right side has 2x; subtracting 2x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.6x − 12 − 2x = 2x + 16 − 2x
4x − 12 = 16 - Add 12 to both sides. This undoes the −12, because −12 + 12 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x − 12 + 12 = 16 + 12
4x = 28 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 284
x = 284 = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
Check. Substitute x = 7 into the original equation. Left side: 6 × (7 − 2) = 6 × 5 = 30. Right side: 2 × (7 + 8) = 2 × 15 = 30. Both sides equal 30, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5(2x − 1) = 3(3x + 1)
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 5(2x − 1):
5 × 2x = 10x
5 × (−1) = −5
Expand 3(3x + 1):
3 × 3x = 9x
3 × 1 = 3
The equation is now 10x − 5 = 9x + 3 - Remove the x term from the right side: subtract 9x from both sides. We want all the x terms on one side. The right side has 9x; subtracting 9x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.10x − 5 − 9x = 9x + 3 − 9x
x − 5 = 3 - Add 5 to both sides. This undoes the −5, because −5 + 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x − 5 + 5 = 3 + 5
x = 8 - Write the solution. x is now on its own, so the equation tells us its value.x = 8
Check. Substitute x = 8 into the original equation. Left side: 5 × (2 × 8 − 1) = 5 × 15 = 75. Right side: 3 × (3 × 8 + 1) = 3 × 25 = 75. Both sides equal 75, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.10 − (x + 2) = 5
- Expand the brackets. Multiply each term inside the bracket by the number outside it. A negative number multiplied by a negative number gives a positive answer, and a negative multiplied by a positive gives a negative.Expand −(x + 2):
(−1) × x = −x
(−1) × 2 = −2
The equation is now 10 − x − 2 = 5 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the numbers 10 − 2 make 8.
The equation is now −x + 8 = 5 - Subtract 8 from both sides. This undoes the +8, because +8 − 8 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.−x + 8 − 8 = 5 − 8
−x = −3 - Divide both sides by −1. −x means −1 multiplied by x. The inverse of multiplying by −1 is dividing by −1, and we do it to both sides to keep the balance. This leaves x on its own. Careful with signs: a positive divided by a negative gives a negative answer, and a negative divided by a negative gives a positive.−x−1 = −3−1
x = −3−1 = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
Check. Substitute x = 3 into the original equation. Left side: 10 − (3 + 2) = 10 − 5 = 5. Right side: 5. Both sides equal 5, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.7 − 2(x − 1) = 3
- Expand the brackets. Multiply each term inside the bracket by the number outside it. A negative number multiplied by a negative number gives a positive answer, and a negative multiplied by a positive gives a negative.Expand −2(x − 1):
(−2) × x = −2x
(−2) × (−1) = 2
The equation is now 7 − 2x + 2 = 3 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the numbers 7 + 2 make 9.
The equation is now −2x + 9 = 3 - Subtract 9 from both sides. This undoes the +9, because +9 − 9 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.−2x + 9 − 9 = 3 − 9
−2x = −6 - Divide both sides by −2. −2x means −2 multiplied by x. The inverse of multiplying by −2 is dividing by −2, and we do it to both sides to keep the balance. This leaves x on its own. Careful with signs: a positive divided by a negative gives a negative answer, and a negative divided by a negative gives a positive.−2x−2 = −6−2
x = −6−2 = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
Check. Substitute x = 3 into the original equation. Left side: 7 − 2 × (3 − 1) = 7 − 2 × 2 = 7 − 4 = 3. Right side: 3. Both sides equal 3, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.20 − 3(x + 1) = 2
- Expand the brackets. Multiply each term inside the bracket by the number outside it. A negative number multiplied by a negative number gives a positive answer, and a negative multiplied by a positive gives a negative.Expand −3(x + 1):
(−3) × x = −3x
(−3) × 1 = −3
The equation is now 20 − 3x − 3 = 2 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the numbers 20 − 3 make 17.
The equation is now −3x + 17 = 2 - Subtract 17 from both sides. This undoes the +17, because +17 − 17 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.−3x + 17 − 17 = 2 − 17
−3x = −15 - Divide both sides by −3. −3x means −3 multiplied by x. The inverse of multiplying by −3 is dividing by −3, and we do it to both sides to keep the balance. This leaves x on its own. Careful with signs: a positive divided by a negative gives a negative answer, and a negative divided by a negative gives a positive.−3x−3 = −15−3
x = −15−3 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 20 − 3 × (5 + 1) = 20 − 3 × 6 = 20 − 18 = 2. Right side: 2. Both sides equal 2, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5 − (x − 3) = 1
- Expand the brackets. Multiply each term inside the bracket by the number outside it. A negative number multiplied by a negative number gives a positive answer, and a negative multiplied by a positive gives a negative.Expand −(x − 3):
(−1) × x = −x
(−1) × (−3) = 3
The equation is now 5 − x + 3 = 1 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the numbers 5 + 3 make 8.
The equation is now −x + 8 = 1 - Subtract 8 from both sides. This undoes the +8, because +8 − 8 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.−x + 8 − 8 = 1 − 8
−x = −7 - Divide both sides by −1. −x means −1 multiplied by x. The inverse of multiplying by −1 is dividing by −1, and we do it to both sides to keep the balance. This leaves x on its own. Careful with signs: a positive divided by a negative gives a negative answer, and a negative divided by a negative gives a positive.−x−1 = −7−1
x = −7−1 = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
Check. Substitute x = 7 into the original equation. Left side: 5 − (7 − 3) = 5 − 4 = 1. Right side: 1. Both sides equal 1, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.12 − 2(x + 3) = 4
- Expand the brackets. Multiply each term inside the bracket by the number outside it. A negative number multiplied by a negative number gives a positive answer, and a negative multiplied by a positive gives a negative.Expand −2(x + 3):
(−2) × x = −2x
(−2) × 3 = −6
The equation is now 12 − 2x − 6 = 4 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the numbers 12 − 6 make 6.
The equation is now −2x + 6 = 4 - Subtract 6 from both sides. This undoes the +6, because +6 − 6 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.−2x + 6 − 6 = 4 − 6
−2x = −2 - Divide both sides by −2. −2x means −2 multiplied by x. The inverse of multiplying by −2 is dividing by −2, and we do it to both sides to keep the balance. This leaves x on its own. Careful with signs: a positive divided by a negative gives a negative answer, and a negative divided by a negative gives a positive.−2x−2 = −2−2
x = −2−2 = 1 - Write the solution. x is now on its own, so the equation tells us its value.x = 1
Check. Substitute x = 1 into the original equation. Left side: 12 − 2 × (1 + 3) = 12 − 2 × 4 = 12 − 8 = 4. Right side: 4. Both sides equal 4, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.9 − 3(x − 2) = 6
- Expand the brackets. Multiply each term inside the bracket by the number outside it. A negative number multiplied by a negative number gives a positive answer, and a negative multiplied by a positive gives a negative.Expand −3(x − 2):
(−3) × x = −3x
(−3) × (−2) = 6
The equation is now 9 − 3x + 6 = 6 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the numbers 9 + 6 make 15.
The equation is now −3x + 15 = 6 - Subtract 15 from both sides. This undoes the +15, because +15 − 15 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.−3x + 15 − 15 = 6 − 15
−3x = −9 - Divide both sides by −3. −3x means −3 multiplied by x. The inverse of multiplying by −3 is dividing by −3, and we do it to both sides to keep the balance. This leaves x on its own. Careful with signs: a positive divided by a negative gives a negative answer, and a negative divided by a negative gives a positive.−3x−3 = −9−3
x = −9−3 = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
Check. Substitute x = 3 into the original equation. Left side: 9 − 3 × (3 − 2) = 9 − 3 × 1 = 9 − 3 = 6. Right side: 6. Both sides equal 6, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Method. Expand properly, then solve the equation.
- Find the mistake. The 3 outside multiplies everything inside. Jo multiplied the x but not the 4.3 × x = 3x and 3 × (−4) = −12, so 3(x − 4) = 3x − 12
- Write the equation again. 3x − 12 = 15
- Solve. Add 12 to both sides, then divide by 3.3x = 27
x = 9
Check. 3 × (9 − 4) = 3 × 5 = 15. ✓
Watch out. Multiply every term inside the bracket.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number I think of.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Add 4 to the number (x + 4), then multiply the result by 3 (so the bracket is multiplied by 3): 3(x + 4). The answer is 33.
Equation: 3(x + 4) = 33 - Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 3(x + 4):
3 × x = 3x
3 × 4 = 12
The equation is now 3x + 12 = 33 - Subtract 12 from both sides. This undoes the +12, because +12 − 12 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 12 − 12 = 33 − 12
3x = 21 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 213
x = 213 = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The number is 7.
Check. 7 + 4 = 11, and 3 × 11 = 33 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the cost of an adult ticket in pounds.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.A child ticket costs £3 less, so it costs x − 3 pounds. 2 adult tickets cost 2x and 3 child tickets cost 3(x − 3). The total is £41.
Equation: 2x + 3(x − 3) = 41 - Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 3(x − 3):
3 × x = 3x
3 × (−3) = −9
The equation is now 2x + 3x − 9 = 41 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms 2x + 3x make 5x.
The equation is now 5x − 9 = 41 - Add 9 to both sides. This undoes the −9, because −9 + 9 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5x − 9 + 9 = 41 + 9
5x = 50 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 505
x = 505 = 10 - Write the solution. x is now on its own, so the equation tells us its value.x = 10
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.An adult ticket costs £10 (and a child ticket costs £7).
Check. 2 × £10 = £20 and 3 × £7 = £21. In total £41 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Substitute the value into both sides of the equation, work each side out, and compare.
- Write the equation and the value to test. A number is a solution of an equation if it makes both sides equal. We test it by substituting it (putting it in place of the letter).2(3x − 5) = 5x − 6, x = 4
- Substitute into the left-hand side. Replace every x with the number. Use brackets around a negative number so that the signs stay clear.Left side = 2 × (3 × 4 − 5) = 2 × 7 = 14
- Substitute into the right-hand side. Do exactly the same to the other side of the equation.Right side = 5 × 4 − 6 = 20 − 6 = 14
- Compare the two sides. If the two sides are equal the number is a solution; if not, it is not.14 = 14, so yes, x = 4 is a solution.
Check. The two sides match, which is exactly what we wanted to see. ✓
Watch out. Work out the left side and the right side separately. Do not use an = sign between the two until you have shown they are equal.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the length in cm.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.The rectangle has height 3 cm and width (x + 5) cm, so its area is 3(x + 5) cm2. The area is 36 cm2.
Equation: 3(x + 5) = 36 - Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 3(x + 5):
3 × x = 3x
3 × 5 = 15
The equation is now 3x + 15 = 36 - Subtract 15 from both sides. This undoes the +15, because +15 − 15 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 15 − 15 = 36 − 15
3x = 21 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 213
x = 213 = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.x = 7, so the width is 12 cm.
Check. Width = 7 + 5 = 12 cm and 3 × 12 = 36 cm2 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2(x + 3) = 10
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 2(x + 3):
2 × x = 2x
2 × 3 = 6
The equation is now 2x + 6 = 10 - Subtract 6 from both sides. This undoes the +6, because +6 − 6 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 6 − 6 = 10 − 6
2x = 4 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 42
x = 42 = 2 - Write the solution. x is now on its own, so the equation tells us its value.x = 2
Check. Substitute x = 2 into the original equation. Left side: 2 × (2 + 3) = 2 × 5 = 10. Right side: 10. Both sides equal 10, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3(x − 1) = 12
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 3(x − 1):
3 × x = 3x
3 × (−1) = −3
The equation is now 3x − 3 = 12 - Add 3 to both sides. This undoes the −3, because −3 + 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x − 3 + 3 = 12 + 3
3x = 15 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 153
x = 153 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 3 × (5 − 1) = 3 × 4 = 12. Right side: 12. Both sides equal 12, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4(x + 2) = 20
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 4(x + 2):
4 × x = 4x
4 × 2 = 8
The equation is now 4x + 8 = 20 - Subtract 8 from both sides. This undoes the +8, because +8 − 8 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x + 8 − 8 = 20 − 8
4x = 12 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 124
x = 124 = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
Check. Substitute x = 3 into the original equation. Left side: 4 × (3 + 2) = 4 × 5 = 20. Right side: 20. Both sides equal 20, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Equations with fractions
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
In Chapter 3 you added and subtracted fractions by changing them to the same bottom number (the common denominator). We use the same idea to solve equations that contain fractions, but we use it to remove the fractions altogether.
Consider x + 32 = 5. The whole of the top, x + 3, is divided by 2. To undo “divide by 2” we multiply both sides by 2: x + 3 = 10, so x = 7. On the left, 2 × x + 32 = x + 3, because the 2 on the top and the 2 on the bottom cancel. Note that the top is a bracket in disguise: the fraction bar acts like a bracket around x + 3.
When there is more than one fraction, for example x2 + x3 = 10, find the lowest common multiple (LCM) of the bottom numbers. Here the LCM of 2 and 3 is 6. Multiply every term on both sides by 6:
6 × x2 = 3x 6 × x3 = 2x 6 × 10 = 60, so 3x + 2x = 60, which gives 5x = 60 and x = 12.
Why does this work? Multiplying both sides of an equation by the same number keeps it balanced. If we choose a number that every bottom divides into, each fraction turns into a whole expression. It is exactly like the way you made equivalent fractions in Chapter 3, only now the aim is to get rid of the bottoms.
The most common slips are (1) forgetting to multiply a term that is not a fraction, such as the 10 above, and (2) forgetting that a numerator with two terms needs a bracket when you multiply: 6 × x + 12 = 3(x + 1), not 3x + 1.
Key ideas
- To solve an equation with fractions, multiply every term on both sides by the lowest common multiple of the bottom numbers.
- This removes the fractions: after you multiply, the bottoms cancel.
- If a numerator has two terms, use a bracket after multiplying, then expand it.
- Multiply every term, including whole numbers. Then solve the ordinary equation that is left, and check in the original equation.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x + 32 = 5
- Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.2 → the lowest common multiple is 2
- Multiply every term, on both sides, by 2. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.2 × x + 32 = (x + 3) (2 goes into 2 exactly 1 time, so the bottom disappears)
2 × 5 = 10
The new equation is (x + 3) = 10 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(x + 3) has nothing in front of it except +1, so the brackets can simply be removed: x + 3
The equation is now x + 3 = 10 - Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 3 − 3 = 10 − 3
x = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
Check. Substitute x = 7 into the original equation. Left side: 7 + 32 = 102 = 5. Right side: 5. Both sides equal 5, so the solution is correct. ✓
Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x − 13 = 5
- Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.3 → the lowest common multiple is 3
- Multiply every term, on both sides, by 3. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.3 × 2x − 13 = (2x − 1) (3 goes into 3 exactly 1 time, so the bottom disappears)
3 × 5 = 15
The new equation is (2x − 1) = 15 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(2x − 1) has nothing in front of it except +1, so the brackets can simply be removed: 2x − 1
The equation is now 2x − 1 = 15 - Add 1 to both sides. This undoes the −1, because −1 + 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x − 1 + 1 = 15 + 1
2x = 16 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 162
x = 162 = 8 - Write the solution. x is now on its own, so the equation tells us its value.x = 8
Check. Substitute x = 8 into the original equation. Left side: 2 × 8 − 13 = 153 = 5. Right side: 5. Both sides equal 5, so the solution is correct. ✓
Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x2 + x3 = 10
- Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.2, 3 → the lowest common multiple is 6
- Multiply every term, on both sides, by 6. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.6 × x2 = 3x (2 goes into 6 exactly 3 times, so the bottom disappears)
6 × x3 = 2x (3 goes into 6 exactly 2 times, so the bottom disappears)
6 × 10 = 60
The new equation is 3x + 2x = 60 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms 3x + 2x make 5x.
The equation is now 5x = 60 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 605
x = 605 = 12 - Write the solution. x is now on its own, so the equation tells us its value.x = 12
Check. Substitute x = 12 into the original equation. Left side: 122 + 123 = 6 + 4 = 10. Right side: 10. Both sides equal 10, so the solution is correct. ✓
Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x3 + 2 = x2
- Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.3, 2 → the lowest common multiple is 6
- Multiply every term, on both sides, by 6. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.6 × x3 = 2x (3 goes into 6 exactly 2 times, so the bottom disappears)
6 × 2 = 12
6 × x2 = 3x (2 goes into 6 exactly 3 times, so the bottom disappears)
The new equation is 2x + 12 = 3x - Remove the x term from the left side: subtract 2x from both sides. We want all the x terms on one side. The left side has 2x; subtracting 2x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.2x + 12 − 2x = 3x − 2x
12 = x - Write the solution. x is now on its own, so the equation tells us its value.x = 12
Check. Substitute x = 12 into the original equation. Left side: 123 + 2 = 4 + 2 = 6. Right side: 122 = 6. Both sides equal 6, so the solution is correct. ✓
Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.
Watch out
Multiply every term. In x2 + 3 = 7, multiplying by 2 gives x + 6 = 14, not x + 3 = 14. And when the top has two terms, treat it like a bracket: 6 × x + 12 = 3(x + 1).Exercise 9F Equations with fractions
Non-calculatorKey words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x + 32 = 5
- Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.2 → the lowest common multiple is 2
- Multiply every term, on both sides, by 2. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.2 × x + 32 = (x + 3) (2 goes into 2 exactly 1 time, so the bottom disappears)
2 × 5 = 10
The new equation is (x + 3) = 10 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(x + 3) has nothing in front of it except +1, so the brackets can simply be removed: x + 3
The equation is now x + 3 = 10 - Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 3 − 3 = 10 − 3
x = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
Check. Substitute x = 7 into the original equation. Left side: 7 + 32 = 102 = 5. Right side: 5. Both sides equal 5, so the solution is correct. ✓
Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x + 13 = 4
- Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.3 → the lowest common multiple is 3
- Multiply every term, on both sides, by 3. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.3 × x + 13 = (x + 1) (3 goes into 3 exactly 1 time, so the bottom disappears)
3 × 4 = 12
The new equation is (x + 1) = 12 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(x + 1) has nothing in front of it except +1, so the brackets can simply be removed: x + 1
The equation is now x + 1 = 12 - Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 1 − 1 = 12 − 1
x = 11 - Write the solution. x is now on its own, so the equation tells us its value.x = 11
Check. Substitute x = 11 into the original equation. Left side: 11 + 13 = 123 = 4. Right side: 4. Both sides equal 4, so the solution is correct. ✓
Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x − 25 = 2
- Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.5 → the lowest common multiple is 5
- Multiply every term, on both sides, by 5. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.5 × x − 25 = (x − 2) (5 goes into 5 exactly 1 time, so the bottom disappears)
5 × 2 = 10
The new equation is (x − 2) = 10 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(x − 2) has nothing in front of it except +1, so the brackets can simply be removed: x − 2
The equation is now x − 2 = 10 - Add 2 to both sides. This undoes the −2, because −2 + 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x − 2 + 2 = 10 + 2
x = 12 - Write the solution. x is now on its own, so the equation tells us its value.x = 12
Check. Substitute x = 12 into the original equation. Left side: 12 − 25 = 105 = 2. Right side: 2. Both sides equal 2, so the solution is correct. ✓
Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x + 74 = 3
- Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.4 → the lowest common multiple is 4
- Multiply every term, on both sides, by 4. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.4 × x + 74 = (x + 7) (4 goes into 4 exactly 1 time, so the bottom disappears)
4 × 3 = 12
The new equation is (x + 7) = 12 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(x + 7) has nothing in front of it except +1, so the brackets can simply be removed: x + 7
The equation is now x + 7 = 12 - Subtract 7 from both sides. This undoes the +7, because +7 − 7 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 7 − 7 = 12 − 7
x = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 5 + 74 = 124 = 3. Right side: 3. Both sides equal 3, so the solution is correct. ✓
Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x − 42 = 6
- Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.2 → the lowest common multiple is 2
- Multiply every term, on both sides, by 2. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.2 × x − 42 = (x − 4) (2 goes into 2 exactly 1 time, so the bottom disappears)
2 × 6 = 12
The new equation is (x − 4) = 12 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(x − 4) has nothing in front of it except +1, so the brackets can simply be removed: x − 4
The equation is now x − 4 = 12 - Add 4 to both sides. This undoes the −4, because −4 + 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x − 4 + 4 = 12 + 4
x = 16 - Write the solution. x is now on its own, so the equation tells us its value.x = 16
Check. Substitute x = 16 into the original equation. Left side: 16 − 42 = 122 = 6. Right side: 6. Both sides equal 6, so the solution is correct. ✓
Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x + 56 = 2
- Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.6 → the lowest common multiple is 6
- Multiply every term, on both sides, by 6. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.6 × x + 56 = (x + 5) (6 goes into 6 exactly 1 time, so the bottom disappears)
6 × 2 = 12
The new equation is (x + 5) = 12 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(x + 5) has nothing in front of it except +1, so the brackets can simply be removed: x + 5
The equation is now x + 5 = 12 - Subtract 5 from both sides. This undoes the +5, because +5 − 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 5 − 5 = 12 − 5
x = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
Check. Substitute x = 7 into the original equation. Left side: 7 + 56 = 126 = 2. Right side: 2. Both sides equal 2, so the solution is correct. ✓
Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x + 13 = 5
- Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.3 → the lowest common multiple is 3
- Multiply every term, on both sides, by 3. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.3 × 2x + 13 = (2x + 1) (3 goes into 3 exactly 1 time, so the bottom disappears)
3 × 5 = 15
The new equation is (2x + 1) = 15 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(2x + 1) has nothing in front of it except +1, so the brackets can simply be removed: 2x + 1
The equation is now 2x + 1 = 15 - Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 1 − 1 = 15 − 1
2x = 14 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 142
x = 142 = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
Check. Substitute x = 7 into the original equation. Left side: 2 × 7 + 13 = 153 = 5. Right side: 5. Both sides equal 5, so the solution is correct. ✓
Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3x − 24 = 4
- Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.4 → the lowest common multiple is 4
- Multiply every term, on both sides, by 4. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.4 × 3x − 24 = (3x − 2) (4 goes into 4 exactly 1 time, so the bottom disappears)
4 × 4 = 16
The new equation is (3x − 2) = 16 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(3x − 2) has nothing in front of it except +1, so the brackets can simply be removed: 3x − 2
The equation is now 3x − 2 = 16 - Add 2 to both sides. This undoes the −2, because −2 + 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x − 2 + 2 = 16 + 2
3x = 18 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 183
x = 183 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
Check. Substitute x = 6 into the original equation. Left side: 3 × 6 − 24 = 164 = 4. Right side: 4. Both sides equal 4, so the solution is correct. ✓
Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5x + 32 = 14
- Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.2 → the lowest common multiple is 2
- Multiply every term, on both sides, by 2. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.2 × 5x + 32 = (5x + 3) (2 goes into 2 exactly 1 time, so the bottom disappears)
2 × 14 = 28
The new equation is (5x + 3) = 28 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(5x + 3) has nothing in front of it except +1, so the brackets can simply be removed: 5x + 3
The equation is now 5x + 3 = 28 - Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5x + 3 − 3 = 28 − 3
5x = 25 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 255
x = 255 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 5 × 5 + 32 = 282 = 14. Right side: 14. Both sides equal 14, so the solution is correct. ✓
Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4x − 13 = 5
- Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.3 → the lowest common multiple is 3
- Multiply every term, on both sides, by 3. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.3 × 4x − 13 = (4x − 1) (3 goes into 3 exactly 1 time, so the bottom disappears)
3 × 5 = 15
The new equation is (4x − 1) = 15 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(4x − 1) has nothing in front of it except +1, so the brackets can simply be removed: 4x − 1
The equation is now 4x − 1 = 15 - Add 1 to both sides. This undoes the −1, because −1 + 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x − 1 + 1 = 15 + 1
4x = 16 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 164
x = 164 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 4 × 4 − 13 = 153 = 5. Right side: 5. Both sides equal 5, so the solution is correct. ✓
Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x + 57 = 3
- Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.7 → the lowest common multiple is 7
- Multiply every term, on both sides, by 7. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.7 × 2x + 57 = (2x + 5) (7 goes into 7 exactly 1 time, so the bottom disappears)
7 × 3 = 21
The new equation is (2x + 5) = 21 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(2x + 5) has nothing in front of it except +1, so the brackets can simply be removed: 2x + 5
The equation is now 2x + 5 = 21 - Subtract 5 from both sides. This undoes the +5, because +5 − 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 5 − 5 = 21 − 5
2x = 16 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 162
x = 162 = 8 - Write the solution. x is now on its own, so the equation tells us its value.x = 8
Check. Substitute x = 8 into the original equation. Left side: 2 × 8 + 57 = 217 = 3. Right side: 3. Both sides equal 3, so the solution is correct. ✓
Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3x + 15 = 2
- Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.5 → the lowest common multiple is 5
- Multiply every term, on both sides, by 5. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.5 × 3x + 15 = (3x + 1) (5 goes into 5 exactly 1 time, so the bottom disappears)
5 × 2 = 10
The new equation is (3x + 1) = 10 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(3x + 1) has nothing in front of it except +1, so the brackets can simply be removed: 3x + 1
The equation is now 3x + 1 = 10 - Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 1 − 1 = 10 − 1
3x = 9 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 93
x = 93 = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
Check. Substitute x = 3 into the original equation. Left side: 3 × 3 + 15 = 105 = 2. Right side: 2. Both sides equal 2, so the solution is correct. ✓
Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x2 + x3 = 10
- Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.2, 3 → the lowest common multiple is 6
- Multiply every term, on both sides, by 6. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.6 × x2 = 3x (2 goes into 6 exactly 3 times, so the bottom disappears)
6 × x3 = 2x (3 goes into 6 exactly 2 times, so the bottom disappears)
6 × 10 = 60
The new equation is 3x + 2x = 60 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms 3x + 2x make 5x.
The equation is now 5x = 60 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 605
x = 605 = 12 - Write the solution. x is now on its own, so the equation tells us its value.x = 12
Check. Substitute x = 12 into the original equation. Left side: 122 + 123 = 6 + 4 = 10. Right side: 10. Both sides equal 10, so the solution is correct. ✓
Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x3 + x4 = 7
- Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.3, 4 → the lowest common multiple is 12
- Multiply every term, on both sides, by 12. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.12 × x3 = 4x (3 goes into 12 exactly 4 times, so the bottom disappears)
12 × x4 = 3x (4 goes into 12 exactly 3 times, so the bottom disappears)
12 × 7 = 84
The new equation is 4x + 3x = 84 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms 4x + 3x make 7x.
The equation is now 7x = 84 - Divide both sides by 7. 7x means 7 multiplied by x. The inverse of multiplying by 7 is dividing by 7, and we do it to both sides to keep the balance. This leaves x on its own.7x7 = 847
x = 847 = 12 - Write the solution. x is now on its own, so the equation tells us its value.x = 12
Check. Substitute x = 12 into the original equation. Left side: 123 + 124 = 4 + 3 = 7. Right side: 7. Both sides equal 7, so the solution is correct. ✓
Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x2 − x5 = 6
- Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.2, 5 → the lowest common multiple is 10
- Multiply every term, on both sides, by 10. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.10 × x2 = 5x (2 goes into 10 exactly 5 times, so the bottom disappears)
10 × (−x5) = −2x (5 goes into 10 exactly 2 times, so the bottom disappears)
10 × 6 = 60
The new equation is 5x − 2x = 60 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms 5x − 2x make 3x.
The equation is now 3x = 60 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 603
x = 603 = 20 - Write the solution. x is now on its own, so the equation tells us its value.x = 20
Check. Substitute x = 20 into the original equation. Left side: 202 − 205 = 10 − 4 = 6. Right side: 6. Both sides equal 6, so the solution is correct. ✓
Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x4 + x2 = 9
- Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.4, 2 → the lowest common multiple is 4
- Multiply every term, on both sides, by 4. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.4 × x4 = x (4 goes into 4 exactly 1 time, so the bottom disappears)
4 × x2 = 2x (2 goes into 4 exactly 2 times, so the bottom disappears)
4 × 9 = 36
The new equation is x + 2x = 36 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms x + 2x make 3x.
The equation is now 3x = 36 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 363
x = 363 = 12 - Write the solution. x is now on its own, so the equation tells us its value.x = 12
Check. Substitute x = 12 into the original equation. Left side: 124 + 122 = 3 + 6 = 9. Right side: 9. Both sides equal 9, so the solution is correct. ✓
Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x3 + 2 = x2
- Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.3, 2 → the lowest common multiple is 6
- Multiply every term, on both sides, by 6. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.6 × x3 = 2x (3 goes into 6 exactly 2 times, so the bottom disappears)
6 × 2 = 12
6 × x2 = 3x (2 goes into 6 exactly 3 times, so the bottom disappears)
The new equation is 2x + 12 = 3x - Remove the x term from the left side: subtract 2x from both sides. We want all the x terms on one side. The left side has 2x; subtracting 2x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.2x + 12 − 2x = 3x − 2x
12 = x - Write the solution. x is now on its own, so the equation tells us its value.x = 12
Check. Substitute x = 12 into the original equation. Left side: 123 + 2 = 4 + 2 = 6. Right side: 122 = 6. Both sides equal 6, so the solution is correct. ✓
Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x5 + 3 = x2
- Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.5, 2 → the lowest common multiple is 10
- Multiply every term, on both sides, by 10. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.10 × x5 = 2x (5 goes into 10 exactly 2 times, so the bottom disappears)
10 × 3 = 30
10 × x2 = 5x (2 goes into 10 exactly 5 times, so the bottom disappears)
The new equation is 2x + 30 = 5x - Remove the x term from the left side: subtract 2x from both sides. We want all the x terms on one side. The left side has 2x; subtracting 2x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.2x + 30 − 2x = 5x − 2x
30 = 3x - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.303 = 3x3
x = 303 = 10 - Write the solution. x is now on its own, so the equation tells us its value.x = 10
Check. Substitute x = 10 into the original equation. Left side: 105 + 3 = 2 + 3 = 5. Right side: 102 = 5. Both sides equal 5, so the solution is correct. ✓
Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x4 + 3 = 8
- Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x4 + 3 − 3 = 8 − 3
x4 = 5 - Multiply both sides by 4. x4 means x divided by 4. The inverse of dividing by 4 is multiplying by 4. We do it to both sides to keep the balance.4 × x4 = 4 × 5
x = 20 - Write the solution. x is now on its own, so the equation tells us its value.x = 20
Check. Substitute x = 20 into the original equation. Left side: 204 + 3 = 5 + 3 = 8. Right side: 8. Both sides equal 8, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x3 − 5 = 1
- Add 5 to both sides. This undoes the −5, because −5 + 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x3 − 5 + 5 = 1 + 5
x3 = 6 - Multiply both sides by 3. x3 means x divided by 3. The inverse of dividing by 3 is multiplying by 3. We do it to both sides to keep the balance.3 × x3 = 3 × 6
x = 18 - Write the solution. x is now on its own, so the equation tells us its value.x = 18
Check. Substitute x = 18 into the original equation. Left side: 183 − 5 = 6 − 5 = 1. Right side: 1. Both sides equal 1, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x6 + 2 = 7
- Subtract 2 from both sides. This undoes the +2, because +2 − 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x6 + 2 − 2 = 7 − 2
x6 = 5 - Multiply both sides by 6. x6 means x divided by 6. The inverse of dividing by 6 is multiplying by 6. We do it to both sides to keep the balance.6 × x6 = 6 × 5
x = 30 - Write the solution. x is now on its own, so the equation tells us its value.x = 30
Check. Substitute x = 30 into the original equation. Left side: 306 + 2 = 5 + 2 = 7. Right side: 7. Both sides equal 7, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x2 − 3 = 4
- Add 3 to both sides. This undoes the −3, because −3 + 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x2 − 3 + 3 = 4 + 3
x2 = 7 - Multiply both sides by 2. x2 means x divided by 2. The inverse of dividing by 2 is multiplying by 2. We do it to both sides to keep the balance.2 × x2 = 2 × 7
x = 14 - Write the solution. x is now on its own, so the equation tells us its value.x = 14
Check. Substitute x = 14 into the original equation. Left side: 142 − 3 = 7 − 3 = 4. Right side: 4. Both sides equal 4, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Half of the number is x2. Adding 3 gives x2 + 3. The result is 11.
Equation: x2 + 3 = 11 - Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x2 + 3 − 3 = 11 − 3
x2 = 8 - Multiply both sides by 2. x2 means x divided by 2. The inverse of dividing by 2 is multiplying by 2. We do it to both sides to keep the balance.2 × x2 = 2 × 8
x = 16 - Write the solution. x is now on its own, so the equation tells us its value.x = 16
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The number is 16.
Check. Half of 16 is 8, and 8 + 3 = 11 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number of pages in the book.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.A third of the pages is x3. Ana has read a third of the book and a further 10 pages, which is 40 pages altogether: x3 + 10 = 40.
Equation: x3 + 10 = 40 - Subtract 10 from both sides. This undoes the +10, because +10 − 10 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x3 + 10 − 10 = 40 − 10
x3 = 30 - Multiply both sides by 3. x3 means x divided by 3. The inverse of dividing by 3 is multiplying by 3. We do it to both sides to keep the balance.3 × x3 = 3 × 30
x = 90 - Write the solution. x is now on its own, so the equation tells us its value.x = 90
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The book has 90 pages.
Check. A third of 90 is 30, and 30 + 10 = 40 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Method. Find the term that Kai forgot to multiply, then solve correctly.
- Look at what Kai did. He multiplied both sides by 2. The x2 became x and the 7 became 14. But the + 3 on the left was left alone.Every term must be multiplied, including 3.
- Multiply every term by 2. This keeps the equation balanced and removes the fraction.2 × x2 = x 2 × 3 = 6 2 × 7 = 14
x + 6 = 14 - Solve. Subtract 6 from both sides.x = 8
Check. 82 + 3 = 4 + 3 = 7. ✓
Watch out. Multiply every term, not only the fractions.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the smallest of the three numbers.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.The numbers are x, x + 4 and x + 8. The mean is the total divided by 3, so the mean is x + (x + 4) + (x + 8)3, which is 12. The top simplifies to 3x + 12.
Equation: 3x + 123 = 12 - Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.3 → the lowest common multiple is 3
- Multiply every term, on both sides, by 3. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.3 × 3x + 123 = (3x + 12) (3 goes into 3 exactly 1 time, so the bottom disappears)
3 × 12 = 36
The new equation is (3x + 12) = 36 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(3x + 12) has nothing in front of it except +1, so the brackets can simply be removed: 3x + 12
The equation is now 3x + 12 = 36 - Subtract 12 from both sides. This undoes the +12, because +12 − 12 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 12 − 12 = 36 − 12
3x = 24 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 243
x = 243 = 8 - Write the solution. x is now on its own, so the equation tells us its value.x = 8
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The numbers are 8, 12 and 16.
Check. (8 + 12 + 16) ÷ 3 = 36 ÷ 3 = 12 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x2 = 4
- Multiply both sides by 2. x2 means x divided by 2. The inverse of dividing by 2 is multiplying by 2. We do it to both sides to keep the balance.2 × x2 = 2 × 4
x = 8 - Write the solution. x is now on its own, so the equation tells us its value.x = 8
Check. Substitute x = 8 into the original equation. Left side: 82 = 4. Right side: 4. Both sides equal 4, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x3 + 1 = 4
- Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x3 + 1 − 1 = 4 − 1
x3 = 3 - Multiply both sides by 3. x3 means x divided by 3. The inverse of dividing by 3 is multiplying by 3. We do it to both sides to keep the balance.3 × x3 = 3 × 3
x = 9 - Write the solution. x is now on its own, so the equation tells us its value.x = 9
Check. Substitute x = 9 into the original equation. Left side: 93 + 1 = 3 + 1 = 4. Right side: 4. Both sides equal 4, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x + 12 = 3
- Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.2 → the lowest common multiple is 2
- Multiply every term, on both sides, by 2. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.2 × x + 12 = (x + 1) (2 goes into 2 exactly 1 time, so the bottom disappears)
2 × 3 = 6
The new equation is (x + 1) = 6 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(x + 1) has nothing in front of it except +1, so the brackets can simply be removed: x + 1
The equation is now x + 1 = 6 - Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 1 − 1 = 6 − 1
x = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 5 + 12 = 62 = 3. Right side: 3. Both sides equal 3, so the solution is correct. ✓
Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Substitute the value into both sides of the equation, work each side out, and compare.
- Write the equation and the value to test. A number is a solution of an equation if it makes both sides equal. We test it by substituting it (putting it in place of the letter).x2 + x3 = 5, x = 6
- Substitute into the left-hand side. Replace every x with the number. Use brackets around a negative number so that the signs stay clear.Left side = 62 + 63 = 3 + 2 = 5
- Substitute into the right-hand side. Do exactly the same to the other side of the equation.Right side = 5
- Compare the two sides. If the two sides are equal the number is a solution; if not, it is not.5 = 5, so yes, x = 6 is a solution.
Check. The two sides match, which is exactly what we wanted to see. ✓
Watch out. Work out the left side and the right side separately. Do not use an = sign between the two until you have shown they are equal.
Forming equations from words
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
Equations are most useful when a real problem is turned into one. A good method has five steps:
- Read the question and decide what you need to find.
- Choose a letter for the unknown and say what it stands for, with units. (“Let x be the cost of one ticket in pounds.”)
- Write an equation using the information. The two sides must have the same value.
- Solve the equation.
- Answer in words, with units, and check that your answer fits the story.
The hard step is turning words into algebra. Learn these phrases. Notice that the order matters for “less than” and “minus”.
| In words | In algebra |
|---|---|
| a number plus 5, 5 more than a number | x + 5 |
| 5 less than a number, a number minus 5 | x − 5 |
| 5 minus a number | 5 − x |
| 5 times a number, double a number (2 times) | 5x, 2x |
| a number divided by 4, a quarter of a number | x4 |
| 3 more than twice a number | 2x + 3 |
| twice the sum of a number and 3 | 2(x + 3) |
Ages. If Ali is x years old and his sister is 5 years older, she is x + 5. In 8 years’ time they will be x + 8 and x + 13. “Three times as old” means 3x.
Costs. If one ticket costs £x, then 4 tickets cost 4x pounds. A fixed charge such as a booking fee is added once, but a charge per item or per hour is multiplied by the number of items or hours.
Finally, check your answer in the words of the question, not just in your equation. If your equation was wrong, an answer that satisfies the equation can still be wrong for the story.
Key ideas
- Always begin with “Let x = …” and give the units.
- The two sides of the equation must be equal in value: use words like “is”, “gives”, “total” or “the same as” to find the = sign.
- “5 less than x” is x − 5 (not 5 − x).
- After solving, answer the question that was asked (with units) and check it against the story.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number I think of.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.I multiply the number by 4 (4x) and subtract 7 (4x − 7). The result is 29.
Equation: 4x − 7 = 29 - Add 7 to both sides. This undoes the −7, because −7 + 7 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x − 7 + 7 = 29 + 7
4x = 36 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 364
x = 364 = 9 - Write the solution. x is now on its own, so the equation tells us its value.x = 9
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The number is 9.
Check. 4 × 9 = 36, and 36 − 7 = 29 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number of hours the plumber works.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.The call-out fee of £35 is charged once. The hourly charge is £25 for each hour, which is 25x. The bill is 35 + 25x = 110.
Equation: 25x + 35 = 110 - Subtract 35 from both sides. This undoes the +35, because +35 − 35 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.25x + 35 − 35 = 110 − 35
25x = 75 - Divide both sides by 25. 25x means 25 multiplied by x. The inverse of multiplying by 25 is dividing by 25, and we do it to both sides to keep the balance. This leaves x on its own.25x25 = 7525
x = 7525 = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The plumber worked for 3 hours.
Check. 25 × 3 = 75, and 75 + 35 = 110 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = Ali's age in years.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.His sister is 5 years older, so she is x + 5. Their ages add up to 31.
Equation: x + (x + 5) = 31 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(x + 5) has nothing in front of it except +1, so the brackets can simply be removed: x + 5
The equation is now x + x + 5 = 31 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms x + x make 2x.
The equation is now 2x + 5 = 31 - Subtract 5 from both sides. This undoes the +5, because +5 − 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 5 − 5 = 31 − 5
2x = 26 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 262
x = 262 = 13 - Write the solution. x is now on its own, so the equation tells us its value.x = 13
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.Ali is 13 years old, and his sister is 18.
Check. 13 + 18 = 31, and 18 is 5 more than 13 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number Jo thinks of.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Jo adds 6 to her number: x + 6. She then doubles the whole result, so the bracket is multiplied by 2: 2(x + 6). The answer is 30.
Equation: 2(x + 6) = 30 - Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 2(x + 6):
2 × x = 2x
2 × 6 = 12
The equation is now 2x + 12 = 30 - Subtract 12 from both sides. This undoes the +12, because +12 − 12 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 12 − 12 = 30 − 12
2x = 18 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 182
x = 182 = 9 - Write the solution. x is now on its own, so the equation tells us its value.x = 9
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.Jo's number is 9.
Check. 9 + 6 = 15, and 2 × 15 = 30 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Watch out
“5 less than x” is x − 5. It is not 5 − x. For example, 5 less than 20 is 15 = 20 − 5. Also read what the question asks: after you find x, you may still need to use it (for example to find the sister’s age, not Ali’s).Exercise 9G Forming and solving equations from words
Non-calculatorMethod. Plus means add.
- Say it in symbols. “A number” is x, and “plus 5” means add 5.x + 5
- Check the order. For addition the order does not change the value.x + 5 = 5 + x
Check. If x = 10, the phrase gives 15 and x + 5 = 15. ✓
Method. “Less than” means take away from the number.
- Start with the number. We take 3 away from the number, not the other way round.x
- Subtract 3. 3 less than 10 is 7 = 10 − 3.x − 3
Check. If x = 10, 3 less than it is 7 and x − 3 = 7. ✓
Watch out. “3 less than x” is x − 3, not 3 − x.
Method. “Twice” means multiply by 2.
- Say it in symbols. Twice means two times, and we do not write the × sign in algebra.2 × x = 2x
- Check. If x = 10, twice the number is 20.2x = 20 ✓
Check. 2 × 10 = 20. ✓
Method. “Divided by” means a fraction, with the number on top.
- Write a division. The number is divided by 4.x ÷ 4
- Write it as a fraction. In algebra division is written as a fraction.x4
Check. If x = 20, 20 ÷ 4 = 5 and 204 = 5. ✓
Method. Do the doubling first, then add 3.
- Twice a number. 2x
- 3 more than that. Add 3 to the doubled number.2x + 3
Check. If x = 10: twice is 20, and 3 more is 23. 2 × 10 + 3 = 23. ✓
Method. The word “sum” means the adding happens first, so it goes in a bracket.
- The sum of a number and 3. Add first.x + 3
- Twice that. Multiply the whole sum by 2, so use a bracket.2(x + 3)
Check. If x = 10: the sum is 13, twice it is 26. 2(10 + 3) = 26. ✓
Watch out. Without the bracket, 2x + 3 would only double the x.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = Max's number.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.He adds 7 to the number: x + 7. The result is 15.
Equation: x + 7 = 15 - Subtract 7 from both sides. This undoes the +7, because +7 − 7 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 7 − 7 = 15 − 7
x = 8 - Write the solution. x is now on its own, so the equation tells us its value.x = 8
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.Max's number is 8.
Check. 8 + 7 = 15 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = Mia's number.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.She subtracts 6 from the number: x − 6. The result is 11.
Equation: x − 6 = 11 - Add 6 to both sides. This undoes the −6, because −6 + 6 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x − 6 + 6 = 11 + 6
x = 17 - Write the solution. x is now on its own, so the equation tells us its value.x = 17
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.Mia's number is 17.
Check. 17 − 6 = 11 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Half of the number is x2. Adding 3 gives x2 + 3 = 11.
Equation: x2 + 3 = 11 - Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x2 + 3 − 3 = 11 − 3
x2 = 8 - Multiply both sides by 2. x2 means x divided by 2. The inverse of dividing by 2 is multiplying by 2. We do it to both sides to keep the balance.2 × x2 = 2 × 8
x = 16 - Write the solution. x is now on its own, so the equation tells us its value.x = 16
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The number is 16.
Check. 16 ÷ 2 + 3 = 11 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.3 times the number is 3x. 5 less than that is 3x − 5. This equals 22.
Equation: 3x − 5 = 22 - Add 5 to both sides. This undoes the −5, because −5 + 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x − 5 + 5 = 22 + 5
3x = 27 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 273
x = 273 = 9 - Write the solution. x is now on its own, so the equation tells us its value.x = 9
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The number is 9.
Check. 3 × 9 − 5 = 22 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number of hours.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.The fee is charged once (35). The hourly charge is 25x. The total is 135.
Equation: 25x + 35 = 135 - Subtract 35 from both sides. This undoes the +35, because +35 − 35 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.25x + 35 − 35 = 135 − 35
25x = 100 - Divide both sides by 25. 25x means 25 multiplied by x. The inverse of multiplying by 25 is dividing by 25, and we do it to both sides to keep the balance. This leaves x on its own.25x25 = 10025
x = 10025 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The plumber worked for 4 hours.
Check. 25 × 4 = 100 and 100 + 35 = 135 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = Cara's age in years.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Ben is x + 4. The total of their ages is 30.
Equation: x + (x + 4) = 30 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(x + 4) has nothing in front of it except +1, so the brackets can simply be removed: x + 4
The equation is now x + x + 4 = 30 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms x + x make 2x.
The equation is now 2x + 4 = 30 - Subtract 4 from both sides. This undoes the +4, because +4 − 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 4 − 4 = 30 − 4
2x = 26 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 262
x = 262 = 13 - Write the solution. x is now on its own, so the equation tells us its value.x = 13
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.Cara is 13 years old.
Check. Cara is 13 and Ben is 17. 13 + 17 = 30 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the cost of one pen in pounds.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.3 pens cost 3x. The ruler costs 2. Altogether 11.
Equation: 3x + 2 = 11 - Subtract 2 from both sides. This undoes the +2, because +2 − 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 2 − 2 = 11 − 2
3x = 9 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 93
x = 93 = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.One pen costs £3.
Check. 3 × £3 = £9 and £9 + £2 = £11 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number of minutes.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.The monthly charge is 1200 pence. The calls cost 5x pence. The bill is 1500 pence.
Equation: 5x + 1200 = 1500 - Subtract 1200 from both sides. This undoes the +1200, because +1200 − 1200 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5x + 1200 − 1200 = 1500 − 1200
5x = 300 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 3005
x = 3005 = 60 - Write the solution. x is now on its own, so the equation tells us its value.x = 60
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.There were 60 minutes of calls.
Check. 5 × 60 = 300p and 300 + 1200 = 1500p ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = her brother's age in years.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Zara is 3x years old. The sum of their ages is x + 3x = 36.
Equation: x + 3x = 36 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms x + 3x make 4x.
The equation is now 4x = 36 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 364
x = 364 = 9 - Write the solution. x is now on its own, so the equation tells us its value.x = 9
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.Zara is 27 years old.
Check. Her brother is 9 and Zara is 27. 9 + 27 = 36 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the cost of one notebook in pounds.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Sam spends 20 − 5 = £15. The spending is 4 notebooks and a £3 pen: 4x + 3 = 15.
Equation: 4x + 3 = 15 - Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x + 3 − 3 = 15 − 3
4x = 12 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 124
x = 124 = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.One notebook costs £3.
Check. 4 × £3 = £12, plus £3 is £15, and £20 − £15 = £5 change ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the smallest number.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Consecutive numbers go up by 1, so the numbers are x, x + 1 and x + 2. They add up to 51.
Equation: x + (x + 1) + (x + 2) = 51 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(x + 1) has nothing in front of it except +1, so the brackets can simply be removed: x + 1
(x + 2) has nothing in front of it except +1, so the brackets can simply be removed: x + 2
The equation is now x + x + 1 + x + 2 = 51 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms x + x + x make 3x and the numbers 1 + 2 make 3.
The equation is now 3x + 3 = 51 - Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 3 − 3 = 51 − 3
3x = 48 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 483
x = 483 = 16 - Write the solution. x is now on its own, so the equation tells us its value.x = 16
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The smallest number is 16.
Check. 16 + 17 + 18 = 51 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number of months.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Gym A costs 20 + 10x pounds. Gym B costs 50 + 5x pounds. We want these to be equal.
Equation: 10x + 20 = 5x + 50 - Remove the x term from the right side: subtract 5x from both sides. We want all the x terms on one side. The right side has 5x; subtracting 5x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.10x + 20 − 5x = 5x + 50 − 5x
5x + 20 = 50 - Subtract 20 from both sides. This undoes the +20, because +20 − 20 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5x + 20 − 20 = 50 − 20
5x = 30 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 305
x = 305 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The total costs are the same after 6 months.
Check. Gym A: 20 + 10 × 6 = £80. Gym B: 50 + 5 × 6 = £80. The same ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Method. Translate the words carefully, then solve.
- Check Hana's equation. “5 less than a number” means we take 5 away from the number, so the number comes first.5 less than 20 is 15 = 20 − 5, so “5 less than x” is x − 5.
- Write the correct equation. The result is 17.x − 5 = 17
- Solve it. Add 5 to both sides.x = 17 + 5 = 22
Check. 22 − 5 = 17. ✓ (With Hana's equation, 5 − x = 17 would give a negative number, x = −12, which does not match the story.)
Watch out. “Less than” reverses the order of the words.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = Kim's number.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.She adds 7: x + 7 = 15.
Equation: x + 7 = 15 - Subtract 7 from both sides. This undoes the +7, because +7 − 7 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 7 − 7 = 15 − 7
x = 8 - Write the solution. x is now on its own, so the equation tells us its value.x = 8
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.Kim's number is 8.
Check. 8 + 7 = 15 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the cost of one pencil in pence.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.£3 = 300p. 6 pencils cost 6x pence.
Equation: 6x = 300 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 3006
x = 3006 = 50 - Write the solution. x is now on its own, so the equation tells us its value.x = 50
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.One pencil costs 50p.
Check. 6 × 50p = 300p = £3 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Equations from geometry: perimeters and angles
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
Diagrams often give lengths or angles as expressions with x in them. To find x we need a fact that tells us what the lengths or angles add up to. This gives us an equation. Here are the facts we will use (you meet them again in Chapters 17 and 18):
- The perimeter of a shape is the total distance around its edges: add up all the sides.
- Opposite sides of a rectangle are equal. All the sides of a square or rhombus are equal. Two sides of an isosceles triangle are equal.
- The angles in a triangle add up to 180°. The angles on a straight line add up to 180°.
- The angles round a point add up to 360°. The angles in a quadrilateral add up to 360°.
- The two base angles of an isosceles triangle are equal.
The method is: (1) say which fact you are using, (2) write an equation, (3) solve it, (4) put x back into the expressions on the diagram to find the length or angle that the question asks for, and (5) check, for example by adding the angles to see that they total 180°.
Diagrams in this book are drawn to scale, but in an exam a diagram may say “not drawn accurately”. Do not measure: use the facts.
Key ideas
- Perimeter = the sum of all the sides. For a rectangle: 2 × length + 2 × width.
- Angles in a triangle: 180°. On a straight line: 180°. At a point: 360°. In a quadrilateral: 360°.
- Use equal sides or equal angles (isosceles, square, rhombus) to write an equation without a total.
- After finding x, find what the question asks for, and check it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the perimeter. The perimeter is the total length around the rectangle. A rectangle has two sides of each length.Perimeter = 2(3x + 2) + 2(x + 4) = 36
Equation: 2(3x + 2) + 2(x + 4) = 36 - Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 2(3x + 2):
2 × 3x = 6x
2 × 2 = 4
Expand 2(x + 4):
2 × x = 2x
2 × 4 = 8
The equation is now 6x + 4 + 2x + 8 = 36 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms 6x + 2x make 8x and the numbers 4 + 8 make 12.
The equation is now 8x + 12 = 36 - Subtract 12 from both sides. This undoes the +12, because +12 − 12 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.8x + 12 − 12 = 36 − 12
8x = 24 - Divide both sides by 8. 8x means 8 multiplied by x. The inverse of multiplying by 8 is dividing by 8, and we do it to both sides to keep the balance. This leaves x on its own.8x8 = 248
x = 248 = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 3. The sides are 3 × 3 + 2 = 11 cm and 3 + 4 = 7 cm.
Check. Perimeter = 2 × 11 + 2 × 7 = 36 cm ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the angles in a triangle. The three angles of a triangle add up to 180°.(x + 20) + 2x + (3x − 20) = 180
Equation: (x + 20) + 2x + (3x − 20) = 180 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(x + 20) has nothing in front of it except +1, so the brackets can simply be removed: x + 20
(3x − 20) has nothing in front of it except +1, so the brackets can simply be removed: 3x − 20
The equation is now x + 20 + 2x + 3x − 20 = 180 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms x + 2x + 3x make 6x and the numbers 20 − 20 make 0.
The equation is now 6x = 180 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 1806
x = 1806 = 30 - Write the solution. x is now on its own, so the equation tells us its value.x = 30
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 30. The angles are 50°, 60° and 70°.
Check. 50 + 60 + 70 = 180 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use angles on a straight line. The angles on a straight line add up to 180°.(2x + 10) + (x + 35) = 180
Equation: (2x + 10) + (x + 35) = 180 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(2x + 10) has nothing in front of it except +1, so the brackets can simply be removed: 2x + 10
(x + 35) has nothing in front of it except +1, so the brackets can simply be removed: x + 35
The equation is now 2x + 10 + x + 35 = 180 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms 2x + x make 3x and the numbers 10 + 35 make 45.
The equation is now 3x + 45 = 180 - Subtract 45 from both sides. This undoes the +45, because +45 − 45 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 45 − 45 = 180 − 45
3x = 135 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 1353
x = 1353 = 45 - Write the solution. x is now on its own, so the equation tells us its value.x = 45
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 45. The angles are 100° and 80°.
Check. 100 + 80 = 180 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the equal sides. In an isosceles triangle two sides are equal. The ticks show that AB = AC.AB = AC, so 3x + 1 = x + 9
Equation: 3x + 1 = x + 9 - Remove the x term from the right side: subtract x from both sides. We want all the x terms on one side. The right side has x; subtracting x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.3x + 1 − x = x + 9 − x
2x + 1 = 9 - Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 1 − 1 = 9 − 1
2x = 8 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 82
x = 82 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 4. AB = AC = 13 cm and BC = 8 cm, so the perimeter is 13 + 13 + 8 = 34 cm.
Check. AB = 3 × 4 + 1 = 13 and AC = 4 + 9 = 13. They are equal ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Watch out
Finding x is often not the end. If the question asks for an angle, a length or a perimeter, substitute x back in. And check: three angles of a triangle must add to 180°, so if yours do not, you have made a mistake.Exercise 9H Equations from perimeters and angles
Non-calculatorKey words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the perimeter. The perimeter of a rectangle is two lengths plus two widths.Perimeter = 2(2x + 1) + 2(x + 3) = 32
Equation: 2(2x + 1) + 2(x + 3) = 32 - Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 2(2x + 1):
2 × 2x = 4x
2 × 1 = 2
Expand 2(x + 3):
2 × x = 2x
2 × 3 = 6
The equation is now 4x + 2 + 2x + 6 = 32 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms 4x + 2x make 6x and the numbers 2 + 6 make 8.
The equation is now 6x + 8 = 32 - Subtract 8 from both sides. This undoes the +8, because +8 − 8 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.6x + 8 − 8 = 32 − 8
6x = 24 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 246
x = 246 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 4. The sides are 9 cm and 7 cm.
Check. 2 × 9 + 2 × 7 = 32 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the perimeter. A square has four equal sides, so the perimeter is 4 times one side.Perimeter = 4(4x − 3) = 52
Equation: 4(4x − 3) = 52 - Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 4(4x − 3):
4 × 4x = 16x
4 × (−3) = −12
The equation is now 16x − 12 = 52 - Add 12 to both sides. This undoes the −12, because −12 + 12 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.16x − 12 + 12 = 52 + 12
16x = 64 - Divide both sides by 16. 16x means 16 multiplied by x. The inverse of multiplying by 16 is dividing by 16, and we do it to both sides to keep the balance. This leaves x on its own.16x16 = 6416
x = 6416 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 4. The side of the square is 13 cm.
Check. 4 × 13 = 52 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the perimeter. Add up all three sides.Perimeter = (x + 5) + 2x + (2x + 1) = 31
Equation: (x + 5) + 2x + (2x + 1) = 31 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(x + 5) has nothing in front of it except +1, so the brackets can simply be removed: x + 5
(2x + 1) has nothing in front of it except +1, so the brackets can simply be removed: 2x + 1
The equation is now x + 5 + 2x + 2x + 1 = 31 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms x + 2x + 2x make 5x and the numbers 5 + 1 make 6.
The equation is now 5x + 6 = 31 - Subtract 6 from both sides. This undoes the +6, because +6 − 6 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5x + 6 − 6 = 31 − 6
5x = 25 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 255
x = 255 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 5. The sides are 10 cm, 10 cm and 11 cm.
Check. 10 + 10 + 11 = 31 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the angles in a triangle. The angles of a triangle add up to 180°.(x + 10) + 2x + (3x + 20) = 180
Equation: (x + 10) + 2x + (3x + 20) = 180 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(x + 10) has nothing in front of it except +1, so the brackets can simply be removed: x + 10
(3x + 20) has nothing in front of it except +1, so the brackets can simply be removed: 3x + 20
The equation is now x + 10 + 2x + 3x + 20 = 180 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms x + 2x + 3x make 6x and the numbers 10 + 20 make 30.
The equation is now 6x + 30 = 180 - Subtract 30 from both sides. This undoes the +30, because +30 − 30 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.6x + 30 − 30 = 180 − 30
6x = 150 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 1506
x = 1506 = 25 - Write the solution. x is now on its own, so the equation tells us its value.x = 25
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 25. The angles are 35°, 50° and 95°. The largest is 95°.
Check. 35 + 50 + 95 = 180 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use angles on a straight line. Angles on a straight line add up to 180°.(3x + 10) + (2x + 20) = 180
Equation: (3x + 10) + (2x + 20) = 180 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(3x + 10) has nothing in front of it except +1, so the brackets can simply be removed: 3x + 10
(2x + 20) has nothing in front of it except +1, so the brackets can simply be removed: 2x + 20
The equation is now 3x + 10 + 2x + 20 = 180 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms 3x + 2x make 5x and the numbers 10 + 20 make 30.
The equation is now 5x + 30 = 180 - Subtract 30 from both sides. This undoes the +30, because +30 − 30 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5x + 30 − 30 = 180 − 30
5x = 150 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 1505
x = 1505 = 30 - Write the solution. x is now on its own, so the equation tells us its value.x = 30
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 30. The angles are 100° and 80°.
Check. 100 + 80 = 180 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use angles round a point. Angles around a point add up to 360°.2x + (3x + 10) + (x + 50) = 360
Equation: 2x + (3x + 10) + (x + 50) = 360 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(3x + 10) has nothing in front of it except +1, so the brackets can simply be removed: 3x + 10
(x + 50) has nothing in front of it except +1, so the brackets can simply be removed: x + 50
The equation is now 2x + 3x + 10 + x + 50 = 360 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms 2x + 3x + x make 6x and the numbers 10 + 50 make 60.
The equation is now 6x + 60 = 360 - Subtract 60 from both sides. This undoes the +60, because +60 − 60 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.6x + 60 − 60 = 360 − 60
6x = 300 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 3006
x = 3006 = 50 - Write the solution. x is now on its own, so the equation tells us its value.x = 50
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 50. The angles are 100°, 160° and 100°.
Check. 100 + 160 + 100 = 360 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the angles in a quadrilateral. The angles of a quadrilateral add up to 360°.x + (x + 30) + (2x + 10) + (2x + 20) = 360
Equation: x + (x + 30) + (2x + 10) + (2x + 20) = 360 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(x + 30) has nothing in front of it except +1, so the brackets can simply be removed: x + 30
(2x + 10) has nothing in front of it except +1, so the brackets can simply be removed: 2x + 10
(2x + 20) has nothing in front of it except +1, so the brackets can simply be removed: 2x + 20
The equation is now x + x + 30 + 2x + 10 + 2x + 20 = 360 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms x + x + 2x + 2x make 6x and the numbers 30 + 10 + 20 make 60.
The equation is now 6x + 60 = 360 - Subtract 60 from both sides. This undoes the +60, because +60 − 60 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.6x + 60 − 60 = 360 − 60
6x = 300 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 3006
x = 3006 = 50 - Write the solution. x is now on its own, so the equation tells us its value.x = 50
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 50. The angles are 50°, 80°, 110° and 120°.
Check. 50 + 80 + 110 + 120 = 360 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the base angles and the angle sum. In an isosceles triangle the two base angles are equal. The angles of a triangle add up to 180°.(2x + 5) + (2x + 5) + 50 = 180
Equation: (2x + 5) + (2x + 5) + 50 = 180 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(2x + 5) has nothing in front of it except +1, so the brackets can simply be removed: 2x + 5
(2x + 5) has nothing in front of it except +1, so the brackets can simply be removed: 2x + 5
The equation is now 2x + 5 + 2x + 5 + 50 = 180 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms 2x + 2x make 4x and the numbers 5 + 5 + 50 make 60.
The equation is now 4x + 60 = 180 - Subtract 60 from both sides. This undoes the +60, because +60 − 60 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x + 60 − 60 = 180 − 60
4x = 120 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 1204
x = 1204 = 30 - Write the solution. x is now on its own, so the equation tells us its value.x = 30
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 30. Each base angle is 65°.
Check. 65 + 65 + 50 = 180 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the equal sides. The ticks show that AB = AC.2x + 5 = 4x − 3
Equation: 2x + 5 = 4x − 3 - Remove the x term from the left side: subtract 2x from both sides. We want all the x terms on one side. The left side has 2x; subtracting 2x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.2x + 5 − 2x = 4x − 3 − 2x
5 = 2x − 3 - Add 3 to both sides. This undoes the −3, because −3 + 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5 + 3 = 2x − 3 + 3
8 = 2x - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.82 = 2x2
x = 82 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 4. AB = AC = 13 cm, and BC = 10 cm, so the perimeter is 36 cm.
Check. 2 × 4 + 5 = 13 and 4 × 4 − 3 = 13 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the equal sides. All four sides of a rhombus are equal, so the two expressions are equal.3x − 1 = x + 9
Equation: 3x − 1 = x + 9 - Remove the x term from the right side: subtract x from both sides. We want all the x terms on one side. The right side has x; subtracting x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.3x − 1 − x = x + 9 − x
2x − 1 = 9 - Add 1 to both sides. This undoes the −1, because −1 + 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x − 1 + 1 = 9 + 1
2x = 10 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 102
x = 102 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 5. Each side is 14 cm, so the perimeter is 4 × 14 = 56 cm.
Check. 3 × 5 − 1 = 14 and 5 + 9 = 14 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Method. Use the angle sum to find x, then look at the three angles.
- Write an equation. The angles of a triangle add up to 180°.x + 2x + 3x = 180
- Collect like terms. 1x + 2x + 3x = 6x.6x = 180
- Divide both sides by 6. x = 180 ÷ 6 = 30
- Find the three angles. Put x = 30 into each expression.30°, 2 × 30 = 60°, 3 × 30 = 90°
- Decide. A right angle is 90°.One angle is 90°, so Zane is correct.
Check. 30 + 60 + 90 = 180. ✓
Watch out. In a “show that” or “is he correct?” question, finish with a sentence that gives the conclusion.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the angles in a triangle. The angles of a triangle add up to 180°.(2x − 10) + (x + 30) + (3x − 20) = 180
Equation: (2x − 10) + (x + 30) + (3x − 20) = 180 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(2x − 10) has nothing in front of it except +1, so the brackets can simply be removed: 2x − 10
(x + 30) has nothing in front of it except +1, so the brackets can simply be removed: x + 30
(3x − 20) has nothing in front of it except +1, so the brackets can simply be removed: 3x − 20
The equation is now 2x − 10 + x + 30 + 3x − 20 = 180 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms 2x + x + 3x make 6x and the numbers −10 + 30 − 20 make 0.
The equation is now 6x = 180 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 1806
x = 1806 = 30 - Write the solution. x is now on its own, so the equation tells us its value.x = 30
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 30. The angles are 50°, 60° and 70°, so the largest is 70°.
Check. 50 + 60 + 70 = 180 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the angles on a straight line. Angles on a straight line add up to 180°.(x + 40) + (2x + 20) = 180
Equation: (x + 40) + (2x + 20) = 180 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(x + 40) has nothing in front of it except +1, so the brackets can simply be removed: x + 40
(2x + 20) has nothing in front of it except +1, so the brackets can simply be removed: 2x + 20
The equation is now x + 40 + 2x + 20 = 180 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms x + 2x make 3x and the numbers 40 + 20 make 60.
The equation is now 3x + 60 = 180 - Subtract 60 from both sides. This undoes the +60, because +60 − 60 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 60 − 60 = 180 − 60
3x = 120 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 1203
x = 1203 = 40 - Write the solution. x is now on its own, so the equation tells us its value.x = 40
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 40. The angles are 80° and 100°.
Check. 80 + 100 = 180 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the perimeter. A rectangle has two lengths and two widths.2x + 2 × 4 = 20
Equation: 2x + 8 = 20 - Subtract 8 from both sides. This undoes the +8, because +8 − 8 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 8 − 8 = 20 − 8
2x = 12 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 122
x = 122 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 6, so the length is 6 cm.
Check. 2 × 6 + 2 × 4 = 20 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Solving equations using graphs, and interpreting solutions
Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page
Equations can also be solved with a picture. The straight line y = 2x + 1 shows, for every value of x, the value of 2x + 1 (you will study these lines in Chapter 12). Suppose we want to solve 2x + 1 = 9. That means: for which x does the height of the line equal 9?
Draw a horizontal line at y = 9. It meets the sloping line at one point. Go straight down to the x-axis from that point and read the value: x = 4. This is the solution.
A graph gives approximate answers, because we read a scale by eye. Algebra gives exact answers. But graphs are useful when an equation is hard to solve by hand, and they help you see why an equation has the solution it does.
Two sloping lines. To solve an equation with x on both sides, such as 3x − 2 = x + 4, draw a line for each side: y = 3x − 2 and y = x + 4. The solution is the x-coordinate of the point where the two lines cross, because that is the one place where both sides have the same value.
Does the answer make sense? A solution of an equation is a number, but in a story the number must also fit the situation. You cannot buy 8.3 tickets, or be −4 years old, or have a rectangle with a negative length. After solving a problem from words, always ask: “Is this a sensible answer for the question?” If not, either you have made a mistake, or the situation is impossible, and you should say so.
Key ideas
- To solve f(x) = number from a graph: go across from the number on the y-axis to the line, then down to the x-axis.
- For an equation with x on both sides, plot a line for each side. The x-coordinate of the crossing point is the solution.
- Graph answers are approximate; algebra gives exact answers.
- Always check that a solution makes sense in the context (whole number? positive?).
Method. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 2x + 1, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals 9.2x + 1 = 9 means y = 9
- Find 9 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = 9.Start at y = 9 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = 4.
Check. Check by substituting: 2 × 4 + 1 = 9. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Method. Draw (or read) the graph of each side of the equation. The x-coordinate of the point where the lines cross is the solution.
- Think of each side of the equation as a straight line. The left-hand side is the line y = 3x − 2 and the right-hand side is the line y = x + 4. The equation is true where the two lines have the same height.y = 3x − 2 and y = x + 4
- Find where the two lines cross. At the crossing point both lines have the same x and the same y, so that x makes the equation true.The lines cross at (3, 7).
- Read off the x-coordinate. The question asks for x, so we use the first number of the point.x = 3
Check. Left side: 3 × 3 − 2 = 7. Right side: 1 × 3 + 4 = 7. Equal. ✓
Watch out. The answer is the x-coordinate only, not the whole point.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number of tickets Ben bought.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Each ticket costs £6, so x tickets cost 6x pounds. Ben spent £50.
Equation: 6x = 50 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 506
x = 506 = 253 - Write the solution. x is now on its own, so the equation tells us its value.x = 253
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.x = 253 = 8.333… This is not a whole number, and Ben can only buy a whole number of tickets. So Ben cannot have spent exactly £50 on tickets. (He could buy 8 tickets for £48.)
Check. Substitute x = 253 into the original equation. Left side: 6 × 253 = 50. Right side: 50. Both sides equal 50, so the solution is correct. ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Watch out
The solution is the x-value (how far across), not the y-value (how far up). When you draw the line y = 9, the 9 is the number on the y-axis that you start from. The answer you read from the x-axis is 4.Exercise 9I Graphs and interpreting solutions
Non-calculatorMethod. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 2x + 1, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals 7.2x + 1 = 7 means y = 7
- Find 7 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = 7.Start at y = 7 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = 3.
Check. Check by substituting: 2 × 3 + 1 = 7. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Method. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 2x + 1, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals 11.2x + 1 = 11 means y = 11
- Find 11 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = 11.Start at y = 11 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = 5.
Check. Check by substituting: 2 × 5 + 1 = 11. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Method. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 2x + 1, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals 3.2x + 1 = 3 means y = 3
- Find 3 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = 3.Start at y = 3 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = 1.
Check. Check by substituting: 2 × 1 + 1 = 3. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Method. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 2x + 1, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals 5.2x + 1 = 5 means y = 5
- Find 5 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = 5.Start at y = 5 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = 2.
Check. Check by substituting: 2 × 2 + 1 = 5. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Method. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 2x + 1, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals 9.2x + 1 = 9 means y = 9
- Find 9 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = 9.Start at y = 9 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = 4.
Check. Check by substituting: 2 × 4 + 1 = 9. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Method. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 2x + 1, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals 8.2x + 1 = 8 means y = 8
- Find 8 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = 8.Start at y = 8 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = 3.5.
Check. Check by substituting: 2 × 72 + 1 = 8. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Method. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 3x − 2, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals 7.3x − 2 = 7 means y = 7
- Find 7 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = 7.Start at y = 7 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = 3.
Check. Check by substituting: 3 × 3 − 2 = 7. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Method. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 3x − 2, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals 10.3x − 2 = 10 means y = 10
- Find 10 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = 10.Start at y = 10 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = 4.
Check. Check by substituting: 3 × 4 − 2 = 10. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Method. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 3x − 2, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals 1.3x − 2 = 1 means y = 1
- Find 1 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = 1.Start at y = 1 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = 1.
Check. Check by substituting: 3 × 1 − 2 = 1. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Method. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 3x − 2, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals -2.3x − 2 = -2 means y = -2
- Find -2 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = -2.Start at y = -2 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = 0.
Check. Check by substituting: 3 × 0 − 2 = −2. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Method. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 3x − 2, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals 4.3x − 2 = 4 means y = 4
- Find 4 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = 4.Start at y = 4 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = 2.
Check. Check by substituting: 3 × 2 − 2 = 4. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Method. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 3x − 2, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals -5.3x − 2 = -5 means y = -5
- Find -5 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = -5.Start at y = -5 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = −1.
Check. Check by substituting: 3 × (−1) − 2 = −5. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Method. Draw (or read) the graph of each side of the equation. The x-coordinate of the point where the lines cross is the solution.
- Think of each side of the equation as a straight line. The left-hand side is the line y = 3x − 2 and the right-hand side is the line y = x + 4. The equation is true where the two lines have the same height.y = 3x − 2 and y = x + 4
- Find where the two lines cross. At the crossing point both lines have the same x and the same y, so that x makes the equation true.The lines cross at (3, 7).
- Read off the x-coordinate. The question asks for x, so we use the first number of the point.x = 3
Check. Left side: 3 × 3 − 2 = 7. Right side: 1 × 3 + 4 = 7. Equal. ✓
Watch out. The answer is the x-coordinate only, not the whole point.
Method. Draw (or read) the graph of each side of the equation. The x-coordinate of the point where the lines cross is the solution.
- Think of each side of the equation as a straight line. The left-hand side is the line y = 2x + 1 and the right-hand side is the line y = −x + 10. The equation is true where the two lines have the same height.y = 2x + 1 and y = −x + 10
- Find where the two lines cross. At the crossing point both lines have the same x and the same y, so that x makes the equation true.The lines cross at (3, 7).
- Read off the x-coordinate. The question asks for x, so we use the first number of the point.x = 3
Check. Left side: 2 × 3 + 1 = 7. Right side: −1 × 3 + 10 = 7. Equal. ✓
Watch out. The answer is the x-coordinate only, not the whole point.
Method. Read the crossing point from the graph, then confirm with algebra.
- Read the graph. The solution is the x-coordinate of the point where the lines cross.The lines cross at (4, 7), so x = 4.
- Solve by algebra. Subtract x from both sides, then add 1 to both sides.2x − 1 = x + 3
2x − x − 1 = 3
x − 1 = 3
x = 4 - Compare. The graph and the algebra agree.x = 4 both ways.
Check. 2 × 4 − 1 = 7 and 4 + 3 = 7. ✓
Watch out. The graph gives only an approximate reading. Algebra gives the exact value.
Method. Check her answer by substitution and find which axis she read.
- Test Ava's answer. Put x = 9 into 2x + 1.2 × 9 + 1 = 19, but we need 9. So x = 9 is wrong.
- Find the mistake. The number 9 is where we start on the y-axis. The solution is where the lines meet, read on the x-axis.The point where the lines meet is (4, 9). The x-coordinate is 4.
- Give the correct solution. x = 4
Check. 2 × 4 + 1 = 9. ✓
Watch out. Read the value that the question asks for: x comes from the horizontal axis.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number of tickets.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Each ticket costs £6, so x tickets cost 6x pounds, and the total is 50.
Equation: 6x = 50 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 506
x = 506 = 253 - Write the solution. x is now on its own, so the equation tells us its value.x = 253
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The solution is x = 253 = 8.33…, which is not a whole number of tickets. No, Ben cannot be right: he can only buy a whole number of tickets.
Check. 6 × 8 = 48 and 6 × 9 = 54, so £50 is between two possible totals ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = Dee's number.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Multiply by 4 (4x) and add 3 (4x + 3). The result is 20.
Equation: 4x + 3 = 20 - Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x + 3 − 3 = 20 − 3
4x = 17 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 174
x = 174 - Write the solution. x is now on its own, so the equation tells us its value.x = 174
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The solution is x = 174 = 4.25, which is not a whole number. No: Dee is not telling the truth, because her number would have to be a whole number.
Check. 4 × 4 + 3 = 19 and 4 × 5 + 3 = 23, so 20 is not possible ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number of extra gigabytes.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.The monthly charge is 15 and the extra cost is 4x. The total is 51.
Equation: 4x + 15 = 51 - Subtract 15 from both sides. This undoes the +15, because +15 − 15 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x + 15 − 15 = 51 − 15
4x = 36 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 364
x = 364 = 9 - Write the solution. x is now on its own, so the equation tells us its value.x = 9
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.x = 9 gigabytes, which is a positive whole number. This is a sensible answer.
Check. 4 × 9 = 36 and 36 + 15 = 51 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number of cups of flour.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.The equation is given: 3x + 8 = 2.
Equation: 3x + 8 = 2 - Subtract 8 from both sides. This undoes the +8, because +8 − 8 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 8 − 8 = 2 − 8
3x = −6 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = −63
x = −63 = −2 - Write the solution. x is now on its own, so the equation tells us its value.x = −2
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.x = −2. A recipe cannot use a negative number of cups of flour, so there is a mistake in the story or the equation.
Check. 3 × (−2) + 8 = 2 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Method. Use the graph: a horizontal line at the value on the right-hand side meets the sloping line where the equation is true.
- Rewrite the question as a graph question. On the graph of y = 2x + 1, the height of the line is the value of the left-hand side of the equation. We want the point where that height equals 8.2x + 1 = 8 means y = 8
- Find 8 on the y-axis and draw a horizontal line across. Every point on this horizontal line has y = 8.Start at y = 8 and go across until you touch the sloping line.
- Go straight down to the x-axis. The x-value of the point where the two lines cross is the solution.Reading the scale gives x = 3.5.
Check. Check by substituting: 2 × 72 + 1 = 8. ✓
Watch out. Read the x value (across), not the y value (up). Check the scale on both axes before you read the value.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x + 1 = 7
- Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 1 − 1 = 7 − 1
2x = 6 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 62
x = 62 = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
Check. Substitute x = 3 into the original equation. Left side: 2 × 3 + 1 = 6 + 1 = 7. Right side: 7. Both sides equal 7, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Chapter summary
| Skill | Method |
|---|---|
| Equation vs expression | An equation has an = sign and can be solved; an expression cannot |
| One-step equations | Do the opposite operation to both sides: + and − undo each other, × and ÷ undo each other |
| Two-step equations | Undo + or − first, then × or ÷. Always do the same to both sides |
| Negative and fractional answers | Keep the sign when dividing. Leave exact fractions (or write mixed numbers); do not round unless asked |
| Unknown on both sides | Subtract the smaller x term from both sides, then move the plain numbers, then divide |
| Brackets | Expand first (multiply every term inside); a minus before a bracket changes every sign inside |
| Equations with fractions | Multiply every term by the LCM of the denominators; use a bracket when the top has two terms |
| Equations from words | Let x = … (with units); write the equation; solve; answer the question in words; check |
| Geometry equations | Use perimeter, angle sum 180° (triangle, straight line), 360° (point, quadrilateral), or equal sides/angles |
| Graphs | Solution is the x-value where the graph meets the line, or where two lines cross |
| Checking | Substitute into the original equation: both sides must be equal |
| Higher: special cases | x in the denominator: multiply by the bottom (reject a value that makes it 0). A false statement (6 = 5) means no solution; a true one (6 = 6) means every x works |
Can you…?
- I can tell an equation from an expression, and explain the balance idea.
- I can solve one-step and two-step equations by doing the same to both sides.
- I can solve equations with negative or fractional solutions and give the answer in a sensible form.
- I can solve equations with the unknown on both sides.
- I can expand brackets and solve equations that contain them.
- I can solve equations that contain fractions (Higher: algebraic fractions and unknowns in denominators).
- I can turn a worded problem into an equation, solve it, and answer in context.
- I can use perimeter and angle facts to form and solve equations.
- I can use a graph to solve an equation approximately, and decide whether a solution makes sense.
- I can check a solution by substitution.
Key terms
- Equation
- A statement that two expressions are equal, such as 3x + 5 = 20. It contains an equals sign.
- Expression
- A collection of numbers and letters with no equals sign, such as 3x + 5.
- Unknown
- A letter standing for a number we want to find.
- Solve
- Find the value of the unknown that makes the equation true.
- Solution
- The value of the unknown that makes an equation true.
- Inverse operation
- The operation that undoes another: + and − are inverses; × and ÷ are inverses.
- Substitute
- Replace a letter by a number.
- Like terms
- Terms with the same letter part, such as 3x and 5x.
- Expand
- Remove a bracket by multiplying every term inside by the number outside.
- Common denominator
- A bottom number that several fractions can all be changed to.
- Perimeter
- The total distance around the edge of a shape.
- Identity (Higher)
- An equation that is true for every value of the unknown, such as 3(x + 2) = 3x + 6.
Exercise 9Z Check-up (non-calculator)
Non-calculatorKey words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x + 9 = 20
- Subtract 9 from both sides. This undoes the +9, because +9 − 9 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 9 − 9 = 20 − 9
x = 11 - Write the solution. x is now on its own, so the equation tells us its value.x = 11
Check. Substitute x = 11 into the original equation. Left side: 11 + 9 = 20. Right side: 20. Both sides equal 20, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4x = 36
- Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 364
x = 364 = 9 - Write the solution. x is now on its own, so the equation tells us its value.x = 9
Check. Substitute x = 9 into the original equation. Left side: 4 × 9 = 36. Right side: 36. Both sides equal 36, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3x + 7 = 25
- Subtract 7 from both sides. This undoes the +7, because +7 − 7 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 7 − 7 = 25 − 7
3x = 18 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 183
x = 183 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
Check. Substitute x = 6 into the original equation. Left side: 3 × 6 + 7 = 18 + 7 = 25. Right side: 25. Both sides equal 25, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5x − 8 = 12
- Add 8 to both sides. This undoes the −8, because −8 + 8 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5x − 8 + 8 = 12 + 8
5x = 20 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 205
x = 205 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 5 × 4 − 8 = 20 − 8 = 12. Right side: 12. Both sides equal 12, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x + 9 = 3
- Subtract 9 from both sides. This undoes the +9, because +9 − 9 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 9 − 9 = 3 − 9
2x = −6 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = −62
x = −62 = −3 - Write the solution. x is now on its own, so the equation tells us its value.x = −3
Check. Substitute x = −3 into the original equation. Left side: 2 × (−3) + 9 = −6 + 9 = 3. Right side: 3. Both sides equal 3, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.7x − 5 = 3x + 11
- Remove the x term from the right side: subtract 3x from both sides. We want all the x terms on one side. The right side has 3x; subtracting 3x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.7x − 5 − 3x = 3x + 11 − 3x
4x − 5 = 11 - Add 5 to both sides. This undoes the −5, because −5 + 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x − 5 + 5 = 11 + 5
4x = 16 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 164
x = 164 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 7 × 4 − 5 = 28 − 5 = 23. Right side: 3 × 4 + 11 = 12 + 11 = 23. Both sides equal 23, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4(x − 3) = 20
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 4(x − 3):
4 × x = 4x
4 × (−3) = −12
The equation is now 4x − 12 = 20 - Add 12 to both sides. This undoes the −12, because −12 + 12 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x − 12 + 12 = 20 + 12
4x = 32 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 324
x = 324 = 8 - Write the solution. x is now on its own, so the equation tells us its value.x = 8
Check. Substitute x = 8 into the original equation. Left side: 4 × (8 − 3) = 4 × 5 = 20. Right side: 20. Both sides equal 20, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A common denominator is a bottom number that all the fractions can be changed to; multiplying every term by it removes the fractions.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x + 53 = 4
- Find the common denominator. To get rid of fractions we multiply by a number that every bottom divides into exactly: the lowest common multiple of the bottoms.3 → the lowest common multiple is 3
- Multiply every term, on both sides, by 3. Doing the same to both sides keeps the equation balanced. Every term must be multiplied, including plain numbers. This removes the fractions.3 × x + 53 = (x + 5) (3 goes into 3 exactly 1 time, so the bottom disappears)
3 × 4 = 12
The new equation is (x + 5) = 12 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(x + 5) has nothing in front of it except +1, so the brackets can simply be removed: x + 5
The equation is now x + 5 = 12 - Subtract 5 from both sides. This undoes the +5, because +5 − 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x + 5 − 5 = 12 − 5
x = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
Check. Substitute x = 7 into the original equation. Left side: 7 + 53 = 123 = 4. Right side: 4. Both sides equal 4, so the solution is correct. ✓
Watch out. Multiply every term by the common denominator, including the terms that are not fractions. Then check that no fraction is left.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number Tom thinks of.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.He multiplies it by 3 (3x) and subtracts 4 (3x − 4). The answer is 17.
Equation: 3x − 4 = 17 - Add 4 to both sides. This undoes the −4, because −4 + 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x − 4 + 4 = 17 + 4
3x = 21 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 213
x = 213 = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.Tom's number is 7.
Check. 3 × 7 = 21, and 21 − 4 = 17 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use angles on a straight line. Angles on a straight line add up to 180°.(4x + 20) + (2x + 40) = 180
Equation: (4x + 20) + (2x + 40) = 180 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(4x + 20) has nothing in front of it except +1, so the brackets can simply be removed: 4x + 20
(2x + 40) has nothing in front of it except +1, so the brackets can simply be removed: 2x + 40
The equation is now 4x + 20 + 2x + 40 = 180 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms 4x + 2x make 6x and the numbers 20 + 40 make 60.
The equation is now 6x + 60 = 180 - Subtract 60 from both sides. This undoes the +60, because +60 − 60 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.6x + 60 − 60 = 180 − 60
6x = 120 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 1206
x = 1206 = 20 - Write the solution. x is now on its own, so the equation tells us its value.x = 20
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 20. The angles are 100° and 80°.
Check. 100 + 80 = 180 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Total 35 marks. Allow about 35 minutes.
Exercise 9X Exam-style questions
Non-calculatorKey words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5x + 3 = 38
- Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5x + 3 − 3 = 38 − 3
5x = 35 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 355
x = 355 = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
Check. Substitute x = 7 into the original equation. Left side: 5 × 7 + 3 = 35 + 3 = 38. Right side: 38. Both sides equal 38, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x4 + 3 = 8
- Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x4 + 3 − 3 = 8 − 3
x4 = 5 - Multiply both sides by 4. x4 means x divided by 4. The inverse of dividing by 4 is multiplying by 4. We do it to both sides to keep the balance.4 × x4 = 4 × 5
x = 20 - Write the solution. x is now on its own, so the equation tells us its value.x = 20
Check. Substitute x = 20 into the original equation. Left side: 204 + 3 = 5 + 3 = 8. Right side: 8. Both sides equal 8, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.7x − 3 = 3x + 17
- Remove the x term from the right side: subtract 3x from both sides. We want all the x terms on one side. The right side has 3x; subtracting 3x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.7x − 3 − 3x = 3x + 17 − 3x
4x − 3 = 17 - Add 3 to both sides. This undoes the −3, because −3 + 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x − 3 + 3 = 17 + 3
4x = 20 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 204
x = 204 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 7 × 5 − 3 = 35 − 3 = 32. Right side: 3 × 5 + 17 = 15 + 17 = 32. Both sides equal 32, so the solution is correct. ✓
Watch out. Do the same to both sides. Take away the smaller x term from both sides so the x term stays positive. Check the sign of each term as you move it.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3(2x − 1) = 21
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 3(2x − 1):
3 × 2x = 6x
3 × (−1) = −3
The equation is now 6x − 3 = 21 - Add 3 to both sides. This undoes the −3, because −3 + 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.6x − 3 + 3 = 21 + 3
6x = 24 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 246
x = 246 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 3 × (2 × 4 − 1) = 3 × 7 = 21. Right side: 21. Both sides equal 21, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5(x − 2) = 3(x + 4)
- Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 5(x − 2):
5 × x = 5x
5 × (−2) = −10
Expand 3(x + 4):
3 × x = 3x
3 × 4 = 12
The equation is now 5x − 10 = 3x + 12 - Remove the x term from the right side: subtract 3x from both sides. We want all the x terms on one side. The right side has 3x; subtracting 3x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.5x − 10 − 3x = 3x + 12 − 3x
2x − 10 = 12 - Add 10 to both sides. This undoes the −10, because −10 + 10 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x − 10 + 10 = 12 + 10
2x = 22 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 222
x = 222 = 11 - Write the solution. x is now on its own, so the equation tells us its value.x = 11
Check. Substitute x = 11 into the original equation. Left side: 5 × (11 − 2) = 5 × 9 = 45. Right side: 3 × (11 + 4) = 3 × 15 = 45. Both sides equal 45, so the solution is correct. ✓
Watch out. Multiply every term inside the bracket by the number outside, not just the first one. Take care with a minus sign in front of a bracket: it changes the sign of every term inside.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use angles on a straight line. Angles on a straight line add up to 180°.(5x + 10) + (3x + 30) + 2x = 180
Equation: (5x + 10) + (3x + 30) + 2x = 180 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(5x + 10) has nothing in front of it except +1, so the brackets can simply be removed: 5x + 10
(3x + 30) has nothing in front of it except +1, so the brackets can simply be removed: 3x + 30
The equation is now 5x + 10 + 3x + 30 + 2x = 180 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms 5x + 3x + 2x make 10x and the numbers 10 + 30 make 40.
The equation is now 10x + 40 = 180 - Subtract 40 from both sides. This undoes the +40, because +40 − 40 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.10x + 40 − 40 = 180 − 40
10x = 140 - Divide both sides by 10. 10x means 10 multiplied by x. The inverse of multiplying by 10 is dividing by 10, and we do it to both sides to keep the balance. This leaves x on its own.10x10 = 14010
x = 14010 = 14 - Write the solution. x is now on its own, so the equation tells us its value.x = 14
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 14. The angles are 80°, 72° and 28°.
Check. 80 + 72 + 28 = 180 ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side. A bracket such as 3(x + 2) means 3 multiplied by everything inside. Expanding the bracket means multiplying each term inside by the number outside.
Method. Use a fact from geometry to write an equation, solve it by doing the same to both sides, then use the value of x to answer the question.
- Use the perimeter. The perimeter is two lengths plus two widths.Perimeter = 2(3x + 1) + 2(2x − 3) = 46
Equation: 2(3x + 1) + 2(2x − 3) = 46 - Expand the brackets. Multiply each term inside the bracket by the number outside it.Expand 2(3x + 1):
2 × 3x = 6x
2 × 1 = 2
Expand 2(2x − 3):
2 × 2x = 4x
2 × (−3) = −6
The equation is now 6x + 2 + 4x − 6 = 46 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms 6x + 4x make 10x and the numbers 2 − 6 make −4.
The equation is now 10x − 4 = 46 - Add 4 to both sides. This undoes the −4, because −4 + 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.10x − 4 + 4 = 46 + 4
10x = 50 - Divide both sides by 10. 10x means 10 multiplied by x. The inverse of multiplying by 10 is dividing by 10, and we do it to both sides to keep the balance. This leaves x on its own.10x10 = 5010
x = 5010 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
- Answer the question that was asked. The value of x is usually only a stepping stone: read the question again to see what is wanted (a length, an angle, a perimeter) and put the units back.x = 5. The sides are 16 cm and 7 cm, so the area is 16 × 7 = 112 cm2.
Check. 2 × 16 + 2 × 7 = 46 cm ✓
Watch out. Do not stop at the value of x. Put it back into the expressions on the diagram to find the angle or length that the question asks for.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the cost of one T-shirt in pounds.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.4 T-shirts cost 4x pounds, and delivery of £3 is added once. The total is £35.
Equation: 4x + 3 = 35 - Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x + 3 − 3 = 35 − 3
4x = 32 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 324
x = 324 = 8 - Write the solution. x is now on its own, so the equation tells us its value.x = 8
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.One T-shirt costs £8.
Check. 4 × £8 = £32, and £32 + £3 = £35 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = Ben's age in years.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Amy is 3 years older than Ben: x + 3. Cal is twice as old as Ben: 2x. The three ages add up to 39.
Equation: x + (x + 3) + 2x = 39 - Expand the brackets. Multiply each term inside the bracket by the number outside it.(x + 3) has nothing in front of it except +1, so the brackets can simply be removed: x + 3
The equation is now x + x + 3 + 2x = 39 - Collect like terms on each side. Like terms are terms with the same letter part: all the x terms go together, and all the plain numbers go together. Doing this makes each side as short as possible.On the left side: the x terms x + x + 2x make 4x.
The equation is now 4x + 3 = 39 - Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x + 3 − 3 = 39 − 3
4x = 36 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 364
x = 364 = 9 - Write the solution. x is now on its own, so the equation tells us its value.x = 9
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.Ben is 9, Amy is 12 and Cal is 18.
Check. 9 + 12 + 18 = 39 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Substitute the value into both sides of the equation, work each side out, and compare.
- Write the equation and the value to test. A number is a solution of an equation if it makes both sides equal. We test it by substituting it (putting it in place of the letter).5x − 7 = 3x + 1, x = 4
- Substitute into the left-hand side. Replace every x with the number. Use brackets around a negative number so that the signs stay clear.Left side = 5 × 4 − 7 = 20 − 7 = 13
- Substitute into the right-hand side. Do exactly the same to the other side of the equation.Right side = 3 × 4 + 1 = 12 + 1 = 13
- Compare the two sides. If the two sides are equal the number is a solution; if not, it is not.13 = 13, so yes, x = 4 is a solution.
Check. The two sides match, which is exactly what we wanted to see. ✓
Watch out. Work out the left side and the right side separately. Do not use an = sign between the two until you have shown they are equal.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number of minutes.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Deal A costs 1000 + 4x pence. Deal B costs 400 + 10x pence. We want the costs to be equal.
Equation: 4x + 1000 = 10x + 400 - Remove the x term from the left side: subtract 4x from both sides. We want all the x terms on one side. The left side has 4x; subtracting 4x removes it. We take away the smaller x term, so that the x term that is left is positive and easier to work with. We must do it to both sides to keep the balance.4x + 1000 − 4x = 10x + 400 − 4x
1000 = 6x + 400 - Subtract 400 from both sides. This undoes the +400, because +400 − 400 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.1000 − 400 = 6x + 400 − 400
600 = 6x - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6006 = 6x6
x = 6006 = 100 - Write the solution. x is now on its own, so the equation tells us its value.x = 100
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.The deals cost the same for 100 minutes.
Check. Deal A: 1000 + 4 × 100 = 1400p. Deal B: 400 + 10 × 100 = 1400p. The same ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Substitute the value into both sides of the equation, work each side out, and compare.
- Write the equation and the value to test. A number is a solution of an equation if it makes both sides equal. We test it by substituting it (putting it in place of the letter).2x + 5 = 4x − 1, x = 3
- Substitute into the left-hand side. Replace every x with the number. Use brackets around a negative number so that the signs stay clear.Left side = 2 × 3 + 5 = 6 + 5 = 11
- Substitute into the right-hand side. Do exactly the same to the other side of the equation.Right side = 4 × 3 − 1 = 12 − 1 = 11
- Compare the two sides. If the two sides are equal the number is a solution; if not, it is not.11 = 11, so yes, x = 3 is a solution.
Check. The two sides match, which is exactly what we wanted to see. ✓
Watch out. Work out the left side and the right side separately. Do not use an = sign between the two until you have shown they are equal.