Linear equations and rearranging formulae
Solving equations and changing the subject of a formula.
Learning objectives
What you need to be able to do
Teacher-mapped phrasing — check against the official Cambridge syllabus for exact wording.
- 2.2.1Solve linear equations, including those with brackets and fractions.
- 2.2.2Rearrange formulae to change the subject.
5 minute read
Linear equations and rearranging formulae
Solving linear equations
The governing rule: whatever you do to one side, do to the other. Work backwards through the operations, undoing them in reverse order. 3x + 7 = 22 → subtract 7 → 3x = 15 → divide by 3 → x = 5
- With brackets: expand first, or divide through if everything is divisible.
- With fractions: multiply every term by the denominator to clear it.
- Unknown on both sides: collect the x terms on the side where the coefficient is larger, which avoids negatives.
Always check by substituting back into the original equation.
Rearranging formulae
Changing the subject uses exactly the same rules. To make x the subject:
- Remove fractions by multiplying through.
- Expand any brackets.
- Collect all terms containing x on one side, everything else on the other.
- Factorise out x if it appears more than once.
- Divide by whatever multiplies x.
Step 4 is the one that distinguishes harder questions. For ax + b = cx + d: ax − cx = d − b → x(a − c) = d − b → x = (d − b)/(a − c) Without factorising, x cannot be isolated at all.
Think of it like this
An equation is a balanced scale. Any operation is legal provided it is applied to both pans; rearranging a formula is the same process performed with letters instead of numbers.
Worked examples
Method, step by step
Make r the subject of the formula A = πr².
- 1Divide both sides by π: A/π = r².
- 2Take the square root of both sides: r = √(A/π).
- 3Since r is a radius it must be positive, so the negative root is discarded.
r = √(A/π)
Make x the subject of y = (3x + 2)/(x − 1).
- 1Multiply both sides by (x − 1): y(x − 1) = 3x + 2.
- 2Expand: xy − y = 3x + 2.
- 3Collect x terms on one side: xy − 3x = y + 2.
- 4Factorise: x(y − 3) = y + 2, so x = (y + 2)/(y − 3).
x = (y + 2)/(y − 3)
Common misconceptions
- Applying an operation to only part of a side. Multiplying by 3 must multiply **every** term, not just the first.
- Failing to factorise when the new subject appears twice, leaving x on both sides.
- Taking a square root and giving only the positive answer. `x² = 16` has solutions `x = ±4`.
In the exam
- When the target letter appears more than once, plan to factorise from the start — it is the whole method.
- Substitute your answer back into the original. It takes ten seconds and catches sign errors reliably.