GCSE Maths: Equations and the balance idea (Higher)
Part of the GCSE Maths linear equations chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Equations and the balance idea
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.