GCSE Maths: Negative and fractional solutions, and checking (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)
Negative and fractional solutions, and checking
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. 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).4x + 5 = 6x + 6, x = −12
- 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 × (−12) + 5 = −2 + 5 = 3
- Substitute into the right-hand side. Do exactly the same to the other side of the equation.Right side = 6 × (−12) + 6 = −3 + 6 = 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 = −12 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.7x + 3 = −1
- 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.7x + 3 − 3 = −1 − 3
7x = −4 - 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 = −47
x = −47 = −47 - Write the solution. x is now on its own, so the equation tells us its value.x = −47
Check. Substitute x = −47 into the original equation. Left side: 7 × (−47) + 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.4 − 3x = 10
- 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 + 4 = 10 - 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 = 10 − 4
−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. 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 = 6−3
x = 6−3 = −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: 4 − 3 × (−2) = 4 + 6 = 10. Right side: 10. Both sides equal 10, 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.9x − 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.9x − 2 + 2 = −7 + 2
9x = −5 - Divide both sides by 9. 9x means 9 multiplied by x. The inverse of multiplying by 9 is dividing by 9, and we do it to both sides to keep the balance. This leaves x on its own.9x9 = −59
x = −59 = −59 - Write the solution. x is now on its own, so the equation tells us its value.x = −59
Check. Substitute x = −59 into the original equation. Left side: 9 × (−59) − 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.6 − 5x = −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: write the x term first (adding can be done in any order).
The equation is now −5x + 6 = −3 - 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 = −3 − 6
−5x = −9 - 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 = −9−5
x = −9−5 = 95 - Write the solution. x is now on its own, so the equation tells us its value.x = 95
Check. Substitute x = 95 into the original equation. Left side: 6 − 5 × 95 = 6 − 9 = −3. Right side: −3. Both sides equal −3, 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.