Class 8 RD Sharma Solutions - Chapter 9 Linear Equation In One Variable - Exercise 9.2 | Set 1

Exercise 9.2 Set 1 not only reinforces the techniques for solving linear equations but also lays the groundwork for more advanced algebraic concepts that students will encounter in higher grades. The skills developed here are essential for solving real-world problems and form a crucial foundation for future mathematical studies. The problems in this set are carefully designed to gradually increase in complexity, helping students build confidence as they progress. This structured approach ensures that students can solidify their understanding of basic concepts before tackling more challenging problems.

Solve each of the following equations and also check your results in each case:

Question 1. (2x + 5)/3 = 3x – 10

Solution:

First simplify the equation,

(2x + 5)/3 – 3x = – 10

By taking LCM

(2x + 5 – 9x)/3 = -10

(-7x + 5)/3 = -10

After cross-multiplication we get,

-7x + 5 = -30

-7x = -30 – 5

-7x = -35

x = -35/-7

= 5

Now verify the equation by putting x = 5.

(2x + 5)/3 = 3x – 10

x = 5

(2×5 + 5)/3 = 3(5) – 10

(10+5)/3 = 15-10

15/3 = 5

5 = 5

Thus, L.H.S. = R.H.S.,
Hence, the equation is verified. 

Question 2. (a - 8)/3 = (a - 3)/2

Solution:

After cross-multiplication we will get,

(a - 8)2 = (a-3)3

2a – 16 = 3a – 9

2a – 3a = -9 + 16

-a = 7

a = -7

Now verify the equation by putting a = -7.

(a - 8)/3 = (a - 3)/2

a = -7

(-7 - 8)/3 = (-7 – 3)/2

-15/3 = -10/2

-5 = -5

Thus, L.H.S. = R.H.S.,
Hence, the equation is verified.

Question 3. (7y + 2)/5 = (6y – 5)/11

Solution:

After cross-multiplication we will get,

(7y + 2)11 = (6y – 5)5

77y + 22 = 30y – 25

77y – 30y = -25 – 22

47y = -47

y = -47/47

y = -1

Now verify the equation by putting y = -1.

(7y + 2)/5 = (6y – 5)/11

y =-1

(7(-1) + 2)/5 = (6(-1) – 5)/11

(-7 + 2)/5 = (-6 – 5)/11

-5/5 = -11/11

-1 = -1

Thus, L.H.S. = R.H.S.,
Hence, the equation is verified.

Question 4. x – 2x + 2 – 16/3x + 5 = 3 – 7/2x

Solution:

x – 2x + 2 – 16/3x + 5 = 3 – 7/2x

First rearrange the equation

x – 2x – 16x/3 + 7x/2 = 3 – 2 – 5

By taking LCM for 2 and 3 which is 6

(6x – 12x – 32x + 21x)/6 = -4

-17x/6 = -4

After cross-multiplying we will get,

-17x = -4×6

-17x = -24

x = -24/-17

x = 24/17

Now verify the equation by putting x = 24/17.

x – 2x + 2 – 16/3x + 5 = 3 – 7/2x

x = 24/17

24/17 – 2(24/17) + 2 – (16/3)(24/17) + 5 = 3 – (7/2)(24/17)

24/17 – 48/17 + 2 – 384/51 + 5 = 3 – 168/34

By taking 51 and 17 as the LCM we get,

(72 – 144 + 102 – 384 + 255)/51 = (102 – 168)/34

-99/51 = -66/34

-33/17 = -33/17

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

Hence, the equation is verified.

Question 5. 1/2x + 7x – 6 = 7x + 1/4

Solution:

1/2x + 7x – 6 = 7x + 1/4

First rearrange the equation

1/2x + 7x – 7x = 1/4 + 6 (by taking LCM)

1/2x = (1+ 24)/4

1/2x = 25/4

After cross-multiplying

4x = 25 × 2

4x = 50

x = 50/4

x = 25/2

Now verify the equation by putting x = 25/2.

1/2x + 7x – 6 = 7x + 1/4

x = 25/2

(1/2) (25/2) + 7(25/2) – 6 = 7(25/2) + 1/4

25/4 + 175/2 – 6 = 175/2 + 1/4

By taking LCM for 4 and 2 is 4

(25 + 350 – 24)/4 = (350+1)/4

351/4 = 351/4

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

Hence, the equation is verified.

Question 6. 3/4x + 4x = 7/8 + 6x – 6

Solution:

3/4x + 4x = 7/8 + 6x – 6

First rearrange the equation

3/4x + 4x – 6x = 7/8 – 6

By taking 4 and 8 as LCM

(3x + 16x – 24x)/4 = (7 – 48)/8

-5x/4 = -41/8

After cross-multiplying

-5x(8) = -41(4)

-40x = -164

x = -164/-40

= 82/20

= 41/10

Now verify the equation by putting x = 41/10.

3/4x + 4x = 7/8 + 6x – 6

x = 41/10

(3/4)(41/10) + 4(41/10) = 7/8 + 6(41/10) – 6

123/40 + 164/10 = 7/8 + 246/10 – 6

(123 + 656)/40 = (70 + 1968 – 480)/80

779/40 = 1558/80

779/40 = 779/40

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

Hence, the equation is verified.

Question 7. 7x/2 – 5x/2 = 20x/3 + 10

Solution:

7x/2 – 5x/2 = 20x/3 + 10

First rearrange the equation

7x/2 – 5x/2 – 20x/3 = 10

By taking LCM for 2 and 3 is 6

(21x – 15x – 40x)/6 = 10

-34x/6 = 10

After cross-multiplying

-34x = 60

x = 60/-34

= -30/17

Now verify the equation by putting x = -30/7.

7x/2 – 5x/2 = 20x/3 + 10

x = -30/7

(7-/2)(-30/17) – (5/2)(-30/17) = (20/3)(-30/17) + 10

-210/34 +150/34 = -600/51 + 10

-30/17 = (-600+510)/51

= -90/51

-30/17 = -30/17

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

Hence, the equation is verified.

Question 8. (6x + 1)/2 + 1 = (7x - 3)/3

Solution:

(6x+1)/2 + 1 = (7x-3)/3

(6x + 1 + 2)/2 = (7x – 3)/3

After cross-multiplying

(6x + 3)3 = (7x – 3)2

18x + 9 = 14x – 6

18x – 14x = -6 – 9

4x = -15

x = -15/4

Now verify the equation by putting x = -15/4.

(6x+1)/2 + 1 = (7x-3)/3

x = -15/4

(6(-15/4) + 1)/2 + 1 = (7(-15/4) – 3)/3

(3(-15/2) + 1)/2 + 1 = (-105/4 -3)/3

(-45/2 + 1)/2 + 1 = (-117/4)/3

(-43/4) + 1 = -117/12

(-43+4)/4 = -39/4

-39/4 = -39/4

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

Hence, the equation is verified.

Question 9. (3a-2)/3 + (2a+3)/2 = a + 7/6

Solution:

(3a-2)/3 + (2a+3)/2 = a + 7/6

First rearrange the equation

(3a-2)/3 + (2a+3)/2 – a = 7/6

By taking LCM for 2 and 3 which is 6

((3a-2)2 + (2a+3)3 – 6a)/6 = 7/6

(6a – 4 + 6a + 9 – 6a)/6 = 7/6

(6a + 5)/6 = 7/6

6a + 5 = 7

6a = 7-5

6a = 2

a = 2/6

a = 1/3

Now verify the equation by putting a = 1/3.

(3a-2)/3 + (2a+3)/2 = a + 7/6

a = 1/3

(3(1/3)-2)/3 + (2(1/3) + 3)/2 = 1/3 + 7/6

(1-2)/3 + (2/3 + 3)/2 = (2+7)/6

-1/3 + (11/3)/2 = 9/6

-1/3 + 11/6 = 3/2

(-2+11)/6 = 3/2

9/6 = 3/2

3/2 = 3/2

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

Hence, the equation is verified.

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

Solution:

x – (x-1)/2 = 1 – (x-2)/3

First rearrange the equation

x – (x-1)/2 + (x-2)/3 = 1

By taking LCM for 2 and 3 which is 6

(6x – (x-1)3 + (x-2)2)/6 = 1

(6x – 3x + 3 + 2x – 4)/6 = 1

(5x – 1)/6 = 1

After cross-multiplying

5x – 1 = 6

5x = 6 + 1

x = 7/5

Now verify the equation by putting x = 7/5.

x – (x-1)/2 = 1 – (x-2)/3

x = 7/5

7/5 – (7/5 – 1)/2 = 1 – (7/5 – 2)/3

7/5 – (2/5)/2 = 1 – (-3/5)/3

7/5 – 2/10 = 1 + 3/15

(14 – 2)/10 = (15+3)/15

12/10 = 18/15

6/5 = 6/5

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

Hence, the equation is verified.

Question 11. 3x/4 – (x-1)/2 = (x-2)/3

Solution:

3x/4 – (x-1)/2 = (x-2)/3

First rearrange the equation

3x/4 – (x-1)/2 – (x-2)/3 = 0

By taking LCM for 4, 2 and 3 which is 12

(9x – (x-1)6 – (x-2)4)/12 = 0

(9x – 6x + 6 – 4x + 8)/12 = 0

(-x + 14)/12 = 0

After cross-multiplying

-x + 14 = 0

x = 14

Now verify the equation by putting x = 14

3x/4 – (x-1)/2 = (x-2)/3

x = 14

3(14)/4 – (14-1)/2 = (14-2)/3

42/4 – 13/2 = 12/3

(42 – 26)/4 = 4

16/4 = 4

4 = 4

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

Hence, the equation is verified.

Question 12. 5x/3 – (x-1)/4 = (x-3)/5

Solution:

5x/3 – (x-1)/4 = (x-3)/5

First rearrange the equation

5x/3 – (x-1)/4 – (x-3)/5 = 0

By taking LCM for 3, 4 and 5 which is 60

((5x × 20) – (x-1)15 – (x-3)12)/60 = 0

(100x – 15x + 15 -12x + 36)/60 = 0

(73x + 51)/60 = 0

After cross-multiplying

73x + 51 = 0

x = -51/73

Now verify the equation by putting x = -51/73

5x/3 – (x-1)/4 = (x-3)/5

x = -51/73

(20x – (x-1)3)/12 = (-51/73 – 3)/5

(20x – 3x + 3)/12 = (-270/73)/5

(17x + 3)/12 = -270/365

(17(-51/73) + 3)/12 = -54/73

(-867/73 + 3)/12 = -54/73

((-867 + 219)/73)/12 = -54/73

(-648)/876 = -54/73

-54/73 = -54/73

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

Hence, the equation is verified.

Question 13. (3x+1)/16 + (2x-3)/7 = (x+3)/8 + (3x-1)/14

Solution:

(3x+1)/16 + (2x-3)/7 = (x+3)/8 + (3x-1)/14

First rearrange the equation

(3x+1)/16 + (2x-3)/7 – (x+3)/8 – (3x-1)/14 = 0

By taking LCM for 16, 7, 8 and 14 which is 112

((3x+1)7 + (2x-3)16 – (x+3)14 – (3x-1)8)/112 = 0

(21x + 7 + 32x – 48 – 14x – 42 – 24x + 8)/112 = 0

(21x + 32x – 14x – 24x + 7 – 48 – 42 + 8)/112 = 0

(15x – 75)/112 = 0

After cross-multiplying

15x – 75 = 0

15x = 75

x = 75/15

x = 5

Now verify the equation by putting x = 5

(3x+1)/16 + (2x-3)/7 = (x+3)/8 + (3x-1)/14

x = 5

(3(5)+1)/16 + (2(5)-3)/7 = (5+3)/8 + (3(5)-1)/14

(15+1)/16 + (10-3)/7 = 8/8 + (15-1)/14

16/16 + 7/7 = 8/8 + 14/14

1 + 1 = 1 + 1

2 = 2

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

Hence, the equation is verified.

Summary

Exercise 9.2 Set 1 focuses on solving basic linear equations in one variable. Students practice isolating variables through inverse operations, dealing with equations involving addition, subtraction, multiplication, and division. The set includes equations with parentheses and fractions, helping students build confidence in manipulating more complex expressions. This exercise reinforces the fundamental skills needed for solving linear equations, preparing students for more advanced algebraic concepts.