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Frank ICSE Mathematics Class 8 Solutions Chapter 12 Linear Equations in One Variable
Welcome to NCTB Solutions. Here with this post we are going to help 8th class students for the Solutions of Frank ICSE Mathematics Class 8 Math Book, Chapter 12, Linear Equations in One Variable. Here students can easily find step by step solutions of all the problems for Linear Equations in One Variable, Exercise 12.1, 12.2, 12.3 and 12.4 Also our mathematics teacher’s are solved all the problems with easily understandable methods with proper guidance so that all the students can understand easily. Here in this post students will get chapter 12 solutions. Here in this post all the solutions are based on latest Syllabus.
Linear Equations in One Variable Exercise 12.1 Solution :
Question no – (1)
Solution :
(a) 4x- 11 = 49
or, 4x = 60
or, x = 60/4 = 15
(b) 3y/4 – 1 = 14
or, 3y/4 = 15
or, 3y = 15 × 4 = 60
∴ y = 60/3 = 20
(c) 8x/5 + 26 = 50
or, 8x/5 = 24
or, x = 24 × 5/8 = 15
(d) 5a/2 – 7 = 3
or, 5a/2 = 10
or, a = 10 × 2/5
= 4
(e) 5x – 13 = 2x + 8
or, 5x – 2x = 8 + 13
or, 3x = 21
or, x = 21/3
= 7
(f) 17x + 12 = 13x + 24
or, 17x – 13x = 24 – 12
or, 4x = 12
or, x = 12/4 = 3
(g) 7 (x + 3) = 5 (2x – 3)
or, 7x + 21 = 10x – 15
or, 10x – 7x = 21 + 15
or, 3x = 36
∴ x = 36/3
= 12
(h) 7y/3 + 11 = 9y/4 + 15
or, 7y/3 – 9y/4 = 15 – 1`1
or, y/12 = 4
or, y = 12 × 4
= 48
(i) 2m – 2/9
= 4m – 4/3
or, 4m – 2m
= 4/3 – 2/9
= 2m = 1/9
or, m = 10/9/2
= 5/9
(j) 6 (4y – 2) = 3 (5 + 3y)
or, 24y – 12 = 15 + 9y
or, 5y = 27
or, y = 27/15
= 9/5
Question no – (2)
Solution :
(a) x – 1/4 = x – 3/5
or, x/4 – 1/4 = x/5 = x/5 – 3/5
or, x/4 – x/5 = 1/4 – 3/5
or, x/20 = – 7/20
or, x = – 7
(b) 2x/3x + 1 =x – 3/5
or, x/4 – 1/4 = x/5 – 3/5
or, x/4 – x/5 = 1/4
or, 1/4 – 3/5
or, x/20 = – 7/20
or, x = – 7
(c) 2n/3x + 1 = 3/5
or, 21x+ 3 = 125 – 40x
or, 61x = 122
or, x = 122
or, x 122/61
= 2
(d) 25 – 8x/7x + 1 = 3/5
= 9x – 9 = 8x
or, 9x – 8x = 9
or, x = 9
(e) 5x – 2/7 = 2x + 1/3
= 15x – 6
= 14x + 7
or, 15 – 14x = 7
or, 15x – 14x = 7 + 6
or, x = 13
(f) 2x + 3/5 = 4x + 1/3
20x – 5 = 6x + 9
or, 14x = 14
or, x = 14/14
= 1
Question no – (3)
Solution :
(a) (x + 2 (2x – 3) = (2x + 1) (x – 2)
or, 2x2 + x – 6
= 2x2 – 3x – 2
or, x – 6 = – 3x – 2
or, x + 3x = 6 – 2
or, 4x = 4
or, 4x = 4
or, x = 4/4
= 1
(b) (6x + 5) (2x + 3) = (4x + 7) (3x + 2)
or, 12x2 + 28x + 15
= 12x2 + 29x + 14
or, 29x – 28x = 15 – 14
or, x = 1
(c) 9x – 7/3x+ 5 = 3x – 4/x + 6
or, 9x2 + 47x – 42 = 9x2 + 3x – 20
or, 47x – 42 = 3x – 20
or, 44x = 42 – 20 = 22
or, 44x = 22
or, x = 22/24 = 1/2
(d) 6x + 7/3x + 4 = 4x + 5/2x + 3
or, 12x2 + 32x + 21
= 12x2 + 31x + 20
or, 32x – 31x = 20 – 21
or, x = – 1
(e) 5x + 4/7x + 3 = 5x- 2/7x – 2
or, 35x2 + 8x – 3
= 35x2 + 18x – 8
or, 8x – 18x = – 8 + 3
or, – 10x = – 5
or, x = – 5/- 10
= 1/2
(f) x + 3/x – 2 = x – 2/x + 3
= x2 + 6x + 9
= x2 – 4x + 4
or, 6x + 4x = 4 – 9
or, 10x = – 5
or, 10x = – 5
or, x = – 5/10
= – 1/2
Question no – (4)
Solution :
(a) 4x + 1/3 + 3x + 1/2 = 3x – 7/4 + 13
or, 4x/3 + 3x/2 3x/4
= 13 – 7/4 – 1/2 – 1/3
or, 16 + 18x – 9x/12
= 156 – 21 – 6 – 4/12
or, 25x = 125
or, x = 125/25
= 5
(b) x – 5/4 = 4x + 3/3 – x – 3/6
or, x/4 – 4x/3 + x/6
= 3/3 + 3/6 + 5/4
or, 3x – 6 + 2x/12
= 12 + 6 + 15/12
or, – 11x = 33
or, x 33/- 11
= – 3
(c) 5x – 4/7 – 3x – 1/14 = 1/2
or, 5x/7 – 3x/14
= 1/2 4/7 – 1/14
or, 10x – 3x/14
= 7 + 8 – 1/14
or, 7x/14 = 14/14
or, x/2 = 1
or, x = 2
(d) 3y – 9/4 + 2y + 1/3
= 7y – 1/6
or, 3y/4 + 2y/3 – 7y/6
= 9/4 – 1/3 -1/6
or, 9y + 8y – 14y/12
= 27 – 4 – 2/12
or, 3y = 21
or, y = 21/3
= 7
(e) x – 1/6 – 3x + 1/6 = x + 4/3
or, x/6 – 3x/6 – x
= 4/3 + 1/6 + 1/6
or, x – 3x – 6x/6
= 8 + 1 + 1/6
or, – 8x = 10
or, x = – 10/8
= – 5/4
(f) 2/3y + 1 = 4/5 (y -1/4)
or, 2/3y – 4/5y
= – 1/5 – 1
or, 10y – 12y/15
= – 1 – 5/5
or, – 2y/15 = – 6/5
or, y 6 × 15/5 × 2
= 9
(g) 5 (y + 12) – 17 (2 – y)/7y – 1 = 8
or, 5y + 60 – 34 + 17y = 56y – 6
or, – 12y + 26 = 56y – 8
or, 5y + 60 – 34 + 17y
= 56y – 8
(h) 5x + 2/6 – x – 5/3 = 3
or, 5x/6 – x/3 = 3 – 5 – 2/6
or, 5x – 2x/6 = 18 – 10 – 2/6
or, 3x = 6
or, x = 6/3
= 2
(i) 2 (x – 4)/(x – 7) = 4/5
or, 12x – 28 = 10x – 20
or, 2x = 8
or, x = 4
(l) 6x – 2/3 – 2x – 1/5 = 1/3
or, 6x/3 – 2x/5
= 1/3 – 2/3 – 1/5
or, 30x – 6x/15
= 5 – 10 – 3/15
or, – 3x = – 8
or, x = 8/3
(k) x – 2/4 + 1/3 = 2x – 1/3
or, x/4 – x + 2x/3
= 1/3 – 1/3 + 1/2
or, 3x – 1x+ 8x/12
= 4 – 4 + 6/12
or, – x = 36
or, x = – 6
(l) 1 – x/6 + 2x/3 – 1 – 7x/4 = 6 2/3
or, x/6 + 2x/6 = 20/3 – 1/6 + 1/4
or, – 2x + 4x + 2x/12 = 80 – 2 + 3/12
or, 23x = 81
or, x = 81/23
(m) 3(x – 1)/5 – 5 (x – 4)/7 = 1
or, 3x/5 – 5x/7 = 1 – 20/7 + 3/5
or, 21x – 25x/35
= 35 – 100 + 21/35
or, – 4x = – 44
or, x = 44/4 = 11
(n) 5(x + 2)/3 – 3(x – 2)/2 = 7
= 5x/3 – 3x/2
= 7 – 10/3 – 6/2
or, 10x – 9x/6
= 42 – 20 – 18/6
or, x = 4
Linear Equations in One Variable Exercise 12.2 Solution :
Question no – (1)
Solution :
Suppose the three numbers are,
= 7 (x – 1), 7x, 7 (x + 1)
= 7x + 7(x – 1) + 7 (x + 1) = 777
or, x + (x – 1) + (x + 1) = 111
or, 3x = 111 or, x = 111/3 = 37
or, 3x = 111
or, x = 111/3 = 37
The numbers are (7 × 36), (7 × 37) (7 × 38) = 252, 259, 266
Question no – (2)
Solution :
Suppose the multiples are 11 (x – 1), 11x 11 (x + 1)
∴ 2 × 11 (x – 1) + (3 × 11x) + 4 × 11 (x + 1) = 814
or, 11 [2x – 2 + 3x + 4x + 4] × 11 × [9x + 2] = 814
or, 9x + 2 = 814/14 = 74
or, 9x + 2 = 74
or, 9x = 74 – 2 = 72
∴ 9x = 72
or, x = 72/9
= 8
Question no – (3)
Solution :
Suppose the parts are x, (300 – x)
= 1/2 x = 1/3 (300 – x)
or, x/2 + x/3 = 300/3
or, 5x/6 = 100
or, x = 100 × 6/5 = 120
The parts are = 120 (300 – 120)
= 180
Question no – (4)
Solution :
Suppose the number is x
∴ 5x – 8 = 4x + 4
or, 5x – 4x = 4 + 8
or, x = 12
Question no – (5)
Solution :
Suppose the integers are 7x, 4x
∴ 7x – 4x = 75
or, 3x = 75
or, x = 75/3 = 25
∴ The integers are 7x = 7 × 25
= 175,
= 4x = 4 × 25
= 100
Question no – (6)
Solution :
Suppose, the number are – 3x, 4x, 5x
∴ (3x + 5x) – 4x
= 64
or, 8x – 4x = 64
or, 4x = 64
or, x = 64/4 = 16
∴ The numbers are – 16 × 4 = 64
= 15 × 5
= 80
Question no – (7)
Solution :
Suppose she thinks of x
∴ 9 (x – 4) = 7x
or, 9x – 36 = 7x
or, 9x – 7x = 36
or, 2x = 36
∴ x = 36/2
= 18
Question no – (8)
Solution :
Suppose initially the fraction was = x/x + 5
= x + 11/x + 5 – 14 = 5
or, x + 11/x – 9 = 5
or, 5x – 45 = x + 11
or, 5x – x = 11 + 45
or, 4x = 56
x = 56/4
= 14
Question no – (9)
Solution :
Suppose the number is x
= x/15 – x + 1 = 1/3
or, 3x = 15 – x + 1
or, 3x + x = 15 + 1
or, 4x = 16
or, = 16/4 = 4
Question no – (10)
Solution :
Suppose 4 years ago son’s age x years
Father’s age was 4x year
= (4x + 4) + (x + 4) = 53
= 5x + 8 = 53
or, 5x = 45
or, x = 45/5 = 9
Son’s present age
= 9 + 4
= 13 years
Fathers present age
= 53 – 13
= 40 year
= 9 years ago
Question no – (11)
Solution :
Suppose Ankit’s son’s age was x years
Akint present age = 2(x + 9) years
Ankit present age
Ankit’s age after 4 years = 4 × x = 4x
∴ 4x = 2 (x + 9) + 4
or, 4x = 2x + 18 + 4
or, 4x – 2x = 22
or, 2x = 22
or, x = 11
Ankit’s son’s present age,
= 11 + 9
= 20 years
Ankit’s son’s age,
= 2 × 20
= 40 years
Question no – (12)
Solution :
Suppose, at the time of wedding the ages
of bride and groom was 3x, 4x years
= 3x + 8/4x + 8 = 4/5
or, 16x + 32 = 15x + 40
or, 16x – 15x
= 40 – 32
or, x = 8 years
The age of bride at the time of wedding
= 3 × 8
= 24 years
Therefore, age of the bride will be 24 years.
Question no – (13)
Solution :
Suppose Lalitha’s and Sarita’s present ages are 5x, 4x, years
∴ 5x + 3/4x + 3 = 11/9
or, 45x + 27 = 44x + 33
or 45x – 44x
= 33 – 27
or, x = 6 years
Saritha’s present age = 4 × 6
= 24 years
Question no – (14)
Solution :
Suppose the breath is x cm
= 2[x + (x + 14) = 180
or, 2x + 14 = 180/2
= 90
or, 2x = 90 – 14 = 76
or, x = 76/2 = 38 cm
∴ Breath = 38 cm,
∴ Length = 38 + 14 = 52 cm
∴ Area = 38 × 52 = 1976 cm2
Question no – (15)
Solution :
Suppose breath, length are x cm (x + 7) cm
∴ x (x + 7) =(x + 3) (x +7 – 4)
or, x2 + 7x = (x + 3) (x + 3)
or, x2 + 7x = x2 + 6x + 9
or, 7x – 6x = 9
or, x = 9
∴ Breath = 9 cm,
∴ Length = (9 + 7)
= 16 cm
Question no – (16)
Solution :
Suppose, the base and heights are
= 1/2 (4x) (3x)= 1/2
(4x + 4) (3x – 2)
or, 12x2 = 12x2 + 12x – 8x – 8
or, 4xx – 8 = 0
or, 4x = 8
or, x = 8/4= 2 cm
∴ Base = 4 × 2= 8 cm
∴ Height = 3 × 2 = 6 cm
Question no – (17)
Solution :
∴ x+ 3x/2 + 3x/2 = 32
or, 2x + 3x + 3x/2 = 32
or, 8x = 32 × 2 = 64
or, x = 64/8 = 8 cm
Length of base = 8 cm
Length of others side 8 × 3/2 = 12 cm
Question no – (18)
Solution :
Suppose, cost of the chair = x rupees
x = + 116x/100 = 1620
or, 100x + 116x/100 = 1620
or, 216x = 1620 × 100
or, x = 1620 × 100/216
= 750 rupees
Cost of chair = 750 rupees
Cost of table = 750 × 116/100
= 870 rupees
Question number – (19)
Solution :
Suppose Srividya gets x rupees
So, she deposits in bank = x/2 rupees
She gave her daughter = x/4 + 6000
She gave her daughter son = x/8 + 3000
∴ (x/8 + 3000) + (x/4 + 6000) = x/2
or, x/8 + x/4 – x/2 = – 6000 – 3000
or, x + 2x – 4x/8 = – 9000
or, – x/8 = – x/8 – 9000
or, x/8= 9000
∴ x = 8 × 900 = 72000
Question no – (20)
Solution :
Suppose Anuj has x coins of 5 rupees
Suppose Anuj has 5x/8 coins of 2 rupees
∴ (5 × x) + (5x/8 × 2) = 1000
or, 5x + 5x/4 = 1000
or, 20x + 5x/4 = 1000
or, 25x = 4 × 1000
or, x = 1000 × 4/25
= 160
= 160 coins of 5 rupees
= 160 × 5/8 = 1000 coins of 2 rupees
Question no – (21)
Solution :
Suppose the labourer came to work x days
∴ He did not come to work = (30 – x) days
∴ (80 × x) – 10 (30 – x) = 1860
or, 80x – 300 + 10x = 1860
or, 90x = 1860 + 300
or, 90x = 2160
or, x = 2160/90
or, = 24
He come to work 24 days
Question no – (22)
Solution :
Let, A’s share = x rupees B’s share = 5x/6
C’s share = 4/5 (5x/6) = 2x/3
∴ x + 5x/6 + 2x/3 = 1500
or, 6x + 5x + 4x = 1500
or, 15x/6 = 1500
or, x = 1500 × 6/15 = 600 rupees
A’s Share = 600 × 5/6 = 500 rupees
C’s Share 500 × 4/5 = 400 rupees
Question no – (23)
Solution :
Suppose x adult tickets were sold
∴ 25x + 15 (210 – x) = 4300
or, 25x + 3150 – 15x = 4300
or, 10x = 4300 – 3150
or, 10x = 1150
or, x = 1150/10 = 115
Question no – (24)
Solution :
Suppose x no of guests attended the party
∴ x – (x/3 + x/5 + x/6) = 27
or, x – (10x + 6x + 5x/30) = 27
or, x – 7x/10 = 27
or, 10x – 7x/10 = 27
or, 3x/10 = 27
or, x, 27 × 10/3
= 90
Question no – (25)
Solution :
Let, the speed of stream be x km/hr
∴ 2(18 + x) = 2 1/2 × (18 – x
or, 2x + 36 = 5/2 (18 – x)
or, 2x + 36 = 45 – 5x/2
or, 2x + 5x/2 = 45 – 36
or, 9x/2 = 9
or, x = 9 × 2/9
= 2 km/hr
Question no – (26)
Solution :
Suppose the speeds of the trains are x km/hr, (x + 10) km/hr respectively
Since they were morning towards each other, their relative speed = (x + x + 10) km/hr
= (2x + 10) km/hr
∴ with (2x + 10) km/hr speed they travelled
= (500 – 45) = 455 km in 3 1/2 = 7/2 hours
∴ 455/7/2 = 2x + 10
or, 2x + 10 = 455 × 2/7 = 130
or, 2x = 120
or, x = 120/2
= 60
Speed of the trains = 60 km/hr
= 70 km/hr
Question no – (27)
Solution :
Suppose Devender bought x apples x bananas
∴ Total cost price of apples = 5x rupees
Total cost price of bananas = 2x rupees
∴ Profit from apples
= 5x × 20/100
= x rupees
Loss from apples
= 2x × 20/100
= 2x/5 rupees
Overall profit = x – 2x/5 = 360
or, 5x – 2x/5 = 360
or, 3x/5 = 360
or, x = 360 × 5/3 = 600
∴ He bought 600 apples.
Linear Equations in One Variable Exercise 12.3 Solution :
Question no – (1)
Solution :
= 5x + 7y = – 9 (i)
= 4x + 11y = 26 – (ii)
= 20x + 28y – 36 – (iii)
+ 20x – 557 = + 130 – (iv)
(-) (+) (-)
————————————————–
83y = – 166
or, Y = – 2
Multiplying eqn (i) with 4 and eqn (ii) with 5 we get
Putting y = – 2 (i) we get
= 5x + 7 (- 2) = – 9
or, 5x – 14x = – 9
or, 5x – 14 = – 9 = 5
∴ x = 5/5 = 1
∴ The solution is = x = 1, y = – 2
Question no – (2)
Solution :
5x + 6y = 13 – (i)
6x 5y = 8 – (ii)
25x + 30y = 65 – (iii)
+ 42x + 3yy = + 48 – (iv)
(- ) (- ) (- )
———————————————————–
= – 17x = 17
Multiplying (i) with 5 and (ii) with 6 we get,
Putting x = – 1 in (i) we get,
= 5(1) + 6y = 13
or, 6y – 5 = 13
or, 6y – 5 = 13
or, 6y = 13 + 5 = 18
∴ y = 18/6 = 3
∴ The solution is x = – 1, y = 3
Question no – (3)
Solution :
= 3x – 5y = – 34 – (i)
= 8x – 7y = – 21 (ii)
= 21x – 35y = – 238
+ 40x – 35y = – 238 – (iii)
————————————————————–
= – 19x = – 133
or, x = – 133/ – 19 = 7
Multiplying (i) 7 and (ii) with 5 we get
Putting x = 7 in (i) we get
= 3 (7) – 5y = – 34
or, 21 – 5y = – 34
or, 5y = 21 + 34 = 55
or, y = 55/5 = 14
∴ The solution is x = 7, y = 11
Question no – (4)
Solution :
= 3a + 8b = 28 – (i)
= 7a + 10b = 22 – (ii)
= 21a + 56b = 196 – (iii)
= 21a + 30b = 66
————————————————————–
= 26b = 130
= 130/26 = 5 Multiplying (i) with 7 and (ii) with 3, we get,
Putting b = 5 in (i) we get,
= 3a + 8(5) = 28
or, 3a = 28 – 40 = – 12
∴ a = – 12/3 = – 4
∴ The solution is a = – 4, b = 3
Question no – (5)
Solution :
= 7m + 8n = 52
= 9m- 11n = 3
= 63m + 72n = 468
= + 63m – 77n = 21
(-) (+) (-)
————————————————–
= 149 = 447
= n = 447/149 = 3
Multiplying (i) with 9 and (ii) with (7) we get
= 7 m + 24 = 52
or, m = 4
∴ The solution is m = 4, n = 3
Question no – (6)
Solution :
= x – 5/3 = y + 2/7
= 7x – 35
= 3y + 6
= 7x – 3y = 47 (i)
= 2x – 3y = 1 (i)
(-) (+) (-)
————————————-
= 5x = 40
∴ x = 40/5 = 8
Putting x = 8 (ii) we get,
= 2 (8) – 3y = 1
or, 3y = 16 – 1 = 15
or, y = 15/3 = 5
Question no – (7)
Solution :
= x/3 + y/2 = 8 (i)
= 3x/4 – y/4 = 7 (ii) × 2
= 3x/2 – y/2 = 14 (iii)
Now, (i) + (iiii) (x/3 + 3x/2) + (y/2 – y/20 = 8+ 14
or, 11x/6 = 22
or, x = 22 × 6/11 = 12
Putting x = 12 in (i) we get, 12/3 + y/2 = 8
or, y/2 = 8 – 4
or, y = 4 × 2= 8
∴ The solution is x = 12, y = 8
Question no – (8)
Solution :
= x/3 – y2 = 1 – (i)
= x/5 + y/4 = 5 – (ii) × 2
= 2x/5 + y/2 = 10 (iii)
Now, (i) + (iii) – (x/3 + 2x/5) + (- y/2) + y/2) = 1 + 10
or, 11x/15 = 11
or, 11 × 15/11 = 15
Putting x = 15 in (i) 15/3 – y/2 = 1
or, y/2 = 5 – 1 = 4
or, y = 8
= [x = 15, y = 8]
Question number – (9)
Solution :
= 3x + 8y = 7 – (i)
= 2x + 9y = 1 (ii)
= 6x + 16y = 14 – (iii)
= 6x + 27y = 3 (iv)
—————————————
= – 11y = 11
or, y = 11/- 11 = – 1
Multiplying 2 with (i) and 3 with (ii) we get,
Putting y = – 1 or, (i) we get
= 3x – 8 = 7
or, 3x = 15
= x = 15/3 = 5
∴ [x – 5, y = – 1]
Question no – (10)
Solution :
= 22x + 21y = 3
= 21x + 22y = – 3
—————————–
= 43x + 43y = 0
or, x + y = 0 – (i)
= 22x + 21y = 3
= 21x + 22y = – 3
(-) (-) (+)
—————————————
= x – y = 6 – (ii)
Now, adding (i) and (ii)
= x + y = 0
= x – y = 6
—————————————
= 2x = 6
∴ x = 3
Putting x = 3 in (ii) we get 3 – y = 6
or, y = – 3
= [x = 3, y = – 3]
Question no – (11)
Solution :
= 4x – 47y = 35
= 4x – 41y = 53
—————————————
= 88x – 88y = 88
or, x – y = 1 (i)
= 41x – 41y = 53
—————————————
6x + 6y = 18
or, x + y = 3
Adding (i) and (ii)
x – y = 1
x + y = 3
—————————————
= 2x = 4
∴ x = 2
Putting x = 2 in (ii) 2 + y = 3
or, y = – 2 = 1
= [x = 2, y = 1]
= [(12) → Similar]
Question no – (13)
Solution :
= 2x – 3y = 4 – (i)
= – x + 3y = 1 – (ii)
—————————————
= x = 5
Putting x = 5 in (ii) we get,
= 3y = 1 + 5 = 6
= y = 6/3 = 2
[x = 5, y = 2]
Question number – (14)
Solution :
= 3x – 7y = 6 – (i)
= – 4x + 6y = 2 – (ii)
= 12x – 28y = 24 – (iii)
= – 12x + 18y = 6 – (iv)
—————————————
= – 10 = 30
= y = – 3
Multiplying (i) with 4 and (ii) with 3, we get
Putting y = – 3 in (i) we get
= 3x – 7 (- 3) = 6
or, 3x = 6 – 21 = – 15
= x – 15/3 = – 5
∴ [x = – 5, y = – 3]
Question no – (15)
Solution :
= 2x + y = 7 – (i)
= 2x/5 + y/3 = 1 – (ii)
Subtracting (iii) from (i) we get,
= 2x + y = 7
= + 2x + 5y/3 = 5
—————————————
= – 2y/3 = 2
or, y = 2 × 3/ – 2 = – 3
Multiplying (ii) with 5 get 2x + 5y/3 = 5 – (iii)
Subtracting (iii) from (i) we get,
putting y = – 3 in (i) we get
= 2x – 3 = 7
or, 2x = 10
or, x = 10/2 = 5
= [x = 5, y = – 3]
Question no – (16)
Solution :
= 57x – 56y = – 169
= 56x – 57y = – 170
—————————————
= 113x – 113y – 339
or, x – y – 3 –(i)
Adding (i) and (ii)
= 57x – 56y = – 169
+ 56x – 57y = – 170
—————————————
= x + y = 1 – (ii)
= x – y = – 3
= x + y = 1
—————————————
or, x = – 2/2 = – 1
Putting x = – 1 in (i) we get,
= 1 – y = – 3
or, y = 3 – 1 = 2
Question no – (17)
Solution :
= 2x – 3y = 3 – (i)
= 2x/3 + 4y = 6 – (ii)
Subtracting (i) from (iii)
= 2x + 12y = 18
= 2x – 3y = 3
————————————
15y = 15
or, y = 1
Multiplying (ii) with 3 we get,
= 2x + 12y = 18 – (iii)
Subtracting y = 1 in (i) we get
= 2x – 3 (i) = 3
or, 2x = 6
or, x = 6/2 = 2
= [x = 3, y = 1]
Question no – (18)
Solution :
= x – 1/y + 1 = 3/4
= 4x – 4 = 3y + 3
= 4x – 3y = 7 (i)
= x + 2/y – 2 = 4/3
= 3x + 6
= 4y – 8
= 3x – 4y = – 14 – (ii)
Multiplying (i) with (3) and (ii) with we get
= 12x – 9y = 21 – (iii)
= + 12x – 16y = – 56 – (ii)
—————————————–
= 7y = 77
or, y = 11
Putting y = 11 in (i) we grt
= 4x – 3 (ii) = 7
or, 4x = 33 + 7 = 40
∴ x = 40/4 = 10
or, y = 11
= [x = 10, y = 11]
Question no – (19)
Solution :
= 3x/2 + y/2 = – 1 – (i)
= x/2 + 3y/2 = 5 – (ii)
Multiplying (ii) with 3 we get
= 3x/2 + 9y/2 = 15 – (iii)
Subtracting (i) from (iii) we get
= (3x/2 – 3x/2) + (9y/2 – y/2) = 15 – (- 1)
or, 9y – y/2 = 16
or, 8y/2 = 16
or, 8y/2 = 16
= y = 2 × 16/8
or, y = 4
Putting y = 4 in (i) 3x/2 + 1 = 1
or, 3x/2 = – 3
or, x = – 3 × 2/3
= – 2
= [x = – 2, y = 4]
Question no – (20)
Solution :
= 371x + 197y = 454
= 197x + 371y = 23
—————————————–
= 568x + 567 = 568
or, x + y = 1 (i)
Adding (i) and (ii)
= x + y = 1
= x – y = 2
—————————————–
= 2x = 3
= 371x + 197y = 545
= 197x + 371y = 23
—————————————–
= 174x – 174y = 522
or, x – y = 2 – (ii)
[∴ x = 3/2, y = – 1/2]
Linear Equations in One Variable Exercise 12.4 Solution :
Question no – (1)
Solution :
Suppose the numbers are x, y
So, x + y = 187
= x – y = 23
—————————-
= 2x = 210
∴ x = 105
Putting x = 105 in (i) we get
= 105 + y = 187
or, 187 – 105 = 82
∴ [x = 105, y = 82]
Question no – (2)
Solution :
= x + y = 53 – (i)
= 2x – 3y = 11 (ii)
Multiplying (i) by 2 we get
= 2x + 2y = 106 – (iii)
Subtracting (ii) from (iii) we get
= 2x + 2y = 106
= 2x – 3y = 11
Putting = y = 19 in (i) we get
= 19 + x = 53
or, x = 53 – 19 = 34
∴ y = 19
= [x= 34, y = 19]
Question no – (3)
Solution :
= x + 3/y = 56
= 6x + 18 = 5y
= 6x – 5y – 18 (i)
= x + 5/y + 4 = 3/4
= 4x + 20 = 3y + 12
or, 4x – 3y – 8 – (ii)
Multiplying (i) with 3 and (ii) with 5 we, get
= 18x – 15y = – 54
= + 20x – 15y = – 40
(-) (+) (+)
—————————————
– 2x = – 14
∴ x = – 14/- 2 = – 17
= [7/12]
Putting x = 7 in (i) we get
= 6 (7) – 5y = – 18
or, 5y = + 18 + 42 = 60
or, y = 60/5 = 12
Question no – (4)
Solution :
= x + 1/y – 1 = 2/3
= 3x + 3 = 2 + 2
= 3x – 2y – 5 (i)
= x + 2/y + 2 = 3/5
= 5x + 10
= 3y + 6
= 5x – 3y
= – 4 – (ii)
Multiplying (i) with 3 and (ii) with2, we get,
9x – 6y = – 15 – (iii)
+ 10x – 6y = – 8 (iv)
————————————-
– x = – 7
∴ x = 7
Putting x = 7 in (i)
= 3 (7) – 2y = – 5
or, 2y = 5 + 21 = 26
or, y = 26/2 = 13
= [7/13]
Question no – (5)
Solution :
= x + 7/5x + 2 + 7 = 1/3
or, 5x + 9
= 3x + 21
or, 5x – 3x = 21 – 9
or, 2x = 12
or, x = 6
Daughter’s present age = 6 years, Anita’s present age
= 5(6) + 2
= 32 years
Question no – (6)
Solution :
Suppose son’s age = x years
= x + 3/66 – x + 3
= 1/3
or, 3x + 9 = 69 – x
or, 3x + x = 69 – 9
or, 4x = 60
or, x = 15
Son’s age = 15 years, fathers age
= 66 – 15
= 51 years
Question no – (7)
Solution :
Suppose Suresh’s age = years
= x/x + 16 + 20 = 1
or, 4x + = x = 36
or, 4x – x = 36
or, 3x = 36
or, x = 36/3 = 12
Suresh’s age = 12 years
Ramesh’s age = 12 + 16
= 28 years
Question no – (8)
Solution :
Suppose their incomes are 8x rupees 7x rupees
Suppose their expenditures are 5y rupees 4y rupees
∴ 8x – 5y = 21000 – (i)
= 7x – 4y = 21000 – (ii)
= 32x – 20y = 84000 – (iii)
= + 35x – 20y = 105000 – (iv)
——————————————————
= – 3x = – 21000
or, x = 7000
Multiplying (i) with 4 and (ii) with 5, we get
Putting x = 7000 in (i) we get
= 8 (7000) – 5y = 21000
or, 5y = 56000 – 21000
or, = 35000/5
= 7000
Their income,
= (8 × 7000) = 56000
= (7 × 7000) = 49000
Question no – (9)
Solution :
= 5x + 27/4x + 27 = 8/7
or, 35x + 189 = 32x + 216
or, 3x = 27
or, x = 27/3 = 9
The numbers are = (9 × 5) = 45
= (9 × 4) = 36
Question no – (10)
Solution :
Suppose costs of a pen = x rupees, cost of a pencil = y rupees
= 18x + 72y = 343 – (i)
= 11x + 96y = 237 – (ii)
Multiplying (i) with 4 and (ii)
with 3 we get
= 72x + 288y = 1296
+ 33x + 288y = 711
—————————————
= 39x = 585
= x = 585/39 = 15
Putting x = 15 in (i) we get
= 18(15) + 72y = 324
or, 12y = 324 – 270 = 54
or, y = 54/72
= 63/4 rupees
[Pen = 15 rupees, pencil = 3/4 rupees]
Question no – (11)
Solution :
Suppose cost of a chair = y rupees cost of a table = x rupees
= 8x + 15y = 5225 – (i)
= 10x + 17y = 6225 – (ii)
= 40x + 75y = 26125
= 4x + 63y = 24900
—————————————
= 7y = 1255
∴ y = 1225/7 = 175
Multiplying (i) with 5 and (ii) with 4 we get
Putting y = 175 we, get
= 8x + 15 (175) – 5225
or, 8x = 5225 – 2625 = 2600
or, ∴ x = 2600/8 = 325
[∴ Table = 325, chair = 175]
Question no – (12)
Solution :
Suppose speed of bus = km/hr
Speed of taxi = y km/hr
= 9x + 4y = 670 – (i)
= 8x + 7y = 785 – (ii)
= 63x + 28y = 4690 – (iii)
+ 32x + 28y = 3140 – (iv)
————————————————-
= 31x = 1550
∴ x = 1550/31 = 50
Multiplying (i) with 7 and (ii) with 4 we get
Putting x = 50 in(i) we get
= 9 (50) + 4y = 670
or, 4y = 670 – 450 = 220
∴ y = 220/4 = 55
[∴ Bus = 50 km/hr, Taxi = 55 km/hr]
Question no – (13)
Solution :
Speed in downstream
= 88/4
= 22 km/hr
Speed in upstream
= 40/4
= 10 km/hr
Suppose speed of the boat I still = x km/hr
Speed of the stream = y km/hr
∴ x + y = 22
= x – y
——————————–
2x = 32
∴ x = 16km/hr
= y = 6km/hr
[Boat = 16km/hr Stream/hr]
Question no – (14)
Solution :
Speed in upstream
= 45/4 1/2
= 45/9/2
= 10 km/hr
Speed in downstream
= 56/4
= 14 km/hr
Suppose : Speed of the boat in still water = x km/hr
∴ x + y =14
x – y = 10
——————————–
= 2x = 24
∴ x = + 2 km/hr
y = 2 km/hr
[Speed of the boat book = 12 km/hr : Speed of the stream = 2 km/hr]
Question no – (15)
Solution :
Speed in upstream = 30/31/3 = 30/10/3 = 9 km/hr
Speed in downstream = 30/2 = km/hr
∴ x + y = 15
= x – y = 0
——————————–
= 2x = 24
∴ x = 12
= y = 3
[Speed of the book = 12 km/hr : Speed of the stream = 3 km/hr]
Question no – (16)
Solution :
= 15 + 2y = 97
= 2x + 5y = 106
——————————–
7x + 7y = 203
or, x + y = 29 – (i)
= 5x + 2y = 97
= 2x + 5y = 106
——————————–
= 3x – 3y = – 9
= x – y = – 3 – (ii)
Adding (i) and (ii)
x + y = 29
x – y = – 3
——————————–
= 2x = 20
∴ x = 26/2 = 13
= y = – 15
[∴ x = 13 y = 15]
Question no – (17)
Solution :
Suppose the tens digit is x
Suppose the tens unit digit is = 4x
∴ (10 × x) + (4x × 1) + 54
= (4x × 10) + (x + 1)
or, 10x + 4x + 54 = 40x + x
or, 41x – 14x = 54
or, 27 x = 54
or, x = 54/27 = 2
∴ The tens digit is 2 unit digit is 8
∴ The number is 28
Question no – (18)
Solution :
Suppose the unit digit is
Suppose the unit tens is (9 – x)
∴ 10(9 – x) + (x × 1) + 27
= 10 (x) + 1 (9 – x)
or, 90 – 10x + x + 27
= 10x + 9 – x
or, 18x = 108
or, x = 6
∴ Unit digit = 6
Tens digit = 3
∴ The number is 36
Question no – (19)
Solution :
Suppose the unit digit is x
∴ Suppose the unit digit = (2x + 1)
∴ 10x + 1 (2x + 1) + 36
= 10 (2x + 1) + (1 ∴ x)
or, 10x + 2x + 1 + 36
= 20x + 10 + x
or, 9x = 27
or, = 27/9 = 3
Tens digit = 3
∴ Unit digit = (3 × 2) + 1 = 7
Question no – (20)
Solution :
Suppose Arun got x rupees Varun got y
∴ x+ y = 76 – (i)
= x – 7 = y + 7
= x – y = 14 – (ii)
Adding (i) and (ii) (x + x) + (y – y)
= (76 + 14)
or, 2x = 90
or, x = 90/2 = 45
= y = 76 – 45 = 31
[Arun got = 45 rupees : Varun got = 31 rupees]
Revision Exercise Questions Solution :
Question no – (1)
Solution :
(a) 6x – 2/3 – 2x – 1/5 = 1/3
or, 6x/3 – 2/3- 2x/5 + 1/5 = 1/3
or, 2x – 2x/5 = 1/3 + 2/3 = 1/5
or, 2x/4 = 4/5
or, x/2 = 4/5
or, x = 8/5
(b) x/3 + 1/2 = x/4 + 1/3
or, x/3 – x/4
= 1/3 – 1/2
or, x/12
= – 1/6
or, x = – 12/6
or, x = – 2
Question no – (2)
Solution :
= 5 – x/7 – x = 2 + x/7 + x
or, x2 – 2x + 35
= x2 + 5x + 14
or, 7x = 21
or, x = 21/7
= 3
Question no – (3)
Solution :
= 3x + 5/2x + 1 = 4/3
or, 9x + 15
= 8x + 4
or, x = 4 – 15
= 11
Question no – (4)
Solution :
= x + 4/x + 6 = x – 2/x – 1
or, x2 + 4x – 12
= x2 + 3x – 4
or, x = 12 – 4
= 8
Question no – (5)
Solution :
= (x + 40)° + (2x – 10)° + (x + 50)° = 180°
or, (x + 2x + x)° + (40 + 50 – 10)° = 180°
or, 4x + 80° = 180°
or, 4x = 180° – 80° = 100°
or, x = 100°/4
= 25°
Question no – (6)
Solution :
Suppose the man’s age is – x years
His father age = 2x years his grandfather age = 3x years
∴ x + 2x + 3x = 150
or, 6x = 150
or, x = 25 years
Question no – (7)
Solution :
Suppose now Sarita is x years old
∴ 4x + 5/x + 5 = 3/1
or, 4x + 5 = x + 15
or, 3x = 10
or, x = 10/3
Question no – (8)
Solution :
Suppose the numbers are 9x, 5x
∴ 9x + 5x = 56
or, 14x = 56
or, x = 56/14
= 4
Question no – (9)
Solution :
Suppose the unit’s digit is x ten’s digit = (7 – x)
∴ 10 (7 – x) + (x × 1) + 27
= (10 × x) + (7 – x)
or, 70x – 10x + x + 27
= 10x + 7 – x
or, 18x = 90
or, x = 90/18
= 5
∴ (7 – x) = 2
The number is 25
Question no – (10)
Solution :
Suppose the tens digit = x
∴ units digit = (x + 2)
∴ x + x + 2 = 16
or, 2x = 14
or, x = 14/2 = 7
∴ (x + 2) = 9
The number is = 79
Question no – (11)
Solution :
Suppose the numerator = x denominator = x + 6
∴ x + 3/x + 6 = 2/3
or, 3x + 9 = 2x + 12
or, 3x – 2x = 12 – 9
or, x = 3
∴ The original fraction = 3/3 + 6
= 3/9
Question no – (12)
Solution :
Suppose he hit x times, missed y times
∴ x+ y = 160 – (i)
4x – y = 440 – (ii)
——————————————-
5x = 600
∴ x = 600/5 = 120
∴ He hit 120 times
Question no – (13)
Solution :
Suppose speed of the stream = x km/hr
= 20/8 + x
= 12/8 – x
or, 96 + 12x = 160 – 20x
or, 12x + 20x – 160 – 96
or, 32x = 64
or, x = 64/32
= 2 km/hr
Question number – (14)
Solution :
Suppose speed of the streams = x km/hr
∴ 9 (x + 1) = 10(x – 1)
or, 10x – 9x
= 10 + 9
or, x = 19 km/hr
∴ The total distance 2 × 9 × 919 + 1)
= 18 × 20
= 360 km
Question no – (15)
Solution :
Suppose three were x number of 50 rupees notes
∴ 50x + 100 (25 – x) = 200
or, 50x – 100x
= 2000 – 2500
or, 50x = 500
or, x = 500/50 = 10
∴ Three were 10 notes of 50 rupees
Question no – (16)
Solution :
Suppose the least one is 2x
∴ (4 × 2x) – 6
= (2x + 2) + (2x + 4) + (2x + 6)
or, 8x – 6 = 6x + 12
or, 8x – 6x
= 12 + 6
or, 2x = 18
∴ x = 18/2 = 9
∴ The numbers are (9 × 2) = 18
= (9 × 2 + 2) = 20
= 20 + 2 = 22
= 22 + 2 = 24
Next Chapter Solution :
👉 Chapter 13 👈