GCSE Revision

Solving linear equations

One-step and two-step equations

To an equation is to find the value of the letter that makes it true. Use (the opposite operation), doing the same to both sides.

  1. Undo any addition or subtraction.
  2. Undo any multiplication or division.

Example: solve 3x + 7 = 22.

  1. Subtract 7: 3x = 15
  2. Divide by 3: x = 5

Example: solve x/4 − 2 = 3.

  1. Add 2: x/4 = 5
  2. Multiply by 4: x = 20

Exam tip

Check by substituting your answer back: 3 × 5 + 7 = 22, so x = 5 is right.

Key terms

Equations with brackets

  1. Expand the brackets.
  2. Solve as a two-step equation.

Answers don't have to be whole numbers: they can be negative, decimals or fractions.

Example: solve 4(2x − 3) = 28.

  1. Expand: 8x − 12 = 28
  2. Add 12: 8x = 40
  3. Divide by 8: x = 5

Example: solve 2(x + 5) = 3.

  1. Expand: 2x + 10 = 3
  2. Subtract 10: 2x = −7
  3. Divide by 2: x = −3.5

Unknowns on both sides

  1. Expand any brackets.
  2. Move the letter terms to the side with more of them, by adding or subtracting.
  3. Move the number terms to the other side.
  4. Divide.

Example: solve 5x − 4 = 2x + 11.

  1. Subtract 2x: 3x − 4 = 11
  2. Add 4: 3x = 15
  3. x = 5

Harder example: solve 3(x − 2) = 5x + 8.

  1. Expand: 3x − 6 = 5x + 8
  2. Subtract 3x: −6 = 2x + 8
  3. Subtract 8: −14 = 2x
  4. x = −7

Check: 3(−7 − 2) = −27, and 5 × −7 + 8 = −27.

Common mistake

Do the same to every term on both sides. Subtracting 2x from 5x − 4 gives 3x − 4, not 3x + 4.

Equations with fractions

Get rid of the fractions first by multiplying by the denominator (or by the LCM of the denominators).

Example: solve (2x + 1)/3 = 5.

  1. Multiply both sides by 3: 2x + 1 = 15
  2. Subtract 1: 2x = 14
  3. x = 7

Example: solve x/2 + x/3 = 10.

  1. The LCM of 2 and 3 is 6. Multiply every term by 6: 3x + 2x = 60
  2. 5x = 60
  3. x = 12

Harder example: solve (x + 3)/4 = (x − 1)/2.

  1. Multiply both sides by 4: x + 3 = 2(x − 1)
  2. Expand: x + 3 = 2x − 2
  3. Subtract x and add 2: x = 5

Forming and solving equations

  1. Choose a letter for the , and write each fact as algebra.
  2. Form an equation and solve it.
  3. Answer the question asked.

Example: I think of a number, multiply it by 4 and subtract 7. I get the same answer as when I add 11 to the number. What is it?

  1. Let the number be n.
  2. 4n − 7 = n + 11
  3. Subtract n: 3n − 7 = 11, so 3n = 18
  4. n = 6

Example: two angles on a straight line are 3x + 10 and 2x − 5 degrees. Find both angles.

  1. Angles on a line add to 180°: 3x + 10 + 2x − 5 = 180
  2. 5x + 5 = 180, so x = 35
  3. 3 × 35 + 10 = 115 and 2 × 35 − 5 = 65: 115° and 65°

Key terms

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