AQA GCSE Mathematics Paper 2 (Foundation), June 2025: Question 11

3 marks · Easy difficulty · Proof

Show that the algebraic expressions 7(x - 2) + 4x + 6 and 5(3x + 2) - 4x - 18 are equivalent.

Practise this question

Question

Question 11 shows two expressions labelled A and B inside rectangular boxes. Expression A is 7(x - 2) + 4x + 6. Expression B is 5(3x + 2) - 4x - 18. Below the boxes, the question asks to 'Show that A and B are equivalent' for 3 marks.

Mark scheme

Show the mark scheme Mark scheme for Question 11 detailing two alternative methods. Alternative 1: M1 for expanding one set of brackets correctly to 7x - 14 or 15x + 10; M1dep for fully simplifying one expression to 11x - 8; A1 for fully expanding and simplifying both expressions to 11x - 8. Alternative 2: compares x terms and constant terms separately. Additional guidance clarifies partial marks and notes that substituting numerical values earns 0 marks.

How to answer it

Showing That Two Algebraic Expressions Are Equivalent

📋 What this question tests

This question assesses your ability to manipulate and simplify algebraic expressions containing single brackets, collect like terms correctly (paying careful attention to directed numbers/negative signs), and construct a clear mathematical argument showing that two distinct expressions simplify to the identical final form.

Question 11 (3 Marks)

Expressions: A = 7(x − 2) + 4x + 6  |  B = 5(3x + 2) − 4x − 18

💡 Key Knowledge

  • Expanding brackets: Multiply the term outside the bracket by every term inside:
    a(b + c) = ab + ac
  • Directed numbers: Remember sign rules:
    7 × (−2) = −14
    −14 + 6 = −8
    10 − 18 = −8
  • Equivalence: Two expressions are equivalent if they can both be rewritten in the exact same simplified form for all values of x .

📐 Step-by-Step Calculation

Step 1: Expand and simplify Expression A
A = 7(x − 2) + 4x + 6
Expand bracket: 7x − 14 + 4x + 6
Collect terms: (7x + 4x) + (−14 + 6) = 11x − 8
Step 2: Expand and simplify Expression B
B = 5(3x + 2) − 4x − 18
Expand bracket: 15x + 10 − 4x − 18
Collect terms: (15x − 4x) + (10 − 18) = 11x − 8
Step 3: Conclusion
Both expressions simplify to 11x − 8 , so A and B are equivalent.

✅ Mark Breakdown

  • M1: Correctly expand at least one bracket (e.g. seeing 7x − 14 or 15x + 10 ).
  • M1 (dep): Fully correct simplification of either A or B to 11x − 8 (with full expansion shown).
  • A1: Both A and B fully correctly simplified to 11x − 8 with all necessary working shown.
Special Case (SC1): Writing 11x − 8 for one or both expressions with no intermediate working scores only 1 mark.

🧠 Exam Technique

  • Work down in two separate columns: Label your work clearly with "Expression A" and "Expression B".
  • Never drop terms halfway: Write the full expanded line before simplifying (e.g. write 7x − 14 + 4x + 6 before writing 11x − 8 ).
  • Do not set up an equation: "Show that" means showing algebraic equivalence, not solving for x .

❌ Common Traps & Examiner Warnings

  • Substituting numbers (Zero marks): Trying to prove equivalence by testing x = 1 or x = 2 scores 0 marks. Proof requires algebraic simplification.
  • Bracket expansion slips: Forgetting to multiply the second term inside the bracket (e.g. writing 7x − 2 instead of 7x − 14 ).
  • Negative number errors: Calculating −14 + 6 as −20 or +8 , or 10 − 18 as +8 .
  • Subsequent solving: Going on to write 11x − 8 = 0 and finding x = 8/11 . While you may keep M marks, do not solve an expression when asked to show equivalence!

Topics

Algebra · 3.2.1 Notation, vocabulary and manipulation

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 2 (Foundation), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.