GCSE Maths: Two-step equations (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)
Two-step equations
Many equations need two steps. Look at 3x + 5 = 20. Two things have been done to x: it was first multiplied by 3, and then 5 was added. A function machine shows this.
To get x back we undo the steps in reverse order, just as you take your shoes off before your socks if you put your socks on first. We undo the adding first (subtract 5), then the multiplying (divide by 3):
3x + 5 = 20 → subtract 5 from both sides: 3x = 15 → divide both sides by 3: x = 5.
Why this order? The “+ 5” is stuck to the whole of 3x. If you divide by 3 first, you have to divide everything (the 5 and the 20 too), which gives fractions and is much harder. Take away the number added or subtracted first, then deal with the multiplication or division.
The same method works when the equation is written the other way round, such as 17 = 2x + 5. The unknown is on the right-hand side, but you do exactly the same steps. When you reach 6 = x you can write it as x = 6, because an equals sign works both ways.
When x is divided by a number, as in x4 − 2 = 3, we still add 2 first, and then multiply by 4 to undo the division.
Key ideas
- Undo the adding or subtracting first, then undo the multiplying or dividing.
- Do the same to both sides each time.
- Write every step on a new line, one operation per line, and keep the equals signs lined up.
- Always check by substituting your answer back into the original equation.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3x + 5 = 20
- Subtract 5 from both sides. This undoes the +5, because +5 − 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 5 − 5 = 20 − 5
3x = 15 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 153
x = 153 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 3 × 5 + 5 = 15 + 5 = 20. Right side: 20. Both sides equal 20, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4x − 7 = 21
- Add 7 to both sides. This undoes the −7, because −7 + 7 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x − 7 + 7 = 21 + 7
4x = 28 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 284
x = 284 = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
Check. Substitute x = 7 into the original equation. Left side: 4 × 7 − 7 = 28 − 7 = 21. Right side: 21. Both sides equal 21, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x5 + 2 = 6
- Subtract 2 from both sides. This undoes the +2, because +2 − 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x5 + 2 − 2 = 6 − 2
x5 = 4 - Multiply both sides by 5. x5 means x divided by 5. The inverse of dividing by 5 is multiplying by 5. We do it to both sides to keep the balance.5 × x5 = 5 × 4
x = 20 - Write the solution. x is now on its own, so the equation tells us its value.x = 20
Check. Substitute x = 20 into the original equation. Left side: 205 + 2 = 4 + 2 = 6. Right side: 6. Both sides equal 6, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.17 = 2x + 5
- Subtract 5 from both sides. This undoes the +5, because +5 − 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.17 − 5 = 2x + 5 − 5
12 = 2x - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.122 = 2x2
x = 122 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
Check. Substitute x = 6 into the original equation. Left side: 17. Right side: 2 × 6 + 5 = 12 + 5 = 17. Both sides equal 17, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Watch out
Do not divide only one term. In 3x + 5 = 20, dividing just the 3x by 3 gives x + 5 = 20, which is wrong. Subtract 5 first: 3x = 15, then divide both sides by 3.Exercise 9B Two-step equations
Non-calculatorKey words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x + 3 = 11
- Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 3 − 3 = 11 − 3
2x = 8 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 82
x = 82 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 2 × 4 + 3 = 8 + 3 = 11. Right side: 11. Both sides equal 11, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3x + 1 = 16
- Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 1 − 1 = 16 − 1
3x = 15 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 153
x = 153 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 3 × 5 + 1 = 15 + 1 = 16. Right side: 16. Both sides equal 16, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5x + 4 = 24
- Subtract 4 from both sides. This undoes the +4, because +4 − 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5x + 4 − 4 = 24 − 4
5x = 20 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 205
x = 205 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 5 × 4 + 4 = 20 + 4 = 24. Right side: 24. Both sides equal 24, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4x + 7 = 31
- Subtract 7 from both sides. This undoes the +7, because +7 − 7 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x + 7 − 7 = 31 − 7
4x = 24 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 244
x = 244 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
Check. Substitute x = 6 into the original equation. Left side: 4 × 6 + 7 = 24 + 7 = 31. Right side: 31. Both sides equal 31, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x + 9 = 21
- Subtract 9 from both sides. This undoes the +9, because +9 − 9 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x + 9 − 9 = 21 − 9
2x = 12 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 122
x = 122 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
Check. Substitute x = 6 into the original equation. Left side: 2 × 6 + 9 = 12 + 9 = 21. Right side: 21. Both sides equal 21, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.6x + 2 = 20
- Subtract 2 from both sides. This undoes the +2, because +2 − 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.6x + 2 − 2 = 20 − 2
6x = 18 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 186
x = 186 = 3 - Write the solution. x is now on its own, so the equation tells us its value.x = 3
Check. Substitute x = 3 into the original equation. Left side: 6 × 3 + 2 = 18 + 2 = 20. Right side: 20. Both sides equal 20, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.2x − 3 = 9
- Add 3 to both sides. This undoes the −3, because −3 + 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.2x − 3 + 3 = 9 + 3
2x = 12 - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.2x2 = 122
x = 122 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
Check. Substitute x = 6 into the original equation. Left side: 2 × 6 − 3 = 12 − 3 = 9. Right side: 9. Both sides equal 9, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.3x − 5 = 13
- Add 5 to both sides. This undoes the −5, because −5 + 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x − 5 + 5 = 13 + 5
3x = 18 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 183
x = 183 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
Check. Substitute x = 6 into the original equation. Left side: 3 × 6 − 5 = 18 − 5 = 13. Right side: 13. Both sides equal 13, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.5x − 1 = 24
- Add 1 to both sides. This undoes the −1, because −1 + 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5x − 1 + 1 = 24 + 1
5x = 25 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 255
x = 255 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 5 × 5 − 1 = 25 − 1 = 24. Right side: 24. Both sides equal 24, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.4x − 9 = 15
- Add 9 to both sides. This undoes the −9, because −9 + 9 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x − 9 + 9 = 15 + 9
4x = 24 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 244
x = 244 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
Check. Substitute x = 6 into the original equation. Left side: 4 × 6 − 9 = 24 − 9 = 15. Right side: 15. Both sides equal 15, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.7x − 2 = 40
- Add 2 to both sides. This undoes the −2, because −2 + 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.7x − 2 + 2 = 40 + 2
7x = 42 - Divide both sides by 7. 7x means 7 multiplied by x. The inverse of multiplying by 7 is dividing by 7, and we do it to both sides to keep the balance. This leaves x on its own.7x7 = 427
x = 427 = 6 - Write the solution. x is now on its own, so the equation tells us its value.x = 6
Check. Substitute x = 6 into the original equation. Left side: 7 × 6 − 2 = 42 − 2 = 40. Right side: 40. Both sides equal 40, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.10x − 6 = 34
- Add 6 to both sides. This undoes the −6, because −6 + 6 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.10x − 6 + 6 = 34 + 6
10x = 40 - Divide both sides by 10. 10x means 10 multiplied by x. The inverse of multiplying by 10 is dividing by 10, and we do it to both sides to keep the balance. This leaves x on its own.10x10 = 4010
x = 4010 = 4 - Write the solution. x is now on its own, so the equation tells us its value.x = 4
Check. Substitute x = 4 into the original equation. Left side: 10 × 4 − 6 = 40 − 6 = 34. Right side: 34. Both sides equal 34, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x2 + 3 = 8
- Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x2 + 3 − 3 = 8 − 3
x2 = 5 - Multiply both sides by 2. x2 means x divided by 2. The inverse of dividing by 2 is multiplying by 2. We do it to both sides to keep the balance.2 × x2 = 2 × 5
x = 10 - Write the solution. x is now on its own, so the equation tells us its value.x = 10
Check. Substitute x = 10 into the original equation. Left side: 102 + 3 = 5 + 3 = 8. Right side: 8. Both sides equal 8, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x3 + 1 = 5
- Subtract 1 from both sides. This undoes the +1, because +1 − 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x3 + 1 − 1 = 5 − 1
x3 = 4 - Multiply both sides by 3. x3 means x divided by 3. The inverse of dividing by 3 is multiplying by 3. We do it to both sides to keep the balance.3 × x3 = 3 × 4
x = 12 - Write the solution. x is now on its own, so the equation tells us its value.x = 12
Check. Substitute x = 12 into the original equation. Left side: 123 + 1 = 4 + 1 = 5. Right side: 5. Both sides equal 5, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x4 − 2 = 3
- Add 2 to both sides. This undoes the −2, because −2 + 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x4 − 2 + 2 = 3 + 2
x4 = 5 - Multiply both sides by 4. x4 means x divided by 4. The inverse of dividing by 4 is multiplying by 4. We do it to both sides to keep the balance.4 × x4 = 4 × 5
x = 20 - Write the solution. x is now on its own, so the equation tells us its value.x = 20
Check. Substitute x = 20 into the original equation. Left side: 204 − 2 = 5 − 2 = 3. Right side: 3. Both sides equal 3, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x5 + 4 = 9
- Subtract 4 from both sides. This undoes the +4, because +4 − 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x5 + 4 − 4 = 9 − 4
x5 = 5 - Multiply both sides by 5. x5 means x divided by 5. The inverse of dividing by 5 is multiplying by 5. We do it to both sides to keep the balance.5 × x5 = 5 × 5
x = 25 - Write the solution. x is now on its own, so the equation tells us its value.x = 25
Check. Substitute x = 25 into the original equation. Left side: 255 + 4 = 5 + 4 = 9. Right side: 9. Both sides equal 9, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x6 − 1 = 2
- Add 1 to both sides. This undoes the −1, because −1 + 1 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x6 − 1 + 1 = 2 + 1
x6 = 3 - Multiply both sides by 6. x6 means x divided by 6. The inverse of dividing by 6 is multiplying by 6. We do it to both sides to keep the balance.6 × x6 = 6 × 3
x = 18 - Write the solution. x is now on its own, so the equation tells us its value.x = 18
Check. Substitute x = 18 into the original equation. Left side: 186 − 1 = 3 − 1 = 2. Right side: 2. Both sides equal 2, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x10 + 7 = 10
- Subtract 7 from both sides. This undoes the +7, because +7 − 7 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x10 + 7 − 7 = 10 − 7
x10 = 3 - Multiply both sides by 10. x10 means x divided by 10. The inverse of dividing by 10 is multiplying by 10. We do it to both sides to keep the balance.10 × x10 = 10 × 3
x = 30 - Write the solution. x is now on its own, so the equation tells us its value.x = 30
Check. Substitute x = 30 into the original equation. Left side: 3010 + 7 = 3 + 7 = 10. Right side: 10. Both sides equal 10, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.13 = 2x + 3
- Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.13 − 3 = 2x + 3 − 3
10 = 2x - Divide both sides by 2. 2x means 2 multiplied by x. The inverse of multiplying by 2 is dividing by 2, and we do it to both sides to keep the balance. This leaves x on its own.102 = 2x2
x = 102 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 13. Right side: 2 × 5 + 3 = 10 + 3 = 13. Both sides equal 13, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.22 = 4x + 2
- Subtract 2 from both sides. This undoes the +2, because +2 − 2 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.22 − 2 = 4x + 2 − 2
20 = 4x - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.204 = 4x4
x = 204 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 22. Right side: 4 × 5 + 2 = 20 + 2 = 22. Both sides equal 22, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.17 = 3x − 4
- Add 4 to both sides. This undoes the −4, because −4 + 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.17 + 4 = 3x − 4 + 4
21 = 3x - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.213 = 3x3
x = 213 = 7 - Write the solution. x is now on its own, so the equation tells us its value.x = 7
Check. Substitute x = 7 into the original equation. Left side: 17. Right side: 3 × 7 − 4 = 21 − 4 = 17. Both sides equal 17, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.30 = 5x + 5
- Subtract 5 from both sides. This undoes the +5, because +5 − 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.30 − 5 = 5x + 5 − 5
25 = 5x - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.255 = 5x5
x = 255 = 5 - Write the solution. x is now on its own, so the equation tells us its value.x = 5
Check. Substitute x = 5 into the original equation. Left side: 30. Right side: 5 × 5 + 5 = 25 + 5 = 30. Both sides equal 30, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Method. Subtract the number that has been added, from both sides.
- Subtract 4 from both sides. This undoes the + 4.6x + 4 − 4 = 40 − 4
- Work out the right side. 40 − 4 = 36
- Write the new equation. 6x = 36
Check. 36 + 4 = 40. ✓
Method. Divide both sides by the number multiplying x.
- Divide both sides by 6. This undoes the multiplication by 6.6x ÷ 6 = 36 ÷ 6
- Work out the right side. 36 ÷ 6 = 6
- Write the answer. x = 6
Check. 6 × 6 + 4 = 40. ✓
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.8x + 5 = 93
- Subtract 5 from both sides. This undoes the +5, because +5 − 5 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.8x + 5 − 5 = 93 − 5
8x = 88 - Divide both sides by 8. 8x means 8 multiplied by x. The inverse of multiplying by 8 is dividing by 8, and we do it to both sides to keep the balance. This leaves x on its own.8x8 = 888
x = 888 = 11 - Write the solution. x is now on its own, so the equation tells us its value.x = 11
Check. Substitute x = 11 into the original equation. Left side: 8 × 11 + 5 = 88 + 5 = 93. Right side: 93. Both sides equal 93, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Solve the equation by using inverse operations (the opposite operation) on both sides, so that the equation stays balanced, until the letter x is on its own.
- Write down the equation. The two sides of an equation are equal, like the two sides of a balanced pair of scales. We want the value of x that keeps them balanced.x7 − 3 = 4
- Add 3 to both sides. This undoes the −3, because −3 + 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.x7 − 3 + 3 = 4 + 3
x7 = 7 - Multiply both sides by 7. x7 means x divided by 7. The inverse of dividing by 7 is multiplying by 7. We do it to both sides to keep the balance.7 × x7 = 7 × 7
x = 49 - Write the solution. x is now on its own, so the equation tells us its value.x = 49
Check. Substitute x = 49 into the original equation. Left side: 497 − 3 = 7 − 3 = 4. Right side: 4. Both sides equal 4, so the solution is correct. ✓
Watch out. Whatever you do to one side of the equation you must do to the other side. Use the opposite operation: the opposite of + is −, and the opposite of × is ÷.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number Mia thought of.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Mia multiplied her number by 4 (4x) and then added 3 (4x + 3). She got 35.
Equation: 4x + 3 = 35 - Subtract 3 from both sides. This undoes the +3, because +3 − 3 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.4x + 3 − 3 = 35 − 3
4x = 32 - Divide both sides by 4. 4x means 4 multiplied by x. The inverse of multiplying by 4 is dividing by 4, and we do it to both sides to keep the balance. This leaves x on its own.4x4 = 324
x = 324 = 8 - Write the solution. x is now on its own, so the equation tells us its value.x = 8
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.Mia's number was 8.
Check. 4 × 8 = 32, and 32 + 3 = 35 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the cost of one ticket in pounds.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.3 tickets cost 3x pounds. A booking fee of £4 is added once, so the total is 3x + 4 = 31.
Equation: 3x + 4 = 31 - Subtract 4 from both sides. This undoes the +4, because +4 − 4 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.3x + 4 − 4 = 31 − 4
3x = 27 - Divide both sides by 3. 3x means 3 multiplied by x. The inverse of multiplying by 3 is dividing by 3, and we do it to both sides to keep the balance. This leaves x on its own.3x3 = 273
x = 273 = 9 - Write the solution. x is now on its own, so the equation tells us its value.x = 9
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.One ticket costs £9.
Check. 3 × £9 = £27, and £27 + £4 = £31 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number of weeks.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Jade has £12 and saves £5 each week. After x weeks she has 12 + 5x pounds. She wants £62.
Equation: 5x + 12 = 62 - Subtract 12 from both sides. This undoes the +12, because +12 − 12 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.5x + 12 − 12 = 62 − 12
5x = 50 - Divide both sides by 5. 5x means 5 multiplied by x. The inverse of multiplying by 5 is dividing by 5, and we do it to both sides to keep the balance. This leaves x on its own.5x5 = 505
x = 505 = 10 - Write the solution. x is now on its own, so the equation tells us its value.x = 10
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.Jade needs 10 weeks.
Check. 12 + 5 × 10 = 12 + 50 = 62 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.
Method. Solve both equations, then compare the two answers.
- Solve Ali's equation. Subtract 3, then divide by 4.4x = 19 − 3 = 16, so x = 16 ÷ 4 = 4
- Solve Beth's equation. Add 1, then divide by 3.3x = 11 + 1 = 12, so x = 12 ÷ 3 = 4
- Compare. Both solutions are 4.Neither is bigger: they get the same answer, so the difference is 0.
Check. 4 × 4 + 3 = 19 and 3 × 4 − 1 = 11. ✓
Watch out. Read the question: it asks who is bigger, and the answer can be “neither”.
Method. Try Zoe's method on every term, then compare it with subtracting first.
- Divide every term by 3 (Zoe's method). We must divide the whole of each side, including the 5 and the 23.3x3 + 53 = 233
x + 53 = 233 - Notice the fractions. Neither 5 nor 23 divides exactly by 3, so we have to work with thirds.x = 233 − 53 = 183 = 6
- Subtract 5 first. The numbers stay whole.3x = 23 − 5 = 18, so x = 18 ÷ 3 = 6
Check. Both methods give 6, and 3 × 6 + 5 = 23. ✓
Watch out. Zoe's method is correct if every term is divided, but the other order avoids fractions.
Key words. An equation says that two things are equal, for example 3x + 5 = 20. To solve it means to find the value of the letter that makes it true. The inverse operation is the operation that undoes another: adding and subtracting undo each other, and multiplying and dividing undo each other. An equation is like a pair of balanced scales: whatever you do to one side you must do to the other side.
Method. Turn the words into an equation, solve it using inverse operations, then use the answer to respond to the question that was asked.
- Choose a letter for the unknown. We do not know the number yet, so we give it a letter. Say clearly what the letter stands for, including the units.Let x = the number of pages Omar reads each day.
- Use the words to write an equation. Translate the sentence into maths, a bit at a time. The two sides of the equation must be equal in value.Omar reads x pages a day for 6 days, so 6x pages, and has read 18 pages before. The book has 90 pages: 6x + 18 = 90.
Equation: 6x + 18 = 90 - Subtract 18 from both sides. This undoes the +18, because +18 − 18 = 0, which leaves the x term on its own. We do it to both sides to keep the balance.6x + 18 − 18 = 90 − 18
6x = 72 - Divide both sides by 6. 6x means 6 multiplied by x. The inverse of multiplying by 6 is dividing by 6, and we do it to both sides to keep the balance. This leaves x on its own.6x6 = 726
x = 726 = 12 - Write the solution. x is now on its own, so the equation tells us its value.x = 12
- Answer the question that was asked. The solution of the equation is the value of the letter. Read the question again to see exactly what it wants, and put the units back.Omar reads 12 pages each day.
Check. 6 × 12 = 72, and 72 + 18 = 90 ✓
Watch out. Always give the answer to the question that was asked, with units. The value of x is not always the final answer.