Free GCSE Maths lesson: Algebra
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Lesson 32 · GCSE / Key Stage 4 · Maths · Algebra

Formulae and Changing the Subject

Learn how to use formulae safely, then rearrange them so the letter you need is on its own.

Qualification: GCSE Key Stage 4 Subject: Maths Strand: Algebra

GCSE specification fit

A core algebra skill that links number, geometry and measures.

Formulae give a rule that connects quantities. GCSE questions may ask you to substitute values, calculate with a formula, or rearrange it so a different variable becomes the subject.

QualificationGCSE Mathematics
Key stageKey Stage 4
StrandAlgebra
Tier guidanceFoundation and Higher

What you will learn

  • How to substitute values into formulae.
  • How to use order of operations, brackets and powers inside formulae.
  • What it means to make a letter the subject.
  • How to rearrange one-step and two-step formulae.
  • How to deal with brackets, fractions and squared variables.
  • How to check a rearranged formula by substitution.

Why this matters

Formulae appear across GCSE Maths: area, volume, speed, density, pressure, interest and straight-line graphs all use them. Rearranging a formula lets you find a missing quantity without guessing.

The safest method is to treat the formula like an equation: do the same inverse operation to both sides until the required letter is isolated.

Prior knowledge

You should already be comfortable with:

  • substituting numbers for letters,
  • using inverse operations,
  • solving simple equations,
  • expanding brackets when needed,
  • working carefully with fractions.

Clear explanation

Substitute first, then calculate

Substitution means replacing each letter with a number. Write the substituted line before calculating so the method is clear.

A = lw If l = 8 and w = 5, then A = 8 × 5 A = 40

The subject is the letter on its own

In the formula C = 2πr, C is the subject because it is already on its own. If a question says make r the subject, the final answer must start with r =.

C = 2πr Divide both sides by 2π r = C2π

Undo operations in reverse order

Work backwards from the operations around the letter. Addition and subtraction are usually undone before multiplication or division.

y = 3x + 4 y − 4 = 3x y − 43 = x x = y − 43

Clear brackets before isolating the letter

If the letter is inside a bracket, decide whether it is easier to divide first or expand first. Keep every step balanced.

A = 5(b + 2) A5 = b + 2 b = A5 − 2

Worked examples

Example 1: Substitute into a formula

Use P = 2l + 2w when l = 9 and w = 4.

P = 2 × 9 + 2 × 4 P = 18 + 8
Answer: P = 26.

Example 2: Make x the subject

Rearrange y = x + 7 to make x the subject.

Subtract 7 from both sides.
Answer: x = y − 7.

Example 3: Make m the subject

Rearrange F = ma to make m the subject.

Divide both sides by a.
Answer: m = Fa.

Example 4: Rearrange a two-step formula

Make p the subject of q = 4p − 9.

q + 9 = 4p q + 94 = p
Answer: p = q + 94.

Quick checks

Choose an answer, then check your thinking.

1. In v = u + at, which letter is the subject?

2. Make a the subject of c = a + b.

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

Practice questions

Question 1

A rectangle has area A = bh. Use the formula to find A when b = 12 cm and h = 7 cm, including units.

Reveal answer and marking guidance

Answer: A = 84 cm².

Marking: Substitute to get A = 12 × 7, calculate accurately, and use square centimetres because this is an area.

Question 2

A taxi fare is modelled by C = 5n + 2, where n is the number of miles and C is the cost in pounds. Find C when n = 6.

Reveal answer and marking guidance

Answer: C = 32, so the cost is £32.

Marking: Substitute n = 6, multiply before adding, and interpret the result as pounds: 5 × 6 + 2 = 32.

Question 3

A temperature model uses y = x − 8. Rearrange the formula to make x the subject so the original temperature can be found from y.

Reveal answer and marking guidance

Answer: x = y + 8.

Marking: Add 8 to both sides to undo the subtraction and leave x on its own.

Question 4

The perimeter of a regular hexagon is p = 6a, where a is the side length. Make a the subject.

Reveal answer and marking guidance

Answer: a = p6.

Marking: Divide both sides by 6 and keep the formula general; do not substitute a number unless the question gives one.

Question 5

Make t the subject of d = 3t + 5.

Reveal answer and marking guidance

Answer: t = d − 53.

Marking: Subtract 5 first, then divide the whole of d − 5 by 3.

Question 6

Make r the subject of A = πr2.

Reveal answer and marking guidance

Answer: r = √Aπ.

Marking: Divide by π first, then take the positive square root because r is a length.

Question 7

Make h the subject of V = 13πr2h.

Reveal answer and marking guidance

Answer: h = 3Vπr2.

Marking: Multiply both sides by 3, then divide by πr2. Keep πr2 together as one factor.

Question 8

Use y = 2x2 − 3x when x = −4.

Reveal answer and marking guidance

Answer: y = 44.

Marking: Substitute with brackets: y = 2(−4)2 − 3(−4) = 32 + 12 = 44.

Question 9

Make x the subject of 3x + y = 2x − 5.

Reveal answer and marking guidance

Answer: x = −y − 5.

Marking: Subtract 2x from both sides to get x + y = −5, then subtract y from both sides.

Question 10

Make x the subject of ax + b = cx + d.

Reveal answer and marking guidance

Answer: x = d − ba − c, where a ≠ c.

Marking: Collect the x-terms first: ax − cx = d − b, then factorise to x(a − c) = d − b and divide by the whole of a − c.

Answers and marking guidance

The exact practice answers are hidden under each question so you can try first. For formulae, marks usually come from writing a clear substitution line, applying order of operations correctly, using balanced inverse operations when changing the subject, collecting like terms before moving on, keeping brackets or fraction bars around the whole expression, and checking that the requested letter is isolated in the final formula.

Common mistakes

  • Substituting without brackets: negative values and powers often need brackets to avoid sign errors.
  • Forgetting order of operations: multiply and square before adding unless brackets say otherwise.
  • Stopping too early: the requested subject must be completely on its own.
  • Dividing only one term: in t = (d − 5) ÷ 3, the whole d − 5 must be divided by 3.
  • Losing the square root: undo squaring with a square root when a formula contains r2.

Extension challenge

Make x the subject of y = 4(2x − 3).

Reveal answer

Answer: x = y + 128.

Expand to y = 8x − 12, then y + 12 = 8x, so x = (y + 12) ÷ 8.

Exam-board guidance

Formulae and changing the subject are assessed by every GCSE Maths board. Questions may be pure algebra or may appear inside geometry, measures, graphs and real-life contexts.

AQA GCSE Maths

Show each inverse operation when changing the subject, bracket substituted negative values, and check that the requested letter is fully isolated.

OCR GCSE Maths

Write the substituted formula before calculating, keep powers and brackets visible, and check a rearranged formula by substituting simple values.

Pearson Edexcel GCSE Maths

Expect formulae in algebra, geometry and compound measures, so rearrange before substituting when the question asks for a different quantity.

Eduqas GCSE Maths

Treat changing the subject like solving an equation, and keep the whole numerator or denominator together when a fraction is created.

WJEC Wales

Formula questions may be practical, so state the rearranged formula first, then substitute values and carry the correct units.

CCEA GCSE Maths

Show the algebraic steps before the final answer, and use a quick substitution check to confirm the rearranged formula works.

Next lesson

Next, practise continuing patterns and finding nth-term formulae.

Previous lesson: Inequalities Next lesson: Sequences and the nth Term Back to GCSE Maths lessons