Selina Concise Class 9 Maths Chapter 6 Simultaneous Equations Exercise 6B Solutions
EXERCISE – 6B
For solving each pair of equations, in this exercise use the method of elimination by equating coefficients
(i) 13 + 2y = 9x
3y = 7x
Solution:
Given equations are,
13 + 2y = 9x —– (i)
3y = 7x —– (ii)
Multiply equation (i) by 3 and equation (ii) by 2, we get
13×3 + 3×2y = 3×9x
2×3y = 2×7x
39 + 6y = 27x
6y = 14x
(-) (-) –
_____________
39 = 13x
39/13 = x
∴ 3 = x
∴ x = 3 Put in equation (i)
We get, 13 + 2y = 9×3
13 + 2y = 27
2y = 27 – 13
2y = 14
y = 14/2
y = 7
∴ The solution is x = 3 and y = 7.
(2) 3x – y = 23
(x/3) + (y/3) = 4
Solution:
Given equations are –
3x – y = 23 —– (i)
x/3 + y/4 = 4 —- (ii)
From equation (ii),
x/3 + y/4 = 4
4x + 3y/12 = 4
4x + 3y = 12×4
4x + 3y = 48 —– (iii)
Multiply equation (i) by 3,
3 × 3x – 3y = 3 × 23
9x – 3y = 69 —– (iv)
From equation (iii) and (iv),
9x – 3y = 69
4x + 3y = 48
(-) (-)
____________
13x = 117
x = 117/13
x = 9
Put x = 9 in equation (i), we get,
3 × 9 – y = 23
27 – y = 23
-y = 23 – 27
-y = -4
y = 4
∴ The solution is x = 9 and y = 4.
(3) (5y/2) – (x/3) = 8
(y/2) + (5x/3) = 12
Solution:
Given equations are
(5y/2) – (x/3) = 8
3×5y-2x/6 = 8
15y – 2x/6 = 8
15y – 2x = 6 × 8
15y – 2x = 48 —– (i)
(y/2) + (5x/3) = 12
3y + 2×5x/6 = 12
3y + 10x = 6 × 12
3y + 10x = 72 —– (ii)
Multiply equation (i) by 5,
5 × 15y – 5 × 2x = 5×48
75y – 10x = 240 —– (iii)
From equations (iii) and (ii), we get,
75y – 10x = 240
3y + 10x = 72
___________________
78y = 312
y = 312/78
y = 4
Put, y = 4 in equation (i), we get
15y – 2x = 48
15 × 4 – 2x = 48
60 – 2x = 48
-2x = 48 – 60
-2x = -12
x = -12/-2
x = 6
∴ The solution is x = 6 and y = 4
(4) 1/5 (x – 2) = 1/4 (1 – y)
26x + 3y + 4 = 0
Solution:
Given, equations are
1/5 (x – 2) = 1/4 (1 – y)
4 (x – 2) = 5 (1 – y)
4x – 8 = 5 – 5y
4x + 5y – 8 – 5 = 0
4x + 5y – 13 = 0
4x + 5y = 13 —– (i)
26x + 3y = -4 —– (ii)
Multiply equation (i) by 3 and equation (ii) by 5, we get
3 × 4x + 3 × 5y = 3 × 13
12x + 15y = 39 —- (iii)
5 × 26x + 5 × 3y = 5 × (-4)
130x + 15y = -20 —- (iv)
From equations (iii) and (iv), we get,
130x + 15y = -20
12x + 15y = 39
(-) (-) (-)
_________________
118x = -59
x = -59/118
x = -1/2
Put x = -1/2 in equation (ii), we get,
26x + 3y = -4
26 × -1/2 + 3y = -4
-13 + 3y = -4
3y = -4 + 13
3y = 9
y = 9/3
y = 3
∴ The solution is x = -1/2 and y = 3
(5) y = 2x – 6
y = 0
Solution:
Given equations are
y = 2x – 6 —– (i) and
y = 0 —– (ii)
From equation (ii) –
y = 0 put in equation (i),
y = 2x – 6
0 = 2x – 6
6 = 2x
6/2 = x
3 = x
∴ x = 3
The solution is x = 3 and y = 0
(6) (x-y)/6 = 2 (4 – x)
2x + y = 3 (x – 4)
Solution:
(x-y)/6 = 2 (4 – x)
x – y = 6 × 2 (4 – x)
x – y = 12 (4 – x)
x – y = 48 – 12x
x + 12x – y = 48
13x – y = 48 —– (i)
2x + y = 3 (x – 4)
2x + y = 3x – 12
2x – 3x + y = – 12
– x + y = -12
Multiply by (-1) on both sides,
x – y = 12 —– (ii)
Subtracting equations (i) and (ii), we get,
13x – y = 48
x – y = 12
(-) (+) (-)
______________
12x = 36
x = 36/12
x = 3
Put x = 3 in equation (ii), we get,
3 – y = 12
-y = 12 – 3
-y = 9
y = – 9
The solution is x = 3 and y = -6
(7) 3 – (x – 5) = y + 2
2 (x + y) = 4 – 3y
Solution:
Given equations are –
3 – (x – 5) = y + 2
3 – x + 5 = y + 2
-x + 8 = y + 2
– x – y = 2 – 8
– x – y = – 6
Multiply by (-1) on both side,
x + y = 6 —— (i)
2 (x + y) = 4 – 3y
2x + 2y = 4 – 3y
2x + 2y + 3y = 4
2x + 5y = 4 —- (ii)
Multiply equation (i) by 2, we get,
2x + 2y = 2 × 6
2x + 2y = 12 —- (iii)
Subtracting equation (ii) and (iii), we get,
2x + 2y = 12
2x + 5y = 4
(-) (-) (-)
______________
-3y = 8
y = -8/3
Put y = -8/3 in equation (ii), we get,
2x + 5y = 4
2x + 5 × (-8/3) = 4
2x – 40/3 = 4
2x = 4 + 40/3
= 12 + 40/3
2x = 52/3
x = 52/2×3
x = 52/6
x = 26/3
∴ The solution is x = 26/3 and y = -8/3
(8) 2x – 3y – 3 = 0
(2x/3) + 4y + 1/2 = 0
Solution:
Given equations are –
2x – 3y – 3 = 0
2x – 3y = 3 —– (i)
and (2x/3) + 4y = -1/2
Multiply by 6 on both sides,
6 × 2x/3 + 6 × 4y = 6 × (- 1/2)
2 × 2 + 24y = -3
4x + 24y = -3 —- (ii)
Multiply equation (i) by 2, we get,
2 × 2x – 2 × 3y = 2 × 3
4x – 6y = 6 —– (iii)
Subtracting equation (ii) and (iii), we get,
4x + 24y = -3
4x – 6y = 6
(-) (+) (-)
____________
30y = -9
y = -9/30
y = -3/10
Put, y = -3/10 in equation (ii), we get
4x + 24y = -3
4x + 24 (-3/10) = -3
4x – 72/10 = -3
4x = -3 + 72/10
= -30+72/10
= 42/10
4x = 42/10
x = 42/10×4
x = 21/10×2
x = 21/20
∴ The solution is x = 21/20 and y = -3/10
(9) 13x + 11y = 70
11x + 13y = 74
Solution:
Given equation are –
13x + 11y = 70 —- (i) and
11x + 13y = 74 —– (ii)
Adding equations (i) and (ii), we get
13x + 11y = 70
11x + 13y = 74
(+) (+) (+)
______________
24x + 24y = 144
Dividing by 24 on both sides,
24x/24x + 24y/24 = 144/24
x + y = 6 —- (iii)
Subtracting equation (i) and (ii),
13x + 11y = 70
11x + 13y = 74
(-) (-) (-)
_________________
2x – 2y = -4
Dividing 2 on both sides,
2x/2 – 2y/2 = -4/2
x – y = -2 —– (iv)
Now, solving equations (iii) and (iv),
Adding equation (iii) and (iv), we get,
x + y = 6
x – y = -2
___________
2x = 4
x = 4/2
x = 2
Put, x = 2 in equation (iv), we get,
2 – y = -2
-y = -2 -4
-y = -4
y = 4
∴ The solution is x = 2 and y = 4.
(10) 41x + 53y = 135
53x + 41y = 147
Solution:
Given equations are –
41x + 53y = 135 —– (i)
and 53x + 41y = 147 —– (ii)
Adding equations (i) and (ii),
41x + 53y = 135
53x + 41y = 147
(+) (+) (+)
_________________
94x + 94y = 282
Dividing 94 on both sides,
94x/94 + 94y/94 = 282/94
x + y = 3 —– (iii)
Subtracting equations (i) and (ii),
41x + 53y = 135
53x + 41y = 147
(-) (-) (-)
__________________
-12x + 12y = -12
Dividing 12 on both sides,
-12x/12 + 12y/12 = -12/12
– x + y = -1 —– (iv)
Now, solving equations (iii) and (iv),
Adding equations (iii) and (iv),
x + y = 3
– x + y = -1
____________
2y = 2
y = 2/2
y = 1
Put y = 1 in equation (iv), we get
-x + 1 = -1
-x = -1-1
-x = -2
x = 2
The solution is x = 2 and y = 1
(11) If 2x + y = 23 and 4x – y = 19
Find the values of x – 3y and 5y – 2x
Solution:
Given equations are –
2x + y = 23 —- (i) and
4x – y = 19 —– (ii)
Adding equations (i) and (ii), we get
2x + y = 23
4x – y = 19
___________
6x = 42
x = 42/6
x = 7
Put, x = 7 in equation (ii), we get
4x – y = 19
4 × 7 – y = 19
28 – y = 19
-y = 19 – 28
-y = -9
y = 9
Now, we have to find the value of
x – 3y and 5y – 2x
First, we have to find –
x – 3y = 7 – 3 × 9
= 7 – 27
x – 3y = -20
and now, we have to find –
5y – 2x = 5 × 9 – 2 × 7
= 45 – 14
5y – 2x = 31
(12) If 10y = 7x – 4 and 12x + 18y = 1,
Find the values of 4x + 6y and 8y – x.
Solution:
Given equations are –
10y = 7x – 4 —- (i)
and 12x + 18y = 1 —- (ii)
First we have to arrange equation (i),
-7x + 10y = -4 —- (iii)
Multiply education (ii) by 7 and equation (iii) by 12, we get,
7 × 12x + 7 × 18y = 7 × 1
84x + 126y = 7 —– (iv)
and 12 × (-7x) + 12 × 10y = 12 × (-4)
– 84x + 120y = -48 —- (v)
Now, solving equations (iv) and (v),
Adding equations (iv) and (v), we get,
84x + 126y = 7
-84x + 120y = -48
_________________
246y = -41
y = -41/246
y = -1/6
Put, y = -1/6 in equation, we get
10y = 7x – 4
10 × 1/6 = 7x – 4
-5/3 = 7x – 4
Multiply by 3 on both sides
3 × -5/3 = 3 × 7x – 3 × 4
-5 = 21x – 12
-5 + 12 = 21x
7 = 21x
7/21 = x
∴ x = 1/3
∴ The solution is x = 1/3 and y = -1/6
(13) Solve for x and y:
(i) (y + 7)/5 = (2y – x)/4 + 3x – 5
(7 – 5x)/2 + (3 – 4y)/6 = 5y – 18
Solution:
Given equations are –
(y+7)/5 = (2y-x)/4 + 3x – 5
Multiply by 20 on both sides,
20 × (y + 7)/5 = 20 × (2y – x)/4 + (3x – 5) × 20
4 × (y + 7) = 5 × (2y – x) + 60x – 100
4y + 28 = 10y – 5x + 60x – 100
4y – 10y + 28 = + 55x – 100
4y – 10y + 28 + 100 – 55x = 0
-6y – 55x + 128 = 0
-55x – 6y + 128 = 0
-55x – 6y = -128
55x + 6y = -128
55x + 6y = 128 —– (i)
(7-5x)/2 + (3 – 4y)/6 = 5y – 18
Multiply by 6 on both sides,
6× (7-5x)/2 + 6 × (3-4y)/6 = 6 (5y – 18)
3 (7 – 5x) + (3 – 4y) = 30y – 108
21 – 15x + 3 – 4y = 30y – 108
-15x – 4y – 30y + 24 + 108 = 0
-15x – 34y + 132 = 0
-15x – 34y = -132
Multiply by (-1) on both side,
15x + 34y = 132 —– (ii)
Now, Solving equations (i) and (ii),
Multiply equation (i) by 3, we get,
3 × 55x + 3 × 6y = 3 × (-128)
165x + 18y = -384 —– (iii)
Multiply equation (ii) by 11, we get
11 × 15x + 11 × 34y = 11 × 132
165x + 374y = 1452 —- (iv)
Subtracting equations (iii) and (iv), we get,
165x + 18y = 384
165x + 374y = 1,452
(-) (-) (-)
____________________
-356y = -1,068
y = -1068/356
y = 3
Put y = 3 in equation (i), we get,
55x + 6y = 128
55x + 6 × 3 = 128
55x + 18 = 128
55x = 128 – 8
55x = 110
x = 110/55
x = 2
∴ The solution is x = 2 and y = 3.
(ii) 4x = 17 – (x – y)/8
2y + x = 2 + (5y+2)/3
Solution:
Given equations are –
4x = 17 – (x – y)/8
4x = 136 – (x – y)/8
4x = 136 – x + y/8
8 × 4x = 136 – x + y
32x = 136 – x + y
32x + x = 136 + y
33x – y = 136 —- (i)
2y + x = 2 + (5y + 2)/3
2y + x = 6+5y+2/3
3 × (2y + x) = 6 + 5y + 2
6y + 3x = 8 + 5y
6y – 5x + 3x = 8
y + 3x = 8
3x + y = 8 —– (ii)
Now, solving equations (i) and (ii),
Adding equations (i) and (ii) we get,
33x – y = 136
3x + y = 8
_____________
36x = 144
x = 144/36
x = 4
Put x = 4 in equation (i), we get,
33x – y = 136
33 × 4 – y = 136
132 – y = 136
-y = 136 – 132
-y = 4
y = -4
∴ The solution is x = 4 and y = -4
(14) Find the value of m, if x = 2, y = 1, is a solution of the equation 2x + 3y = m
Solution:
Given: x = 2 and y = 1
∴ Also given equation is –
2x + 3y = m
Put x = 2 and y = 1
2 × 2 + 3 × 1 = m
4 + 3 = m
7 = m
∴ m = 7
∴ The value of m = 7
(16) The value of expression mx – ny is 3 when x = 5 and y = 6. And it’s value is 8 when x = 6 and y = 5. Find the values of m and n.
Solution:
Given that,
mx – ny = 3
When x = 5 and y = 6
∴ m (5) – n (6) = 3
5m – 6n = 3 —- (i)
Also, mx – ny = 8
If x = 6 and y = 5
∴ m (6) – n (6) = 8
6m – 5n = 8 —- (ii)
Now, solving equations (i) and (ii), multiply equation (i) by 6, we get,
6 × 5m – 6 × 6n = 6 × 3
30m – 36n = 18 —– (iii)
Multiply equation (ii) by 5, we get,
5 × 6m – 5 × 5n = 5 × 8
30m – 25n = 40 —- (iv)
Subtracting equations (iii) and (iv), we get,
30m – 36n = 18
30m – 25n = 40
(-) (+) (-)
________________
-11n = -22
n = -22/-11
n = 2
Put n = 2 in equation (i), we get
5m – 6 × 2 = 3
5m – 12 = 3
5m = 3 + 12
5m = 15
m = 15/5
m = 3
∴ The value of m = 3 and n = 2
(17) Solve:
11 (x – 5) + 10 (y – 2) + 54 = 0
7 (2x – 1) + 9 (3y – 1) = 25
Solution:
Given equations are –
11 (x – 5) + 10 (y – 2) + 54 = 0
7 (2x – 1) + 9 (3y – 1) = 25
11 (x – 5) + 10 (y – 2) = -54
11x – 55 + 10y – 20 = -54
11x + 10y – 75 = -54
11x + 10y = -54 + 75
11x + 10y = 21 —– (i)
7 (2x – 1) + 9 (3y – 1) = 25
14x – 7 + 27y – 9 = 25
14x + 27y – 16 = 25
14x + 27y = 25 + 16
14x + 27y = 41 —- (ii)
Now, solving equations (i) and (ii),
Multiplying equation (i) by 14, we get,
14 × 11x + 14×10y = 14×21
154x + 140y = 294 —– (iii)
Multiplying equation (ii) by 11, we get,
11×14x + 11×27y = 11×41
154x + 297y = 451 —– (iv)
Subtracting equations (iii) and (iv), we get
154x + 140y = 294
154x + 297y = 451
(-) (-) (-)
________________
-157y = -157
y = -157/-157
y = 1
Put y = 1 in equation (i), we get,
11x + 10y = 21
11x + 10×1 = 21
11x + 10 = 21
10x = 21 – 11
10x = 10
x = 10/10
x = 1
∴ The solution is x = 1 and y = 1
(18) Solve: (7+x)/5 – (2x-y)/4 = 3y – 5
(5y-7)/2 + (4x-3)/6 = 18 – 5x
Solution:
Given equations are –
(7+x)/5 – (2x-y)/4 = 3y – 5
Multiply by 20 on both sides,
20× (7+x)/5 – 20 × (2x-y)/4 = 20 (3y – 5)
4 (7+x) – 5 × (2x – y) = 60y – 100
28 + 4x – 10x + 5y = 60y – 100
-6x + 5y – 60y + 28 + 100 = 0
-6x – 55y + 128 = 0
-6x – 55y = -128
Multiply by (-1) on both sides,
6x + 55y = 128 —– (i)
(5y-7)/2 + (4x-3)/6 = 18 – 5x
Multiply by 6 on both sides,
6 × (5y-7)/2 + 6 × (4x-3)/6 = 6 × (18 – 5x)
3 (5y – 7) + (4x – 3) = 108 – 30x
15y – 21 + 4x – 3 = 108 – 30x
15y + 4x + 30x – 24 – 108 = 0
15y + 34x – 132 = 0
15x + 34x = 132
34x + 15y = 132 —– (ii)
Solving equations (i) and (ii),
Multiplying equation (i) by 3, we get,
3 × 6x + 3 × 55y = 3 × 128
18x + 165y = 384 —– (iii)
Multiplying equation (ii) by 11, we get,
11 × 34x + 11 × 15y = 11 × 132
374x + 165y = 1,452 —- (iv)
Solving equation (iii) and (iv), we get,
subtracting equation (iii) and (iv),
18x + 165y = 384
374x + 165y = 1,452
(-) (-) (-)
___________________
-356x = 1,068
x = -1,068/-356
x = 3
Put x = 3 in equation (i),
6x + 55y = 128
6 × 3 + 55y = 128
18 + 55y = 128
55y = 128 – 18
55y = 110
y = 110/55
y = 2
∴ The solution is x = 3 and y = 2
(19) Solve: 4x + (x-y)/8 = 17
2y + x – (5y+2)/3 = 2
Solution:
Given equations are –
4x + (x – y)/8 = 17
Multiply by 8 on both sides,
8 × 4x + 8 × (x – y) = 8 × 17
32x + x – y = 136
33x – y = 136 —– (i)
2y + x – (5y+2)/3 = 3
Multiply by 3 on both sides,
3 × 2y + 3x – 3 × (5y+2)/3 = 2 × 3
6y + 3x – 5y – 2 = 6
y + 3x = 6 + 2
y + 3x = 8
3x + y = 8 —— (ii)
Now, solving equations (i) and (ii),
Adding equations (i) and (ii),
we get,
33x + y = 136
3x + y = 8
_____________
36x = 144
x = 144/36
x = 4
Put x = 4 in equation (i), we get,
33x – y = 136
33 × 4 – y = 136
132 – y = 136
-y = 136 – 132
-y = +4
y = -4
∴ The solution is x = 4 and y = -4
Here is your solution of Selina Concise Class 9 Maths Chapter 6 Simultaneous Equations Exercise 6B
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