Simultaneous Equations Worksheet with Answers

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GCSE · Algebra

5.11 Simultaneous Equations

EDEXCEL 1MA1 · Calculator allowed · about 100 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

1

Solve the simultaneous equations:
3x + y = 11
x - y = 5

(a)Solve the simultaneous equations:
3x + y = 11
x - y = 5(3)

(Total for Question 1 is 3 marks)

2

Solve the simultaneous equations:
y = x + 2
2x + y = 11

(a)Solve the simultaneous equations:
y = x + 2
2x + y = 11(3)

(Total for Question 2 is 3 marks)

3

Solve the simultaneous equations:
2x + 3y = 16
x + y = 6

(a)Solve the simultaneous equations:
2x + 3y = 16
x + y = 6(4)

(Total for Question 3 is 4 marks)

4

Solve the simultaneous equations:
3x + 2y = 13
5x + 3y = 20

(a)Solve the simultaneous equations:
3x + 2y = 13
5x + 3y = 20(4)

(Total for Question 4 is 4 marks)

5

Tia and Ben go to the cinema on different days.
Tia buys 2 adult tickets and 3 child tickets. She pays a total of £33.50.
Ben buys 3 adult tickets and 1 child ticket. He pays a total of £31.00.
Let a = the cost of an adult ticket in pounds and c = the cost of a child ticket in pounds.

(a)Write down a pair of simultaneous equations in terms of a and c.(2)

(b)Solve your equations to find the cost of an adult ticket and the cost of a child ticket.(3)

(Total for Question 5 is 5 marks)

6

Which pair of values satisfies both of the simultaneous equations 3x + y = 10 and x - y = 2?

(a)Circle the correct answer.(1)

A) x = 2, y = 4
B) x = 3, y = 1
C) x = 4, y = -2
D) x = 1, y = 7

(Total for Question 6 is 1 mark)

7

Show that x = 5, y = -2 satisfies both of the simultaneous equations:
3x + 2y = 11
2x - y = 12

(a)Show that x = 5, y = -2 satisfies both of the simultaneous equations:
3x + 2y = 11
2x - y = 12(2)

(Total for Question 7 is 2 marks)

8

Solve the simultaneous equations:
4x + y = 5
2x - 3y = 13

(a)Solve the simultaneous equations:
4x + y = 5
2x - 3y = 13(3)

(Total for Question 8 is 3 marks)

9

The graphs of y = 2x - 1 and x + y = 5 are drawn on the grid below.

(a)Write down the coordinates of the point where the two lines intersect.(1)

(b)Show algebraically that x = 2, y = 3 is the solution to the simultaneous equations y = 2x - 1 and x + y = 5.(2)

(Total for Question 9 is 3 marks)

10

The sum of two numbers is 17. Three times the larger number is 19 more than the smaller number.
Let x = the larger number and y = the smaller number.

(a)Form a pair of simultaneous equations, using x for the larger number and y for the smaller number.(2)

(b)Solve your equations to find the two numbers.(2)

(Total for Question 10 is 4 marks)

11

The diagram shows a rectangle with length l cm and width w cm.
The perimeter of the rectangle is 54 cm.
The length is 3 cm more than twice the width.

(a)Form a pair of simultaneous equations in terms of l and w.(2)

(b)Solve your equations to find the length and width of the rectangle.(3)

(Total for Question 11 is 5 marks)

12

Solve the simultaneous equations:
1.5x + y = 9
0.5x + y = 5

(a)Solve the simultaneous equations:
1.5x + y = 9
0.5x + y = 5(3)

(Total for Question 12 is 3 marks)

13

A student tries to solve the simultaneous equations:
2x + 3y = 6
4x + 6y = 15
Explain why this pair of simultaneous equations has no solution.

(a)Explain why this pair of simultaneous equations has no solution.(2)

(Total for Question 13 is 2 marks)

14

Two mobile phone companies charge for calls as follows.
Company A: a fixed monthly charge of £15 plus 5p per minute of calls.
Company B: a fixed monthly charge of £9 plus 8p per minute of calls.
Let C = the total monthly cost in pounds and m = the number of minutes of calls used in a month.

(a)Write down an equation for C in terms of m for each company.(2)

(b)Find the number of minutes of calls, m, for which both companies charge the same monthly amount, and state this common cost.(3)

(Total for Question 14 is 5 marks)

15

A boat travels 36 km downstream in 2 hours. The same boat travels the same 36 km upstream in 3 hours.
Let s = the speed of the boat in still water in km/h and w = the speed of the water current in km/h.
(Downstream speed = s + w. Upstream speed = s - w.)

(a)Write down two equations connecting s and w.(2)

(b)Solve your equations to find the speed of the boat in still water and the speed of the current.(3)

(Total for Question 15 is 5 marks)

16

Solve the simultaneous equations:
y = x + 2
y = x² - 4x + 6
Give your answers as coordinate pairs (x, y).

(a)Solve the simultaneous equations:
y = x + 2
y = x² - 4x + 6
Give your answers as coordinate pairs (x, y).(5)

(Total for Question 16 is 5 marks)

17

Solve the simultaneous equations:
(x + y) / 2 = 5
2x - y = 2

(a)Solve the simultaneous equations:
(x + y) / 2 = 5
2x - y = 2(4)

(Total for Question 17 is 4 marks)

18

A circle has equation x² + y² = 25.
A line has equation y = x + 1.
Find the coordinates of the points where the line intersects the circle.

(a)Find the coordinates of the points where the line intersects the circle.(6)

(Total for Question 18 is 6 marks)

19

The sum of two numbers is 15. The sum of the squares of the two numbers is 113.
Let x and y be the two numbers.

(a)Form a pair of simultaneous equations, using x and y for the two numbers.(2)

(b)Solve your equations to find the two numbers.(3)

(Total for Question 19 is 5 marks)

20

Five years ago, Kwame was three times as old as his daughter Ama.
In eleven years' time, Kwame will be twice as old as Ama will be then.
Let k = Kwame's present age and a = Ama's present age.

(a)Form a pair of simultaneous equations in terms of k and a.(2)

(b)Solve your equations to find Kwame's and Ama's present ages.(3)

(Total for Question 20 is 5 marks)

Mark scheme · 5.11 Simultaneous Equations

Question 1

(a) M1 adds the two equations to eliminate y (or equivalent valid elimination)
(a) A1 x = 4 cao
(a) A1 y = -1 (ft from their x, both values required)
(a) Answer: x = 4, y = -1

Question 2

(a) M1 substitutes y = x + 2 into the second equation
(a) A1 x = 3 cao
(a) A1 y = 5 (ft from their x)
(a) Answer: x = 3, y = 5

Question 3

(a) M1 multiplies x + y = 6 by 2 (or equivalent) to match a coefficient
(a) M1 subtracts the equations correctly to eliminate a variable
(a) A1 x = 2 cao
(a) A1 y = 4 cao
(a) Answer: x = 2, y = 4

Question 4

(a) M1 multiplies both equations to equate the coefficients of x or y (e.g. x3 and x2)
(a) M1 subtracts the equations correctly to eliminate a variable
(a) A1 x = 1 cao
(a) A1 y = 5 cao
(a) Answer: x = 1, y = 5

Question 5

(a) B1 2a + 3c = 33.5 oe
(a) B1 3a + c = 31 oe
(a) Answer: 2a + 3c = 33.50 and 3a + c = 31.00
(b) M1 multiplies 3a + c = 31 by 3 to match the coefficient of c, and eliminates c
(b) A1 a = 8.50 cao
(b) A1 c = 5.50 cao
(b) Answer: adult ticket = £8.50, child ticket = £5.50

Question 6

(a) B1 B cao
(a) Answer: B

Question 7

(a) B1 substitutes into 3x + 2y and shows 3(5) + 2(-2) = 15 - 4 = 11
(a) B1 cso, substitutes into 2x - y and shows 2(5) - (-2) = 10 + 2 = 12
(a) Answer: Both equations are satisfied (shown)

Question 8

(a) M1 rearranges y = 5 - 4x and substitutes into the second equation (or equivalent elimination)
(a) A1 x = 2 cao
(a) A1 y = -3 cao
(a) Answer: x = 2, y = -3

Question 9

(a) B1 (2, 3) cao
(a) Answer: (2, 3)
(b) M1 substitutes x = 2 into both equations
(b) A1 cso, shows y = 2(2) - 1 = 3 and 2 + 3 = 5, so both equations are satisfied
(b) Answer: Confirmed: y = 2(2) - 1 = 3 and 2 + 3 = 5

Question 10

(a) B1 x + y = 17 oe
(a) B1 3x = y + 19 oe (e.g. 3x - y = 19)
(a) Answer: x + y = 17 and 3x - y = 19
(b) M1 adds the equations to eliminate y
(b) A1 x = 9 and y = 8 (both values required, cao)
(b) Answer: x = 9, y = 8

Question 11

(a) B1 l + w = 27 oe (from 2(l + w) = 54)
(a) B1 l = 2w + 3 oe
(a) Answer: l + w = 27 and l = 2w + 3
(b) M1 substitutes l = 2w + 3 into l + w = 27
(b) A1 w = 8 cao
(b) A1 l = 19 cao
(b) Answer: length = 19 cm, width = 8 cm

Question 12

(a) M1 subtracts the equations to eliminate y
(a) A1 x = 4 cao
(a) A1 y = 3 cao
(a) Answer: x = 4, y = 3

Question 13

(a) B1 multiplies the first equation by 2 to get 4x + 6y = 12 (or otherwise shows the coefficients of x and y are proportional)
(a) B1 correct conclusion: since 4x + 6y cannot equal both 12 and 15, the equations are inconsistent (the lines are parallel), so there is no solution
(a) Answer: No solution, because the equations are inconsistent (the lines are parallel)

Question 14

(a) B1 C = 15 + 0.05m oe
(a) B1 C = 9 + 0.08m oe
(a) Answer: C = 15 + 0.05m and C = 9 + 0.08m
(b) M1 sets 15 + 0.05m = 9 + 0.08m
(b) M1 rearranges correctly to solve for m
(b) A1 m = 200 minutes and C = £25 (both required, cao)
(b) Answer: m = 200 minutes, common cost = £25

Question 15

(a) B1 s + w = 18 oe (from 36 / 2)
(a) B1 s - w = 12 oe (from 36 / 3)
(a) Answer: s + w = 18 and s - w = 12
(b) M1 adds the two equations to eliminate w
(b) A1 s = 15 cao
(b) A1 w = 3 cao
(b) Answer: s = 15 km/h, w = 3 km/h

Question 16

(a) M1 sets x + 2 = x² - 4x + 6
(a) M1 rearranges correctly to x² - 5x + 4 = 0
(a) M1 factorises or otherwise solves to get x = 1 and x = 4
(a) A1 y = 3 and y = 6 (ft from their x values)
(a) A1 states both coordinate pairs correctly, (1, 3) and (4, 6)
(a) Answer: (1, 3) and (4, 6)

Question 17

(a) M1 multiplies the first equation by 2 to get x + y = 10
(a) M1 adds the equations to eliminate y
(a) A1 x = 4 cao
(a) A1 y = 6 cao
(a) Answer: x = 4, y = 6

Question 18

(a) M1 substitutes y = x + 1 into the circle equation
(a) M1 correctly expands (x + 1)² to get x² + 2x + 1
(a) M1 rearranges to a three-term quadratic equal to zero, x² + x - 12 = 0 oe
(a) M1 factorises or otherwise solves to get x = -4 and x = 3
(a) A1 y = -3 and y = 4 (ft from their x values)
(a) A1 states both coordinate pairs correctly, (-4, -3) and (3, 4)
(a) Answer: (-4, -3) and (3, 4)

Question 19

(a) B1 x + y = 15 oe
(a) B1 x² + y² = 113 oe
(a) Answer: x + y = 15 and x² + y² = 113
(b) M1 rearranges to y = 15 - x and substitutes into the second equation, expanding correctly
(b) M1 rearranges to a three-term quadratic equal to zero, x² - 15x + 56 = 0 oe, and solves to get x = 7 and x = 8
(b) A1 states the two numbers are 7 and 8 (cao)
(b) Answer: 7 and 8

Question 20

(a) B1 k - 5 = 3(a - 5) oe (e.g. k = 3a - 10)
(a) B1 k + 11 = 2(a + 11) oe (e.g. k = 2a + 11)
(a) Answer: k - 5 = 3(a - 5) and k + 11 = 2(a + 11)
(b) M1 rearranges both equations to make k the subject and sets them equal (or equivalent elimination)
(b) A1 a = 21 cao
(b) A1 k = 53 cao
(b) Answer: Kwame is 53, Ama is 21

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