Simultaneous Equations - Past Paper Questions - ANSWERS
1)
3x + 2y = 17
2x + 5y = 4
×5
×(-2)
15x + 10y = 85
-4x – 10y = -8
11x
= 77
x=7
21 + 2y = 17
2y = -4
y = -2
2)
2x – y = 10
4x + 5y = 6
×5
10x – 5y = 50
4x + 5y = 6
14x
= 56
x=4
16 + 5y = 6
5y = -10
y = -2
3)
2x – 5y = 24
7x + 8y = 33
×8
×5
16x – 40y = 192
35x + 40y =165
51x
= 357
x=7
49 + 8y = 33
8y = -16
y = -2
4)
3x – 2y = 11
2x + 5y = 1
×5
×2
15x – 10y = 55
4x + 10y = 2
19x
= 57
x=3
6 + 5y = 1
5y = -5
y = -1
5)
x + 2y = –5
3x – y = 13
×2
x + 2y = -5
6x – 2y = 26
7x
= 21
x=3
3 + 2y = -5
2y = -8
y = -4
6)
y = 2x – 3
x + 2y = 14
7)
a)
b)
c)
5a + 3c = 158·25
3a + 2c = 98
5a + 3c = 158·25
3a + 2c = 98
x + 2(2x – 3) = 14
x + 4x – 6 = 14
5x = 20
x=4
×2
×-3
y=2×4–3
y=5
10a + 6c = 316·50
-9a – 6c = -294
a
= 22·50
Adult ticket = £22·50, Child Ticket = £15·25
8)
a)
b)
c)
3n + 2b = 145
5n + 3b = 240
3n + 2b = 145
5n + 3b = 240
×5
×-3
15n + 10b = 725
-15n – 9b = 720
b=5
A breakfast costs £5
P(4,5)
67.50 + 2c = 98
2c = 30·50
c = 15·25
9)
a)
b)
c)
4p + 3g = 1·30
2p + 4g = 1·20
4p + 3g = 1·30
2p + 4g = 1·20
×(-1)
×2
-4p – 3g = -1·30
4p + 8g = 2·40
5g = 1·10
g = 0·22
4p + 0·66 = 1·30
4p = 0·64
p = 0·16
3 peaches and 2 grapefruit cost 3 × 0·16 + 2 × 0·22 = 0·92 , £0·92 or 92p
10)
a)
b)
c)
2g + 5s = 125
4g + 3s = 145
2g + 5s = 125
4g + 3s = 145
×2
×(-1)
4g + 10s = 250
-4g – 3s = -145
7s = 105
s = 15
2g + 75 = 125
2g = 50
g = 25
Gold charm costs £25, silver costs £15
11)
a)
b)
c)
6x + 2y = 3148
5x + 3y = 3022
6x + 2y = 3148
5x + 3y = 3022
×3
×(-2)
18x + 6y = 9444
-10x – 6y = -6044
8x
= 3400
x = 425
2550 + 2y = 3148
2y = 598
y = 299
2 adults and 4 children cost 2 × 425 + 4 × 299 = 2046
The group have been overcharged by £10.
12)
a)
b)
c)
24x + 6y = 60
20x + 10y = 40
24x + 6y = 60
20x + 10y = 40
×10
×(-6)
240x + 60y = 600
-120x – 60y = -240
120x
= 360
x=3
72 + 6y = 60
6y = -12
y = -2
David scores 17 × 3 + 13 × (-2) = 25
13)
a)
b)
c)
2a + 4c = 56
a + 3c = 36
2a + 4c = 56
a + 3c = 36
×(-1)
×2
-2a – 4c = -56
2a + 6c = 72
2c = 16
c=8
a + 24 = 36
a = 12
Adult ticket costs £12, child costs £8
14)
A straight line has equation y = mx + c, where m and c are constants.
a)
7 = 2m + c
b)
17 = 4m + c
c)
7 = 2m + c
×2
14 = 4m + 2c
17 = 4m + c
×(-1)
-17 = -4m – c
-3 = c
d)
gradient = 5
7 = 2m – 3
10 = 2m
m=5
15)
a)
b)
c)
14x + 60y = 344·30
21x + 40y = 368·95
14x + 60y = 344·30
21x + 40y = 368·95
×(-2)
×3
-28x – 120y = -688·60
63x + 120y = 1106·85
35x
= 418·25
x
= 11·95
167·30 + 60y = 344·30
60y = 177
y = 2·95
A car costs £11·95, a passenger £2·95
16)
a)
b)
c)
280x + 70y = 52·50
210x + 40y = 38
280x + 70y = 52·50
210x + 40y = 38
-1120x – 280y = -210
1470x + 280y = 266
350x
= 56
x = 0·16
×4
×7
44·80 + 70y = 52·50
70y = 7·70
y = 0·11
A call costs £0·16 per minute a text costs £0·11
17)
a)
b)
c)
14x + 4y = 55
13x + 6y = 54·50
14x + 4y = 55
13x + 6y = 54·50
×3
×(-2)
42x + 12y = 165
-26x – 12y = 109
16x
= 56
x = 3·50
49 + 4y = 55
4y = 6
y = 1·50
Adult fee = £3·50, Child fee = £1·50
18)
a)
b)
c)
x + y = 150
30x + 50y = 6000
x + y = 150
30x + 50y = 6000
×(-30)
-30x – 30y = 4500
30x + 50y = 6000
20y = 1500
y = 75
x + 75 = 150
x = 75
75 £30 seats and 75 £50 seats
19)
2b + 5p = 5·2
3b + 2p = 5·6
×(-2)
×5
-4b – 10p = -10·4
15b + 10p = 28
11b
= 17·6
b = 1·6
A bead is 1·6cm and a pearl is 0·4cm
3·2 + 5p = 5·2
5p = 2
p = 0·4