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Simultaneous equations Quiz

Grade 5 • 5L • 10 Questions (34 Marks)

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Simultaneous equations — Notes & Worked Examples Show / Hide

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
Question 1
[1 mark]
Solve the simultaneous equations:
2x + y = 11
4x - y = 19
Worked Solution & Answer
1) Multiply so the coefficients of x or y match
2) Add or subtract the equations to eliminate that letter
x = 5, y = 1
Answer: x = 5, y = 1
Question 2
[1 mark]
Solve the simultaneous equations:
2x + y = 4
x + 2y = -4
Worked Solution & Answer
1) Multiply so the coefficients of x or y match
2) Add or subtract the equations to eliminate that letter
x = 4, y = -4
Answer: x = 4, y = -4
Question 3 AQA Jun 2025 1H Q1
[3 marks]
Solve the simultaneous equations
2x + 5y = 18
2x + y = 6
Worked Solution & Answer
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
Question Paper (PDF) • Mark Scheme (PDF)
Question 4 AQA Nov 2023 3F Q25 / 3H Q11
[3 marks]
Solve these simultaneous equations.
7x + 2y = 100
3x + 2y = 48
Worked Solution & Answer
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
Question Paper (PDF) • Mark Scheme (PDF)
Question 5
[3 marks]
Solve the simultaneous equations
3x + y = -4
3x - 4y = 6
Worked Solution & Answer
Subtract 2nd equation from 1st: 5y = –10 M1
⇒ y = –2. Substitute y = –2 into 3x + y = –4: 3x – 2 = –4 M1
⇒ 3x = –2
⇒ x = –2/3. A1
Answer: x = -23, y = -2
Question Paper (PDF) • Mark Scheme (PDF)
Question 6 IGCSE Jun 2025 2F Q25 / 2H Q8
[3 marks]
Solve the simultaneous equations
3x + 5y = 8
4x + y = -3.5
Show clear algebraic working.
Worked Solution & Answer
20x + 5y = –17.5 M1
Subtract 3x + 5y = 8: 17x = –25.5 ⇒ x = –1.5 M1
y = –3.5 – 4(–1.5) = 2.5 A1
Answer: x = –1.5, y = 2.5
Question Paper (PDF) • Mark Scheme (PDF)
Question 7
[4 marks]
Use algebra to solve the simultaneous equations
4x - 5y = 20
6x + 7y = -57
You must show all your working.
Worked Solution & Answer
12x – 15y = 60 and 12x + 14y = –114 ⇒ 29y = –174 ⇒ y = –6 M3
4x – 5(–6) = 20 ⇒ 4x = –10 ⇒ x = –2.5 A1
Answer: x = –2.5, y = –6
Question Paper (PDF) • Mark Scheme (PDF)
Question 8
[4 marks]
Solve the simultaneous equations
5x - 2y = 23
2x - 3y = 18
Worked Solution & Answer
15x – 6y = 69 and 4x – 6y = 36 M3
Subtracting gives 11x = 33 ⇒ x = 3. Substituting x = 3: 15 – 2y = 23 ⇒ 2y = –8 ⇒ y = –4. A1
Answer: x = 3, y = –4
Question Paper (PDF) • Mark Scheme (PDF)
Question 9
[5 marks]
A pattern is made from four identical rectangles.
The sides of the rectangles are parallel to the axes.
xyOA(3, 4)CB(11, 20)
Point A has coordinates (3, 4)
Point B has coordinates (11, 20)
Point C is marked on the diagram.
Work out the coordinates of C.
You must show all your working.
Worked Solution & Answer
Let length = L and width = W M2
Horizontal change from A(3,4) to B(11,20): 2L – W = 11 – 3 = 8. Vertical change from A(3,4) to B(11,20): 2L + W = 20 – 4 = 16 M2
Adding gives 4L = 24 ⇒ L = 6, W = 4. Therefore C = (3 + L, 4 + W) = (3 + 6, 4 + 4) = (9, 8). A1
Answer: (9, 8)
Question Paper (PDF) • Mark Scheme (PDF)
Question 10 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 & Answer
(a)
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)
(b)
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 £
Question Paper (PDF) • Mark Scheme (PDF)
Score: 0 / 10 (0 / 34 marks)
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Grade 5 Quizzes

25 Topics
5A - Combining ratios5B - Direct and inverse proportion
5C - Reverse percentages5D - Standard form
5E - Speed5F - Changing the subject
5G - Quadratic expressions5H - Solving quadratic equations
5J - Drawing quadratic graphs5K - Cubic/reciprocal graphs
5L - Simultaneous equations5M - Sector areas and arc lengths
5N - Sim. equ. graphically5P - Midpoint of a line segment
5Q - Gradient of a line5R - Equation of a line
5S - Spheres and cones5T - Similar shapes (lengths)
5U - Trigonometry (SOHCAHTOA)5V - Exact trig. values
5W - Probability trees5X - Vectors
5Y - Venn diagrams5Z - Density
5AA - Pressure