GCSE Exam questions – Simultaneous Equations

1.	Solve the simultaneous equations
	2x + 3y = 6
	3x − 2y = 22
x = ……………………
y = ……………………
(Total 4 marks)

2.	Solve
2x – 3y = 11
5x + 2y = 18
x = ......................
y = ......................

(Total 4 marks)

7.	Solve 
x + 2y = 4
3x – 4y = 7
x = ………………..
y = ………………..
(Total 3 marks)

10.	Solve the simultaneous equations
3x + 7y = 26
4x + 5y = 13
x = .....................................
y = .....................................
(Total 4 marks)

13.	Solve the simultaneous equations
3x – 4y = 11
5x + 6y = 12
x = ……………………………
y = ……………………………
(Total 4 marks)

4.	Solve the simultaneous equations
6x – 2y = 33
4x + 3y = 9
x = ....................
y = ....................
(Total 4 marks)

11.	Solve the simultaneous equations.
	5a + 3b = 9 
2a – 3b = 12
a =.....................................
b = .....................................
(Total 3 marks)

12.	Solve the simultaneous equations
	2x + y  = 4
	5x - y  = 17

x  =  …………….
 y  =  …………….
(Total 2 marks)

3.	Solve the simultaneous equations
x2 + y2 = 29
y – x = 3
………………………………………………………
(Total 7 marks)

5.	Bill said that the line y = 6 cuts the curve x2 + y2 = 25 at two points.
(a)	By eliminating y show that Bill is incorrect.

(2)

(b)	By eliminating y, find the solutions to the simultaneous equations
x2 + y2 = 25
y = 2x – 2

x = ........................ y = ...................
or x = ........................ y = ...................
(6)
(Total 8 marks)

6.	(a)	On the grid below, draw the graphs of
x2 + y2 = 100
	and	2y = 3x – 4
(3)
(b)	Use the graphs to estimate the solutions of the simultaneous equations
x2 + y2 = 100

	and	2y = 3x – 4
.............................
.............................
(2)
	For all the values of x
x2 + 6x = (x + 3)2 – q

(c)	Find the value of q.

q = .............................
(2)
	One pair of integer values which satisfy the equation
x2 + y2 = 100
	is x = 6 and y = 8
(d)	Find one pair of integer values which satisfy
x2 + 6x + y2 – 4y – 87 = 0

x = .............., y = ...............
(3)
(Total 10 marks)

8.	

	The diagram shows a circle of radius 5 cm, centre the origin.

	Draw a suitable straight line on the diagram to find estimates of the solutions to the pair of equations
	x2 + y2 = 25
y = 2x + 1

x = ......................., y = .......................
x = ......................., y = .......................
(Total 3 marks)

9.	

	The diagram shows a sketch of the graph of y = x2 – x
(a)	On the same diagram, sketch and label where the graph of y = (x – 1)2 – (x – 1)
Show clearly where this graph crosses the x-axis and where it crosses the y-axis.
(3)
(b)	On the same diagram, sketch and label the graph of y = 3(x2 – x)
(1)

	The line y = 4 – 4x intersects the curve y = 3(x2 – x) at the points A and B.
(c)	Use an algebraic method to find the coordinates of A and B.

( ............ , ............ )
( ............ , ............ )
(5)
(Total 9 marks)

14.	Solve the simultaneous equations
x + y = 4
x2 + y2 = 40

x =................., y = .................
or
x =................., y = .................
(Total 7 marks)

15.	
Diagram NOT accurately drawn
	The diagram shows two straight lines intersecting at point A.
The equations of the lines are
	y = 4x – 8
y = 2x + 3
	Work out the coordinates of A.

(.................., ..................)
(Total 3 marks)

16.	By eliminating x, find the solutions to the simultaneous equations
	x – 2y = 1
x2 + y2 = 13

x = ………..,   y = ……….
or      x = ………..,   y = ……….
(Total 7 marks)

17.	By eliminating y, find the solutions to the simultaneous equations
	y – 2x = 3
	x2 + y2 = 18
x =……………………. y =…………………….
or x =……………………. y =…………………….
(Total 7 marks)