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Rearranging Formulae: How to Change the Subject

Make any letter the subject of a formula using inverse operations, with worked examples including squares and square roots.

  • ⏱ 3 min read
  • ✏️ 3 worked examples
  • 🎯 5 practice questions
  • 📶 Intermediate
GCSEHigh School
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Curriculum fit: GCSE Maths Algebra (Foundation and Higher; formulae with squares and roots are mainly Higher). US: Algebra I rearranging formulas.

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What Is It?

The subject of a formula is the letter on its own on one side, such as v in v = u + at. Changing the subject means rearranging the formula so a different letter is on its own.

You do this the same way as solving an equation: whatever you do to one side, you do to the other, using inverse operations in the reverse order.

y = 2x + 3y − 3 = 2xx = (y − 3) ÷ 2subtract 3from both sidesdivide bothsides by 2Inverse operations+ and − × and ÷ squared and square root
Undo operations in the reverse order to the order of operations, and do the same to both sides to keep the formula balanced.

Key Formulas and Ideas

  • Inverse of + is −, of × is ÷, of squaring is square root.
  • Undo operations in the reverse order to the order of operations.
  • Keep the formula balanced: do the same to both sides.

Worked Examples

Example 1

Two Steps

Question

Make x the subject of y = 3x + 5.

  1. Subtract 5 from both sides: y − 5 = 3x.
  2. Divide both sides by 3: x = (y − 5) ÷ 3.

✓ Answer x = (y − 5)/3

Example 2

Physics Formula

Question

Make a the subject of v = u + at.

  1. Subtract u: v − u = at.
  2. Divide by t: a = (v − u)/t.

✓ Answer a = (v − u)/t

Example 3

With a Square Root

Question

Make r the subject of A = πr².

  1. Divide by π: A/π = r².
  2. Take the square root: r = √(A/π).

✓ Answer r = √(A/π)

Common Mistakes to Avoid

  • Dividing only one term on a side, such as turning y − 5 = 3x into x = y − 5/3 instead of (y − 5)/3.
  • Doing the steps in the wrong order.
  • Forgetting to apply the square root to the whole side.
  • Moving a term across without changing its sign.

Practice Questions

0 of 5 answered
Q1Make x the subject of y = x + 7.
Q2Make b the subject of A = bh.
Q3Make x the subject of y = 2x − 6.
Q4Make t the subject of d = vt.
Q5Make x the subject of y = x²/4.

Frequently Asked Questions

What if the letter I want appears twice?

Collect all terms containing it on one side, factorise, then divide. This is mainly Higher tier work.

How can I check my rearranged formula?

Substitute simple numbers into both the original and your result and check they agree.

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Sources and Review

Reviewed by StudyMorrow editorial review on 5 Nov 2026.