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Changing the subject of a formula

2026-10-16Equations & Inequalities Mr Yip

Rearrange formulae with roots, fractions and products so a chosen variable becomes the subject.

Key formulas

\text{Inverse steps undo the formula}

If y = mx + c then x = (y - c)/m

If A = \pi r^{2} then r = \sqrt{A/\pi }

Think in inverse operations

A formula builds the subject through a sequence of operations. To change the subject, undo those operations in reverse order — just as with linear equations.

Fractions and roots

Clear a denominator by multiplying both sides. Undo a square with a square root (consider the sign required by context — lengths are positive). Undo multiplication with division.

Subject appearing twice

If the new subject appears in more than one term, collect those terms first and factorise before dividing.

Exam habit

State the target subject. After rearranging, substitute a simple number into both forms to verify.

Worked example

Make r the subject

Problem. Make r the subject of A = 4πr².

Solution. r² = A/(4π). Since r is a length, r = √(A/(4π)).

Review quiz

1 / 4

y = 3x − 7. Make x the subject.

Practice questions

Make x the subject of y = 3x − 7.
Make r the subject of C = 2πr.
Make b the subject of A = (1/2)bh.

Study notes

When do I take ± square roots?

Algebraically both signs exist; in geometry contexts keep the positive root for lengths.

Can I swap sides freely?

Yes — equality is symmetric. Keep the new subject on the left by convention.