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GCSE linear equations practice and worked examples

Build from one-step equations to equations with brackets or unknowns on both sides. A balanced calculation keeps the two sides equal.

Foundation and Higher · Useful for students building fluency and applying the method in GCSE questions.

Core pathway. Core GCSE methods are useful across Foundation and Higher. Start with the core examples and questions.

Higher extension. No questions in this selected set are labelled Higher extension. These core methods also support Higher revision.

3 worked examples and 8 original practice questions · Allow 25–40 minutes · Read online or print. Higher extensions are labelled. No account or email needed.

Learn the methodTry the questions

Printing includes the method, examples and space for working, followed by a separate answer section.

A useful approach

  1. Write the equation clearly and expand any brackets.
  2. Undo operations on both sides, collecting variable terms on one side and constants on the other.
  3. Divide by the coefficient, then substitute the answer into the original equation.

A mistake to watch for

Moving a term is shorthand for doing the same operation to both sides. Forgetting the sign changes the equation.

Learn each method, then practise it

Read each line of working and explain why it follows from the previous line.

Undo one operation

Use inverse operations on both sides. The equality must stay balanced.

Worked example 1

Solving One Step Equations · Core GCSE skill · No calculator · 2 marks

  1. Solve  4m = 28
  2. Solve  t ÷ 5 = 6
  1. a) Divide both sides by 4: m = 28 ÷ 4 = 7.
  2. b) Multiply both sides by 5: t = 6 × 5 = 30.
Answer: (a) m = 7   (b) t = 30
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Now practise this method: Question 1 · Question 2

Brackets and several steps

Expand first or divide away a common factor. Then isolate the unknown and substitute back into the original equation.

Worked example 2

Solving Equations · Core GCSE skill · Calculator allowed · 2 marks

Solve 3(x + 4) = 27

  1. Divide both sides by 3: x + 4 = 9
  2. Subtract 4 from both sides: x = 5
Answer: x = 5
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Now practise this method: Question 3 · Question 4 · Question 5

Form an equation from a context

Define the unknown, translate each relationship into algebra and use the total or geometric fact given. Check the answer in the context.

Worked example 3

Forming and Solving Equations · Core GCSE skill · Calculator allowed · 4 marks

A rectangle has length (3x + 2) cm and width (x − 1) cm.

The perimeter of the rectangle is 50 cm.

Work out the length of the rectangle.

  1. Perimeter = 2(3x + 2) + 2(x - 1) = 6x + 4 + 2x - 2 = 8x + 2
  2. 8x + 2 = 50
  3. 8x = 48, so x = 6
  4. Length = 3(6) + 2 = 20 cm
  5. Check: width = 5 cm, perimeter = 2(20) + 2(5) = 50 cm
Answer: 20 cm
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Now practise this method: Question 6 · Question 7 · Question 8

Your practice questions

Write your working on paper. Marks indicate how much working to show; these questions are self-marked and do not change saved practice results. Use squared paper for drawing questions.

Worked answers: Linear equations

Compare the reasoning as well as the final answer. Another correct method is valid. If a step is unclear, revisit an example before trying a similar question.

Answer 1

Show answer and working for question 1
  1. Subtract 7 from both sides.
  2. x = 12 - 7 = 5.
Answer: x = 5
Back to question 1

Answer 2

Show answer and working for question 2
  1. Add 9 to both sides.
  2. y = 14 + 9 = 23.
Answer: y = 23
Back to question 2

Answer 3

Show answer and working for question 3
  1. Expand the bracket: 8x - 12 = 36
  2. Add 12 to both sides: 8x = 48
  3. Divide both sides by 8: x = 6
Answer: x = 6
Back to question 3

Answer 4

Show answer and working for question 4
  1. Multiply both sides by 3: 2x + 5 = 21
  2. Subtract 5 from both sides: 2x = 16
  3. Divide both sides by 2: x = 8
Answer: x = 8
Back to question 4

Answer 5

Show answer and working for question 5
  1. Let n be the number, so 4n - 9 = 35
  2. Add 9 to both sides: 4n = 44
  3. Divide both sides by 4: n = 11
  4. Check: 4 times 11 = 44, 44 - 9 = 35
Answer: 11
Back to question 5

Answer 6

Show answer and working for question 6
  1. Angles in a triangle add to 180: (x + 10) + 2x + (3x - 10) = 180
  2. 6x = 180, so x = 30
  3. The angles are 40, 60 and 80 degrees
  4. Check: 40 + 60 + 80 = 180
  5. The largest angle is 80 degrees
Answer: 80°
Back to question 6

Answer 7

Show answer and working for question 7
  1. Let d be the number of sweets Dan gets, so Ella gets d + 7
  2. d + d + 7 = 45
  3. 2d = 38, so d = 19
  4. Check: 19 + 26 = 45
  5. Dan is wrong, he got 19 sweets, not 20
Answer: No, Dan got 19 sweets
Back to question 7

Answer 8

Show answer and working for question 8
  1. Angles in a quadrilateral add to 360: x + (x + 30) + (2x - 10) + 100 = 360
  2. 4x + 120 = 360
  3. 4x = 240, so x = 60
  4. The angles are 60, 90, 110 and 100 degrees
  5. Check: 60 + 90 + 110 + 100 = 360
  6. The largest angle is 110 degrees
Answer: 110°
Back to question 8

Practise a specific skill

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Use this in class

Give students time to write the method, then compare their working with the answer section. Ask which step they would check first if their answer differs.

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Related GCSE practice

Simplifying, expanding and factorisingSubstitutionSimultaneous equations
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