Simultaneous equations: IGCSE Mathematics questions with answers
Free Simultaneous equations questions for IGCSE Mathematics (0580), part of Algebra & graphs. Each question has the answer and the marking points an examiner would look for.
A rectangle has area 27 cm² and perimeter 24 cm. Its sides are x cm and y cm, where x > y.
(a)
Write down two equations in x and y and show that x2 − 12x + 27 = 0.
[3]
(b)
Find x and y.
[2]
(c)
A second rectangle has the same perimeter but an area of 28 cm². Show that its sides cannot be whole numbers of centimetres, and find the longer side correct to 2 decimal places.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
xy = 27, x + y = 12
xy = 27 B1 2x + 2y = 24 B1 substitutes y = 12 − x and rearranges A1
3
(b)
x = 9, y = 3
factorise / formula M1 x = 9, y = 3 A1
2
(c)
8.83 cm
x2 − 12x + 28 = 0 has discriminant 32, not a square number M1 formula M1 8.83 A1
3
Question 2Structured · 5 marksA* challenge
Solve the simultaneous equations. y = x2 − 5x + 6 y = −2x + 4
[5]
Show answers and mark scheme
Part
Answer and marking points
Marks
x = 1, y = 2 and x = 2, y = 0
equate and rearrange to a quadratic = 0 M1 x2 − 3x + 2 = 0 A1 factorise or formula M1 x = 1, 2 A1 y = 2, 0 A1
5
Question 3Structured · 4 marks
Solve the simultaneous equations. You must show all your working. 7x − 6y = 39 5x + 5y = 0
[4]
Show answers and mark scheme
Part
Answer and marking points
Marks
x = 3, y = −3
Correct multiplication to make one pair of coefficients equal M1 Correct addition/subtraction to eliminate one variable M1 x = 3 A1 y = −3 A1
4
Question 4Structured · 4 marksA* challenge
At a cinema, 2 adult tickets and 6 child tickets cost $24. 2 adult tickets and 7 child tickets cost $26. Find the cost of an adult ticket and the cost of a child ticket.
[4]
Show answers and mark scheme
Part
Answer and marking points
Marks
adult $6, child $2
2a + 6c = 24 and 2a + 7c = 26 B1 correct method to eliminate one variable M1 a = 6 A1 c = 2 A1
4
Question 5Structured · 8 marksA* challenge
A rectangle has area 40 cm² and perimeter 26 cm. Its sides are x cm and y cm, where x > y.
(a)
Write down two equations in x and y and show that x2 − 13x + 40 = 0.
[3]
(b)
Find x and y.
[2]
(c)
A second rectangle has the same perimeter but an area of 41 cm². Show that its sides cannot be whole numbers of centimetres, and find the longer side correct to 2 decimal places.
[3]
Show answers and mark scheme
Part
Answer and marking points
Marks
(a)
xy = 40, x + y = 13
xy = 40 B1 2x + 2y = 26 B1 substitutes y = 13 − x and rearranges A1
3
(b)
x = 8, y = 5
factorise / formula M1 x = 8, y = 5 A1
2
(c)
7.62 cm
x2 − 13x + 41 = 0 has discriminant 5, not a square number M1 formula M1 7.62 A1
3
Question 6Structured · 5 marksA* challenge
Solve the simultaneous equations. y = x2 − 5x − 5 y = −3x + 3
[5]
Show answers and mark scheme
Part
Answer and marking points
Marks
x = 4, y = −9 and x = −2, y = 9
equate and rearrange to a quadratic = 0 M1 x2 − 2x − 8 = 0 A1 factorise or formula M1 x = 4, −2 A1 y = −9, 9 A1
Riyaz Shaikh has mentored IGCSE, IB and A Level students for 15+ years, one-on-one, online and in Mumbai. The hardest questions on this site are the ones he uses to find a student's weak points.
Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in