AQA GCSE Mathematics Paper 1 (Higher), June 2025: Question 1

Mark scheme

Show the mark scheme

Q Answer Mark Comments

Correct method to eliminate one eg 5y – y = 18 – 6

variable

or 18 – 5y = 6 – y

M1 or 5 × 2x – 2x = 5 × 6 – 18

18 − 2x

or = 6 – 2x

4y = 12 or –4y = –12 or y = 3 oe equation in form ax = b or cy = d

or M1dep

8x = 12 or –8x = –12 or x = 1.5

x = 1.5 and y = 3 A1 oe

Additional Guidance

Intention to subtract is sufficient for M1

eg 2x + 5y = 18

– M1

2x + y = 6

Correct values embedded in both equations M2A0

Correct values embedded in one equation M1M0A0

How to answer it

Solving Linear Simultaneous Equations by Elimination

📌 What this question tests

This question assesses your ability to solve a pair of simultaneous linear equations algebraically. Key skills include recognising matching coefficients, eliminating one unknown variable via subtraction, solving the resulting linear equation, and substituting back to find the second unknown.

Question 1 (3 Marks)

Solve: 2x + 5y = 18 and 2x + y = 6

📐 Step-by-Step Solution

Label the two equations:

(1) 2x + 5y = 18
(2) 2x + y = 6

1 Eliminate x by subtracting (2) from (1):

(2x - 2x) + (5y - y) = 18 - 6

4y = 12

2 Solve for y:

y = 12 / 4 = 3

3 Substitute y = 3 into equation (2):

2x + 3 = 6

2x = 6 - 3

2x = 3

x = 1.5 (or 3/2)

4 Check in equation (1):

2(1.5) + 5(3) = 3 + 15 = 18 ✓

✅ Final Answer & Mark Breakdown

Values:

x = 1.5 (or 3/2, 1 ½)
y = 3

Mark Scheme Breakdown:

[M1] Correct method to eliminate one variable (e.g. clearly setting up subtraction: 5y - y = 18 - 6 ).
[M1 dep] Obtaining an equation in one variable in the form ax = b or cy = d (e.g. 4y = 12 or finding y = 3 ).
[A1] Both values fully correct: x = 1.5 and y = 3 .

💡 Key Knowledge: The Elimination Rule

Same Signs Subtract (SSS): Both equations have +2x . Because the signs match, subtract the equations to eliminate x .
Opposite Signs Add (OSA): If one had +2x and the other -2x , you would add them together.
Remember to subtract every term: the x-terms, the y-terms, and the constant numbers on the right-hand side.

🧠 Exam Technique & Examiner Insight

Show your working clearly: Even writing down the column subtraction with a minus sign awards the first M1 mark, even if you make an arithmetic error later.
Always substitute back to verify: Test both answers in the original equation you didn't use to find the second variable. If 2(1.5) + 5(3) = 18 works, you know with 100% certainty you have full marks!
Don't stop at one variable: A very common slip under exam pressure is finding y = 3 and forgetting to calculate x .

❌ Common Errors to Avoid

Adding instead of subtracting: Adding gives 4x + 6y = 24 , which does not eliminate any variable and scores 0 marks.
Subtracting only one side: Forgetting to subtract the constants: e.g. writing 4y = 18 instead of 18 - 6 = 12 .
Embedded answers without final statements: Showing values that work in the equations but leaving the answer lines blank costs the final accuracy mark (maximum M2 A0).

Topics

Algebra · 3.2.3 Solving equations and inequalities