Selina Concise Class 9 Maths Chapter 6 Simultaneous Equations 6B Solutions

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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Updated: February 22, 2022 — 3:08 pm

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