Class 8 RD Sharma Solutions - Chapter 9 Linear Equation In One Variable - Exercise 9.2 | Set 2

Exercise 9.2 Set 2 builds upon the concepts introduced in Set 1, presenting more challenging linear equations in one variable. This set is designed to deepen students' understanding of algebraic manipulation and problem-solving techniques. It introduces more complex scenarios, including equations with variables on both sides, multiple fractions, and decimal coefficients. The problems in this set aim to enhance student's ability to apply logical reasoning and algebraic skills to solve increasingly sophisticated equations.

Question 14. (1-2x)/7 – (2-3x)/8 = 3/2 + x/4

Solution:

(1-2x)/7 – (2-3x)/8 = 3/2 + x/4

First rearrange the equation

(1-2x)/7 – (2-3x)/8 – x/4 = 3/2

By taking LCM for 7, 8 and 4 which is 56

((1-2x)8 – (2-3x)7 – 14x)/56 = 3/2

(8 – 16x – 14 + 21x – 14x)/56 = 3/2

(-9x – 6)/56 = 3/2

After cross-multiplying

2(-9x-6) = 3(56)

-18x – 12 = 168

-18x = 168+12

-18x = 180

x = 180/-18

x = -10

Now verify the equation by putting x = -10

(1-2x)/7 – (2-3x)/8 = 3/2 + x/4

x = -10

(1-2(-10))/7 – (2-3(-10))/8 = 3/2 + (-10)/4

(1+20)/7 – (2+30)/8 = 3/2 – 5/2

21/7 – 32/8 = 3/2 – 5/2

3 – 4 = -2/2

-1 = -1

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

Question 15. (9x+7)/2 – (x – (x-2)/7) = 36

Solution:

(9x+7)/2 – (x – (x-2)/7) = 36

First simplify the given equation

(9x+7)/2 – (7x-x+2)/7 = 36

(9x+7)/2 – (6x+2)/7 = 36

By taking LCM for 2 and 7 is 14

(7(9x+7) – 2(6x+2))/14 = 36

(63x+49 – 12x – 4)/14 = 36

(51x + 45)/14 = 36

After cross-multiplying

51x + 45 = 36(14)

51x + 45 = 504

51x = 504-45

51x = 459

x = 459/51

x = 9

Now verify the equation by putting x = 9

(9x+7)/2 – (x – (x-2)/7) = 36

(9x+7)/2 – (6x+2)/7 = 36

x = 9

(9(9)+7)/2 – (6(9)+2)/7 = 36

(81+7)/2 – (54+2)/7 = 36

88/2 – 56/7 = 36

44 – 8 = 36

36 = 36

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

Question 16. 0.18(5x – 4) = 0.5x + 0.8

Solution:

0.18(5x – 4) = 0.5x + 0.8

First rearrange the given equation

0.18(5x – 4) – 0.5x = 0.8

0.90x – 0.72 – 0.5x = 0.8

0.90x – 0.5x = 0.8 + 0.72

0.40x = 1.52

x = 1.52/0.40

x = 3.8

Now verify the equation by putting x = 3.8

0.18(5x – 4) = 0.5x + 0.8

x = 3.8

0.18(5(3.8)-4) = 0.5(3.8) + 0.8

0.18(19-4) = 1.9 + 0.8

2.7 = 2.7

Thus L.H.S. = R.H.S.,

Hence, the equation is verified

Question 17. 2/3x – 3/2x = 1/12

Solution:

2/3x – 3/2x = 1/12

By taking LCM for 3x and 2x which is 6x

((2×2) – (3×3))/6x = 1/12

(4-9)/6x = 1/12

-5/6x = 1/12

After cross-multiplying

6x = -60

x = -60/6

x = -10

Now verify the equation by putting x = -10

2/3x – 3/2x = 1/12

x = -10

2/3(-10) – 3/2(-10) = 1/12

-2/30 + 3/20 = 1/12

((-2×2) + (3×3))/60 = 1/12

(-4+9)/60 = 1/12

5/60 = 1/12

1/12 = 1/12

Thus L.H.S. = R.H.S.,

Hence the equation is verified.

Question 18. 4x/9 + 1/3 + 13x/108 = (8x+19)/18

Solution:

4x/9 + 1/3 + 13x/108 = (8x+19)/18

First rearrange the given equation

4x/9 + 13x/108 – (8x+19)/18 = -1/3

By taking LCM for 9, 108 and 18 which is 108

((4x×12) + 13x×1 – (8x+19)6)/108 = -1/3

(48x + 13x – 48x – 114)/108 = -1/3

(13x – 114)/108 = -1/3

After cross-multiplying

(13x – 114)3 = -108

39x – 342 = -108

39x = -108 + 342

39x = 234

x = 234/39

x = 6

Now verify the equation by putting x = 6

4x/9 + 1/3 + 13x/108 = (8x+19)/18

x = 6

4(6)/9 + 1/3 + 13(6)/108 = (8(6)+19)/18

24/9 + 1/3 + 78/108 = 67/18

8/3 + 1/3 + 13/18 = 67/18

((8×6) + (1×6) + (13×1))/18 = 67/18

(48 + 6 + 13)/18 = 67/18

67/18 = 67/18

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

Question 19. (45-2x)/15 – (4x+10)/5 = (15-14x)/9

Solution:

(45-2x)/15 – (4x+10)/5 = (15-14x)/9

First rearranging the given equation

(45-2x)/15 – (4x+10)/5 – (15-14x)/9 = 0

By taking LCM for 15, 5 and 9 which is 45

((45-2x)3 – (4x+10)9 – (15-14x)5)/45 = 0

(135 – 6x – 36x – 90 – 75 + 70x)/45 = 0

(28x – 30)/45 = 0

After cross-multiplying

28x – 30 = 0

28x = 30

x = 30/28

x = 15/14

Now verify the equation by putting x = 15/14

(45-2x)/15 – (4x+10)/5 = (15-14x)/9

x = 15/14

(45-2(15/14))/15 – (4(15/14) + 10)/5 = (15 – 14(15/14))/9

(45- 15/7)/15 – (30/7 + 10)/5 = (15-15)/9

300/105 – 100/35 = 0

(300-300)/105 = 0

0 = 0

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

Question 20. 5(7x + 5)/3 – 23/3 = 13 – (4x-2)/3

Solution:

5(7x+5)/3 – 23/3 = 13 – (4x-2)/3

First rearrange the given equation

(35x + 25)/3 + (4x – 2)/3 = 13 + 23/3

(35x + 25 + 4x – 2)/3 = (39+23)/3

(39x + 23)/3 = 62/3

After cross-multiplying

(39x + 23)3 = 62(3)

39x + 23 = 62

39x = 62 – 23

39x = 39

x = 1

Now verify the equation by putting x = 1

5(7x+5)/3 – 23/3 = 13 – (4x-2)/3

x = 1

(35x + 25)/3 – 23/3 = 13 – (4x-2)/3

(35+25)/3 – 23/3 = 13 – (4-2)/3

60/3 – 23/3 = 13 – 2/3

(60-23)/3 = (39-2)/3

37/3 = 37/3

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

Question 21. (7x-1)/4 – 1/3(2x – (1-x)/2) = 10/3

Solution:

(7x-1)/4 – 1/3(2x – (1-x)/2) = 10/3

when we expand the given equation,

(7x-1)/4 – (4x-1+x)/6 = 10/3

(7x-1)/4 – (5x-1)/6 = 10/3

By taking LCM for 4 and 6 is 24

((7x-1)6 – (5x-1)4)/24 = 10/3

(42x – 6 – 20x + 4)/24 = 10/3

(22x – 2)/24 = 10/3

After cross-multiplying

22x – 2 = 10(8)

22x – 2 = 80

22x = 80+2

22x = 82

x = 82/22

x = 41/11

Now verify the equation by putting x = 41/11

(7x-1)/4 – 1/3(2x – (1-x)/2) = 10/3

x = 41/11

(7x-1)/4 – (5x-1)/6 = 10/3

(7(41/11)-1)/4 – (5(41/11)-1)/6 = 10/3

(287/11 – 1)/4 – (205/11 – 1)/6 = 10/3

(287-11)/44 – (205-11)/66 = 10/3

276/44 – 194/66 = 10/3

69/11 – 97/33 = 10/3

((69×3) – (97×1))/33 = 10/3

(207 – 97)/33 = 10/3

110/33 = 10/3

10/3 = 10/3

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

Question 22. 0.5(x-0.4)/0.35 – 0.6(x-2.71)/0.42 = x + 6.1

Solution:

0.5(x-0.4)/0.35 – 0.6(x-2.71)/0.42 = x + 6.1

First simplify the given equation

(0.5/0.35)(x – 0.4) – (0.6/0.42)(x – 2.71) = x + 6.1

(x – 0.4)/0.7 – (x – 2.71)/0.7 = x + 6.1

(x – 0.4 – x + 2.71)/0.7 = x + 6.1

-0.4 + 2.71 = 0.7(x + 6.1)

0.7x = 2.71 – 0.4 – 4.27

= -1.96

x = -1.96/0.7

x = -2.8

Now verify the equation by putting x = 5

0.5(x-0.4)/0.35 – 0.6(x-2.71)/0.42 = x + 6.1

x = -2.8

0.5(-2.8 – 0.4)/0.35 – 0.6(-2.8 – 2.71)/0.42 = -2.8 + 6.1

-1.6/0.35 + 3.306/0.42 = 3.3

-4.571 + 7.871 = 3.3

3.3 = 3.3

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

Question 23. 6.5x + (19.5x – 32.5)/2 = 6.5x + 13 + (13x – 26)/2

Solution:

6.5x + (19.5x – 32.5)/2 = 6.5x + 13 + (13x – 26)/2

First rearrange the equation

6.5x + (19.5x – 32.5)/2 – 6.5x – (13x – 26)/2 = 13

(19.5x – 32.5)/2 – (13x – 26)/2 = 13

(19.5x – 32.5 – 13x + 26)/2 = 13

(6.5x – 6.5)/2 = 13

6.5x – 6.5 = 13×2

6.5x – 6.5 = 26

6.5x = 26+6.5

6.5x = 32.5

x = 32.5/6.5

x = 5

Now verify the equation by putting x = 5

6.5x + (19.5x – 32.5)/2 = 6.5x + 13 + (13x – 26)/2

x= 5

6.5(5) + (19.5(5) – 32.5)/2 = 6.5(5) + 13 + (13(5) – 26)/2

32.5 + (97.5 – 32.5)/2 = 32.5 + 13 + (65 – 26)/2

32.5 + 65/2 = 45.5 + 39/2

(65 + 65)/2 = (91+39)/2

130/2 = 130/2

65 = 65

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

Question 24. (3x – 8) (3x + 2) – (4x – 11) (2x + 1) = (x – 3) (x + 7)

Solution:

(3x – 8) (3x + 2) – (4x – 11) (2x + 1) = (x – 3) (x + 7)

First simplify the given equation

9x² + 6x – 24x – 16 – 8x² – 4x + 22x + 11 = x² + 7x – 3x – 21

9x² + 6x – 24x – 16 – 8x² – 4x + 22x + 11 – x² – 7x + 3x + 21 = 0

9x² – 8x² – x² + 6x – 24x – 4x + 22x – 7x + 3x – 16 + 21 + 11 = 0

-4x + 16 = 0

-4x = -16

x = 4

Now verify the equation by putting x = 4

(3x – 8) (3x + 2) – (4x – 11) (2x + 1) = (x – 3) (x + 7)

x = 4

(3(4) – 8) (3(4) + 2) – (4(4) – 11) (2(4) + 1) = (4 – 3) (4 + 7)

(12-8) (12+2) – (16-11) (8+1) = 1(11)

4 (14) – 5(9) = 11

56 – 45 = 11

11 = 11

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

Question 25. [(2x+3) + (x+5)]² + [(2x+3) – (x+5)]² = 10x² + 92

Solution:

[(2x+3) + (x+5)]- + [(2x+3) – (x+5)]² = 10x² + 92

First simplify the given equation 

[3x + 8]² + [x – 2]² = 10x² + 92

Now apply the formula (a+b)2

9x² + 48x + 64 + x² – 4x + 4 = 10x² + 92

After rearranging the equation

9x² – 10x² + x² + 48x – 4x = 92 – 64 – 4

44x = 24

x = 24/44

x = 6/11

Now verify the equation by putting x = 6/11

[(2x+3) + (x+5)]² + [(2x+3) – (x+5)]² = 10x² + 92

x = 6/11

[2(6/11) + 3 + (6/11) + 5]² + [2(6/11) + 3 – (6/11) – 5]² = 10(6/11)² + 92

[(12/11 + 3) + (6/11 + 5)]² + [(12/11 + 3) – (6/11 + 5)]² = 10(6/11)² + 92

[(12+33)/11 + (6+55)/11]² + [(12+33)/11- (6+55)/11]² = 10(6/11)² + 92

[(45/11)+ (61/11)]² + [(45/11) – (61/11)]² = 360/121 + 92

(106/11)² + (-16/11)² = (360 + 11132)/121

11236/121 + 256/121 = 11492/121

11492/121 = 11492/121

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

Summary

Exercise 9.2 Set 2 challenges students with more complex linear equations, requiring advanced algebraic manipulation skills. It covers equations with variables on both sides, multiple fractions, and decimal coefficients. The set emphasizes the importance of systematic problem-solving, including simplifying expressions, combining like terms, and using inverse operations to isolate variables. Students practice handling equations with parentheses, dealing with fractional equations, and working with decimal values. This exercise set helps solidify students' understanding of linear equations and prepares them for more advanced mathematical concepts.