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Simultaneous equations

Free Printable Simultaneous equations PDF Worksheets, Revision Guide & GCSE Past Paper Questions (Grade 5)

Download free simultaneous equations worksheets (PDF), GCSE maths past paper questions and worked solutions for Simultaneous equations (5L). This is a Grade 5 algebra topic from the GCSE questions by topic collection for Foundation and Higher tier crossover GCSE Maths (Edexcel, AQA, OCR and IGCSE).

The free PDF worksheets table below brings together printable question booklets (Qs) and worked answers or mark schemes (Ans) from IGCSE MT, Maths Genie, Maths4Everyone, Corbett Maths, Maths Made Easy, Dr Frost, Naiker Maths, 1st Class Maths, Dr Austin, Metatutor, E Classroom and PMT, along with past paper question booklets.

Below the worksheet links you can read our simultaneous equations revision notes and worked examples with diagrams, try real GCSE past paper simultaneous equations questions with step-by-step solutions, or download a PDF of all simultaneous equations worksheet links.

Free Simultaneous equations PDF Worksheets & Answers

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GCSE Maths Grade 5 Revision Notes & Method

Guide to Simultaneous equations: Explanation, Diagrams & Worked Examples

Download Printable One Page Guide (PDF)

Explanation:

To solve two linear simultaneous equations by elimination, multiply one or both equations so that one letter has the same coefficient size in both equations.

Use the rule SSS / DAA: if the signs in front of the matching terms are the Same, Subtract the equations; if the signs are Different, Add the equations. Once you find the first letter, substitute it back into an original equation to find the second letter.

Example 1: Solve 3x + 2y = 19 and 5x − 2y = 13.

Solution:

The 2y terms have different signs (+2y and −2y), so add:
(3x + 5x) + (2y − 2y) = 19 + 13 → 8x = 32 → x = 4
Substitute x = 4 into 3x + 2y = 19:
12 + 2y = 19 → 2y = 7 → y = 3.5
Answer: x = 4, y = 3.5

Example 2: Solve 2x + 3y = 18 (1) and 3x + 5y = 29 (2).

Solution:

Multiply (1) by 3 and (2) by 2 to match 6x:
6x + 9y = 54 and 6x + 10y = 58
Same signs (+6x), so subtract: y = 4
Sub y = 4 into (1): 2x + 12 = 18 → 2x = 6 → x = 3
Answer: x = 3, y = 4
Scanned GCSE Exam Questions & Worked Solutions

GCSE Past Paper Simultaneous equations Questions & Answers

Past Paper Example 1 Eduqas Nov 2018 1F Q24 / 1H Q7 7 marks
(a) Use the diagram to solve the simultaneous equations
y - x = 3
x + 2y = 1
Give your answers correct to 1 decimal place.
(b) A theme park sells entrance tickets.
2 adult tickets and 3 child tickets would cost a total of £72.
3 adult tickets and 1 child ticket would cost a total of £66.
A family ticket costs £45 and allows entry for 2 adults and 2 children.
How much cheaper is it to buy a family ticket than it would be to buy 2 adult and 2 child tickets?
(a) Use the diagram to solve the simultaneous equations
y - x = 3
x + 2y = 1
Give your answers correct to 1 decimal place.
[2 marks]
xy–3–2–1012–2–112345y - x = 3x + 2y = 1
(b) A theme park sells entrance tickets.
2 adult tickets and 3 child tickets would cost a total of £72.
3 adult tickets and 1 child ticket would cost a total of £66.
A family ticket costs £45 and allows entry for 2 adults and 2 children.
How much cheaper is it to buy a family ticket than it would be to buy 2 adult and 2 child tickets?
[5 marks]
Worked Solution & Mark Scheme (7 marks)
Part (a) Solution:
Locate the intersection of the two straight lines on the graph. M1
x-coordinate is roughly -1.7
y-coordinate is roughly 1.3. A1
xy–3–2–1012–2–112345y - x = 3x + 2y = 1(-1.7, 1.3)
Answer (a): x = -1.7, y = 1.3 (accept x = -1.6 to -1.7, y = 1.3 to 1.4)
Part (b) Solution:
Let a = price of adult ticket, c = price of child ticket M1
(1) 2a + 3c = 72 M1
(2) 3a + c = 66 ⇒ c = 66 - 3a
Substitute into (1): 2a + 3(66 - 3a) = 72 ⇒ 2a + 198 - 9a = 72 ⇒ -7a = -126 ⇒ a = 18 M1
Then c = 66 - 3(18) = 66 - 54 = 12 M1
Cost of 2 adult and 2 child tickets = 2(18) + 2(12) = 36 + 24 = £60
Saving with family ticket = 60 - 45 = £15. A1
Answer (b): £ £15 cheaper £
Past Paper Example 2 AQA Jun 2025 1H Q1 3 marks
Solve the simultaneous equations
2x + 5y = 18
2x + y = 6
Worked Solution & Mark Scheme (3 marks)
Subtract equations: (2x + 5y) – (2x + y) = 18 – 6 ⇒ 4y = 12 ⇒ y = 3. M2
Substitute y = 3 into 2x + y = 6: 2x + 3 = 6 ⇒ 2x = 3 ⇒ x = 1.5. A1
Answer: x = 1.5, y = 3
Past Paper Example 3 AQA Nov 2023 3F Q25 / 3H Q11 3 marks
Solve these simultaneous equations.
7x + 2y = 100
3x + 2y = 48
Worked Solution & Mark Scheme (3 marks)
Subtract equations: 4x = 52 ⇒ x = 13. M2
3(13) + 2y = 48 ⇒ 39 + 2y = 48 ⇒ 2y = 9 ⇒ y = 4.5. A1
Answer: x = 13, y = 4.5

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