Name:
Exam Style Questions
Simultaneous Equations
Equipment needed: Calculator, pen
Guidance
1. Read each question carefully before you begin answering it.
2. Check your answers seem right.
3. Always show your workings
Video Tutorial
www.corbettmaths.com/contents
Video 295
Answers and Video Solutions
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1.
Solve the simultaneous equations
5x + 3y = 41
2x + 3y = 20
5x + 3y = 41
2x + 3y = 20
3y = 41 - 5x
3y = 20 - 2x
41 - 5x = 20 - 2x
41 - 20 = -2x + 5x
21 = 3x
x = 21/3 = 7
3y = 20 - 2x
3y = 20 - 2(7)
= 20 - 14
3y = - 6
y = -6/3 = -2
2.
7
-2
x = ......................... y = ..........................
(3)
Solve the simultaneous equations
5x + y = 11
3x − y = 9
5x + y = 11
3x - y = 9
5x + y = 11
5(5/2) + y = 11
5x - 11 = -y
-y = 9 - 3x
5x - 11 = 9 - 3x
25/2 + y = 11
y = 11 - 12.5
= - 1.5
5x + 3x = 9 + 11
8x = 20
x = 20/8 = 10/4 = 5/2
5/2
-1.5
x = ......................... y = ..........................
(3)
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3.
Solve the simultaneous equations
x + 7y = 64
x + 3y = 28
x + 7y = 64
x + 3y = 28
x = 64 - 7y
x = 28 - 3y
64 - 7y = 28 - 3y
64 - 28 = -3y + 7y
36 = 4y
36/4 = y
y=9
x + 3(9) = 28
x + 27 = 28
x = 28 - 27 = 1
9
1
x = ......................... y = ..........................
(3)
4.
Solve the simultaneous equations
4x − 4y = 24
x − 4y = 3
4x - 4y = 24
x - 4y = 3
-4y = 24 - 4x
-4y = 3 - x
24 - 4x = 3 - x
24 - 3 = -x + 4x
21 = 3x
21/3 = x
x=7
-4y = 3 - x
=3-7
-4y = - 4
4y = 4
y = 4/4 = 1
1
7
x = ......................... y = ..........................
(3)
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5.
Solve the simultaneous equations
2x + 4y = 14
4x − 4y = 4
2x + 4y = 14
4x - 4y = 4
4y = 14 - 2x
-4y = 4 - 4x
4y = (4 - 4x)/-1
= -4 + 4x
14 - 2x = -4 + 4x
14 + 4 = 4x + 2x
2(3) + 4y = 14
6 + 4y = 14
4y = 14 - 6
4y = 8
y = 8/4 = 2
18 = 6x
18/6 = x
x=3
6.
2
3
x = ......................... y = ..........................
(3)
David buys 2 scones and 2 coffees in a shop and the cost is £18.
Ellie buys 3 scones and 2 coffees in the same shop and they cost £22.
Form two equations and solve to find the cost of each scone and each coffee.
2s + 2c = $18
3s + 2c = $22
2(4) + 2c = $18
8 + 2c = $18
2c = 18 - 2s
2c = 22 - 3s
2c = 18 - 8
= 10
18 - 2s = 22 - 3s
18 - 22 = -3s + 2s
-4 = -s
s = $4
c = 10/2 = $5
4
5
Scone = £......................... Coffee = £..........................
(4)
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7.
Alan and Connor have £6.70 in total.
Alan has £1.70 more than Connor.
Let a be the amount of money Alan has.
Let c be the amount of money Connor has.
Set up a pair of simultaneous equations and solve to find out how much each
person has.
A = 2.50 + 1.70
A + C = $6.70
= $4.20
C + 1.70 + C = $6.70
2C + 1.70 = $6.70
2C = 6.70 - 1.70
=5
C = 5/2 =$2.50
$4.20
$2.50
Alan = ......................... Connor = ..........................
(3)
8.
Solve the simultaneous equations
6x + y = − 2
6x − 3y = 14
6x + y = -2
6x - 3y = 14
6x = -2 - y
6x = 14 + 3y
-2 - y = 14 + 3y
6x = -2 - (-4)
= -2 + 4
=2
x = 6/2 = 3
-2 - 14 = 3y + y
-16 = 4y
-16/4 = y
y = -4
-4
3
x = .........................
y = ..........................
(3)
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9.
Solve the simultaneous equations
2x + 4y = 26
3x − y = 4
2x + 4y = 26
3x - y = 4
4y = 26 - 2x
-y = 4 - 3x
-4y = 4(4 - 3x)
= 16 - 12x
4y = (16 - 12x)/-1
3x - y = 4
3(3) - y = 4
9-y=4
9-4=y
y=5
= -16 + 12x
-16 + 12x = 26 - 2x
10.
-16 - 26 = -2x - 12x
-42 = - 14x
42 = 14x
42/14 = x
x=3
Solve the simultaneous equations
3
5
x = ......................... y = ..........................
(3)
3x + 2y = 16
2x − 3y = 2
Do not use trial and improvement
3x + 2y = 16
2x - 3y = 2
3x = 16 - 2y
x = 1 + 3(2)/2
2x = 2 + 3y
= 1 + 6/2
x = (2 + 3y)/2
=1+3=4
= 1 + 3y/2
3x = 3(1 + 3y/2)
= 3 + 9y/2
16 - 2y = 3 + 9y/2
16 - 3 = 9y/2 + 2y
13 = 9y/2 + 4y/2
= 13y/2
2(13) = 13y
26 = 13y
26/13 = y
y=2
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4
2
x = ......................... y = ..........................
(4)
11.
Solve the simultaneous equations
3x − 2y = 14
x + 2y = 10
3x - 2y = 14
x + 2y = 10
-2y = 14 - 3x
2y = 10 - x
-2y = -1(10 - x)
= -10 + x
14 - 3x = -10 + x
14 + 10 = x + 3x
x + 2y = 10
6 + 2y = 10
2y = 10 - 6
=4
y = 4/2 = 2
24 = 4x
24/4 = x
x=6
12.
2
6
x = ......................... y = ..........................
(3)
Solve the simultaneous equations
3x + 5y = 1
2x − 3y = 7
3x + 5y = 1
2x - 3y = 7
3x = 1 - 5y
2x = 7 + 3y
x = (7 + 3y)/2
= 7/2 + 3y/2
3x = 3(7/2 + 3y/2)
= 21/2 + 9y/2
1 - 5y = 21/2 + 9y/2
1 = 21/2 + 9y/2 + 5y
1 = 21/2 + 9y/2 + 10y/2
= (21 + 9y + 10y)/2
2(1) = 21 + 9y + 10y
2 = 21 + 19y
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2 - 21 = 19y
-19 = 19y
19y = -19
y = -19/19 = -1
2x - 3(-1) = 7
2x + 3 = 7
2x = 7 - 3
=4
x = 4/2 = 2
2
-1
x = ......................... y = ..........................
(4)
13.
Solve the simultaneous equations
3x − y = 23
2x + 3y = 8
3x - y = 23
2x + 3y = 8
-y = 23 - 3x
3y = 8 - 2x
y = (23 - 3x)/-1
= -23 + 3x
3y = 3(-23 + 3x)
= -69 + 9x
3(78/11) - y = 23
234/11 - y = 23
234/11 - 23 = y
234/11 - 253/11 = y
-19/11 = y
8 - 2x = -69 + 9x
8 + 69 = 9x + 2x
78 = 11x
-19/11
78/11
x = ......................... y = ..........................
(3)
78/11 = x
14.
Solve the simultaneous equations
2y − 5x = 9
4y + 3x = 5
2y - 5x = 9
4y + 3x = 5
4y + 3x = 5
4y + 3(-1) = 5
2y = 9 + 5x
4y = 5 - 3x
4y = 2(9 + 5x)
4y - 3 = 5
4y = 5 + 3
=8
= 18 + 10x
5 - 3x = 18 + 10x
5 - 18 = 10x + 3x
-13 = 13x
-13/13 = x
y = 8/4 = 2
x = -1
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2
-1
x = .........................
y = ..........................
(3)
15.
Solve the simultaneous equations
2x + 9y = 43
3x + 2y = 7
2x + 9y = 43
3x + 2y = 7
-115 = -23y
2x = 43 - 9y
3x = 7 - 2y
x = (43 - 9y)/2
= 43/2 - 9y/2
3x = 2(43/2 - 9y/2)
= 129/2 - 27y/2
7 - 2y = 129/2 - 27y/2
7 = 129/2 - 27y/2 + 2y
7 = 129/2 - 27y/2 + 4y/2
7 = (129 - 27y + 4y)/2
2(7) = 129 - 27y + 4y
14 = 129 - 23y
14 - 129 = -23y
16.
23y = 115
y = 115/23 = 5
3x + 2(5) = 7
3x + 10 = 7
3x = 7 - 10
3x = -3
x = -3/3
= -1
5
-1
x = ......................... y = ..........................
(3)
Solve the simultaneous equations
5x − 3y = 24
2x − 4y = 4
5x - 3y = 24
2x - 4y = 4
2x - 4y = 4
2x - 4(2) = 4
5x = 24 + 3y
2x = 4 + 4y
x = (4 + 4y)/2
2x - 8 = 4
2x = 4 + 8
2x = 12
= 2 + 2y
5x = 5(2 + 2y)
= 10 + 10y
24 + 3y = 10 + 10y
x = 12/2 = 6
24 - 10 = 10y - 3y
14 = 7y
14/7 = y
y=2
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6
2
x = ......................... y = ..........................
(3)
17.
A museum sells adult tickets or child tickets.
Fozia buys 4 adult tickets and 1 child ticket for £120
Sami buys 5 adult tickets and 3 child tickets for £171
Work out the cost of each type of ticket.
Adult ticket £ ..........................
Child ticket £ ..........................
(4)
18.
Solve the simultaneous equations
4x + 3y = 7.5
3x − 5y = 10.7
x = ......................... y = ..........................
(3)
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19.
Solve the simultaneous equations
2y = 8x + 11
2x + 8y = 27
x = ......................... y = ..........................
(3)
20.
Find the coordinates of the point where the straight lines below cross.
y − 3x = 3
x − 2y = 4
(.................... , .....................)
(4)
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21.
Solve the simultaneous equations
3a + c = 8
2a − c = 7
a = ......................... c = ..........................
(3)
22.
Solve the simultaneous equations
9x − 6y = 114
5x − 9y = 30.75
x = ......................... y = ..........................
(4)
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23.
Solve the simultaneous equations
2y = x + 10
y = 2x − 7
x = ......................... y = ..........................
(3)
24.
Solve the simultaneous equations
4x − y = 17
y = x−2
x = ......................... y = ..........................
(3)
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25.
Three bananas and two pears cost £2.07
Five bananas and three pears cost £3.33
Find the cost of ten bananas and ten pears.
..........................
(4)
26.
Solve the simultaneous equations
5x + 2y = − 34
4x − 3y = − 41
x = ......................... y = ..........................
(4)
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27.
Albie is training for a marathon.
He jogs either route A or route B.
During April, he jogs route A nine times and route B five times.
Route B is 8 miles longer than route A.
In total, he jogs 89 miles in April.
In May, he will start jogging route C.
Route C is 20% longer than route B.
Work out the length of route C.
..........................miles
(6)
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28.
Solve the simultaneous equations
6x + 2y = 13c
x + 2y = − 2c
where c is a constant
Give your answers in terms of c.
x = ......................... y = ..........................
(4)
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29.
Shown below is a parallelogram.
Each side is measured in centimetres.
Work out the perimeter of the parallelogram.
..........................cm
(6)
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