Solving Linear Equations

GCSE Maths · Algebra

The golden rule

An equation is a balanced scale. Whatever you do to one side, do to the other. Your goal is to isolate the letter by doing inverse operations in reverse order.

Operation	Inverse
+	−
×	÷
square (x²)	square root (√)

Example 1 — two steps

Solve 3x + 4 = 19

3x + 4 = 19      (subtract 4 from both sides)
3x     = 15      (divide both sides by 3)
 x     = 5

Example 2 — unknown on both sides

Solve 5x − 2 = 2x + 7

5x − 2 = 2x + 7    (subtract 2x from both sides)
3x − 2 = 7         (add 2 to both sides)
3x     = 9         (divide by 3)
 x     = 3

Example 3 — with brackets

Solve 4(x − 3) = 20

4(x − 3) = 20      (expand, or divide by 4 first)
 x − 3   = 5       (add 3)
 x       = 8

Example 4 — fractions

Solve (x + 1)/2 = 5 → multiply both sides by 2 → x + 1 = 10 → x = 9.

Common mistakes

Only doing the operation to one side.
Sign errors — move the smaller letter term to avoid negatives.
Forgetting to expand brackets correctly.

Exam tip

Always show every line of working — method marks are awarded even if the final answer is wrong. Check by substituting your answer back in.