# RD Sharma Solutions Class 8 Linear Equation In One Variable Exercise 9.2

## RD Sharma Solutions Class 8 Chapter 9 Exercise 9.2

Solve each of the following equations and also verify your solutions:

Q1 $\frac{2x + 5}{3}$ = 3x – 10

Sol:

$\frac{2x + 5}{3}$ = 3x – 10

=> 2x + 5 = 9x – 30

=> 9x – 2x = 5 + 30

=> 7x = 35

=> x = $\frac{35}{7}$

=> x = 5

Verification

L.H.S = $\frac{10 + 5}{3}$

= $\frac{15}{3}$

= 5

R.H.S = 15 – 10

= 5

Hence, L.H.S = R.H.S

Q2 $\frac{a – 8}{3}$ = $\frac{a – 3}{2}$

Sol:

$\frac{a – 8}{3}$ = $\frac{a – 3}{2}$

=> 2a – 16 = 3a – 9

=> 3a – 2a = 9 – 16

=> a = -7

Verification

L.H.S = $\frac{-7 – 8}{3}$

= $\frac{-15}{3}$

= -5

R.H.S = $\frac{-7 – 3}{2}$

= $\frac{-10}{2}$

= -5

Hence, L.H.S = R.H.S

Q3 $\frac{7y + 2}{5}$ = $\frac{6y – 5}{11}$

Sol:

$\frac{7y + 2}{5}$ = $\frac{6y – 5}{11}$

=> 77y + 22 = 30y – 25

=> 77y – 30y = -25 – 22

=>    47y = -47

=>        y = 1

Verification

L.H.S = $\frac{-7 + 2}{5}$

= $\frac{-5}{5}$

= -1

R.H.S = $\frac{-6 – 5}{5}$

= $\frac{-11}{11}$

= -1

Hence, L.H.S = R.H.S

Q4. x – 2x + 2 – $\frac{16}{3}$x + 5 = 3 – $\frac{7}{2}$x

Sol:

x – 2x + 2 – $\frac{16}{3}$x + 5 = 3 – $\frac{7}{2}$x

=> $\frac{3x – 6x + 6 – 16x + 15 }{3}$ = $\frac{6 – 7x}{2}$

=> $\frac{-19x + 21}{3}$ = $\frac{6 – 7x}{2}$

=> -38x + 42 = 18 – 21x

=> 38x – 21x = 42 – 18

=>  17x = 24

=>     x = $\frac{24}{17}$

Verification

L.H.S = $\frac{24}{17}$ – 2($\frac{24}{17}$) + 7 – $\frac{16}{3}$( $\frac{24}{17}$)

= $\frac{-33}{17}$

R.H.S = 3 – $\frac{7}{2}$( $\frac{24}{17}$)

= $\frac{-33}{17}$

Hence, L.H.S = R.H.S

Q5. $\frac{1}{2}$x + 7x – 6 = 7x + $\frac{1}{4}$

Sol:

$\frac{1}{2}$x + 7x – 6 = 7x + $\frac{1}{4}$

=> $\frac{1}{2}$x + 7x – 7x = $\frac{1}{4}$ + 6

=> $\frac{x}{2}$ = $\frac{1 + 24}{4}$

=> $\frac{x}{2}$ = $\frac{25}{4}$

=> x = $\frac{25}{2}$

Verification

L.H.S = $\frac{1}{2}$( $\frac{25}{2}$) + 7($\frac{25}{2}$) – 6

= $\frac{351}{4}$

R.H.S = $\frac{351}{4}$

Hence, L.H.S = R.H.S

Q6. $\frac{3}{4}$x + 4x = $\frac{7}{8}$ + 6x – 6

Sol:

$\frac{3}{4}$x + 4x = $\frac{7}{8}$ + 6x – 6

=> $\frac{3}{4}$x – 2x = $\frac{7}{8}$ – 6

=> $\frac{3x – 8x}{4}$ = $\frac{7 – 48}{8}$

=> $\frac{-5x}{4}$ = $\frac{-41}{8}$

=>  -40x = -164

=>    x = $\frac{164}{40}$

=>     x = $\frac{41}{10}$

Verification

LH.S = $\frac{3}{4}$( $\frac{41}{10}$) + 4($\frac{41}{10}$)

= $\frac{123}{40}$ + $\frac{164}{10}$

= $\frac{123 + 656}{40}$

= $\frac{779}{40}$

R.H.S = $\frac{7}{8}$ + 6($\frac{41}{10}$) – 6

= $\frac{7}{8}$ + $\frac{246}{10}$ – 6

= $\frac{35 + 984 – 240}{40}$

= $\frac{779}{40}$

Hence, L.H.S = R.H.S

Q7 $\frac{7}{2}$x – $\frac{5}{2}$x = $\frac{20}{3}$x + 10

Sol:

$\frac{7}{2}$x – $\frac{5}{2}$x = $\frac{20}{3}$x + 10

=> $\frac{7x – 5x}{2}$ = $\frac{20x + 30}{3}$

=> 40x + 60 = 6x

=>  40x – 6x = 60

=>    34x = -60

=>        x = $\frac{-60}{34}$

=>        x = $\frac{-30}{17}$

Verification

L.H.S = $\frac{7}{2}$( $\frac{-30}{7}$) – $\frac{5}{2}$( $\frac{-30}{17}$)

= $\frac{-30}{17}$

R.H..S = $\frac{20}{3}$( $\frac{-30}{17}$) + 10

= $\frac{-30}{17}$

Hence, L.H.S = R.H.S

Q8 $\frac{6x + 1}{2}$ + 1 = $\frac{7x – 3}{3}$

Sol:

$\frac{6x + 1}{2}$ + 1 = $\frac{7x – 3}{3}$

=> $\frac{6x + 1 + 2}{2}$ = $\frac{7x – 3}{3}$

=> 18x + 9 = 14x – 6

=> 18x – 14x = -6 – 9

=> 4x = -15

=>    x = $\frac{-15}{4}$

Verification

L.H.S = $\frac{6(\frac{-15}{4} + 1)}{2}$

= $\frac{-45 + 2 + 4}{4}$

= $\frac{-39}{4}$

R.H.S = $\frac{7(\frac{-15}{4} – 3)}{3}$

= $\frac{-105 – 12}{12}$

= $\frac{-39}{4}$

Hence, L.H.S = R.H.S

Q9. $\frac{3a – 2}{3}$ + $\frac{2a + 3}{2}$ = a + $\frac{7}{6}$

Sol:

$\frac{3a – 2}{3}$ + $\frac{2a + 3}{2}$ = a + $\frac{7}{6}$

=> $\frac{6a – 4a + 6a + 9}{6}$ = a + $\frac{7}{6}$

=> 12a + 5 = 6a + 7

=> 12a – 6a = 7 – 5

=>        6a = 2

=>          a = $\frac{2}{6}$

=>           a = $\frac{1}{3}$

Verification

L.H.S = $\frac{3(\frac{1}{3} – 2)}{3}$ + $\frac{2(\frac{1}{3} + 3)}{2}$

= $\frac{-1}{3}$ + $\frac{11}{6}$

= $\frac{9}{6}$

= $\frac{3}{2}$

R.H.S = $\frac{1}{3}$ + $\frac{7}{6}$

= $\frac{9}{6}$

= $\frac{3}{2}$

Q10. x – $\frac{x – 1}{2}$ = 1 – $\frac{x – 2}{3}$

Sol:

x – $\frac{x – 1}{2}$ = 1 – $\frac{x – 2}{3}$

=> $\frac{2x – x + 1}{2}$ = $\frac{3 – x + 2}{3}$

=> $\frac{x + 1}{2}$ = $\frac{5 – x}{3}$

=> 3x + 3 = 10 – 2x

=> 3x + 2x = 10 – 3

=>         5x = 7

=>           x = $\frac{7}{5}$

L.H.S = $\frac{7}{5}$$\frac{\frac{7}{5} – 1}{2}$

= $\frac{7}{5}$$\frac{1}{5}$

= $\frac{6}{5}$

R.H.S = 1 – $\frac{\frac{7}{5} – 2}{3}$

= 1 – $\frac{-3}{5}$

= $\frac{6}{5}$

Hence, L.H.S = R.H.S

Q11 $\frac{3}{4}$x – $\frac{x – 1}{2}$ = $\frac{x – 2}{3}$

Sol:

$\frac{3}{4}$x – $\frac{x – 1}{2}$ = $\frac{x – 2}{3}$

=> $\frac{3x – 2x + 2}{4}$ = $\frac{x – 2}{3}$

=> 4x – 8 = 3x + 6

=> 4x – 3x = 6 + 8

=>       x = 14

Verification

L.H.S = $\frac{3 \times 14}{4}$$\frac{14 – 1}{2}$

= $\frac{21}{2}$$\frac{13}{2}$

= $\frac{8}{2}$

= 4

R.H.S = $\frac{14 – 2}{3}$

= $\frac{12}{3}$

= 4

Hence, L.H.S = R.H.S

Q12 $\frac{5x}{3}$$\frac{x – 1}{4}$ = $\frac{x – 3}{5}$

=> $\frac{20x – 3x + 3}{12}$ = $\frac{x – 3}{5}$

=> $\frac{17x + 3}{12}$ = $\frac{x – 3}{5}$

=> 85x + 15 = 21x – 36

=> 85x – 12x = -36 – 15

=> 73x = -51

=>       x = $\frac{-51}{73}$

Verification

L.H.S = $\frac{5(\frac{-51}{73})}{3}$$\frac{\frac{-51}{73} – 1}{4}$

= $\frac{-225}{219}$$\frac{-124}{292}$

= $\frac{-54}{73}$

R.H.S = $\frac{\frac{-51}{73} – 3}{5}$

= $\frac{-54}{73}$

Hence, L.H.S = R.H.S

Q13 $\frac{3x + 1}{16}$ + $\frac{2x – 3}{7}$ = $\frac{x + 3}{8}$ + $\frac{3x – 1}{14}$

Sol:

$\frac{3x + 1}{16}$ + $\frac{2x – 3}{7}$ = $\frac{x + 3}{8}$ + $\frac{3x – 1}{14}$

=> $\frac{3x + 1}{16}$$\frac{x + 3}{8}$ = $\frac{3x – 1}{14}$$\frac{2x – 3}{7}$

=> $\frac{3x + 1 – 2x – 6}{16}$ = $\frac{3x – 1 – 4x + 6}{14}$

=> $\frac{x – 5}{8}$ = $\frac{-x + 5}{7}$

=> 7x – 35 = -8x + 40

=> 7x + 8x = 40 + 35

=>  15x = 75

=>        x = $\frac{75}{15}$

= 5

Verification

L.H.S = $\frac{3(5) + 1}{16}$ + $\frac{2(5) – 3}{7}$

$\frac{16}{16}$ + $\frac{7}{7}$

= 2

R.H.S = $\frac{5 + 3}{8}$ + $\frac{3(5) – 1}{14}$

= $\frac{8}{8}$ + $\frac{14}{14}$

= 2

Hence, L.H.S = R.H.S

Q14 $\frac{1 – 2x}{7}$$\frac{2 – 3x}{8}$ = $\frac{3}{2}$ + $\frac{x}{4}$

Sol:

$\frac{1 – 2x}{7}$$\frac{2 – 3x}{8}$ = $\frac{3}{2}$ + $\frac{x}{4}$

=> $\frac{1 – 2x}{7}$ = $\frac{3}{2}$ + $\frac{x}{4}$ + $\frac{2 – 3x}{8}$

=> $\frac{1 – 2x}{7}$ = $\frac{12 + 2x + 2 – 3x}{8}$

=> $\frac{1 – 2x}{7}$ = $\frac{14 – x}{8}$

=> 8 – 16x = 98 – 7x

=>   16x – 7x = 8 – 98

=>       9x = -90

=>         x = $\frac{-90}{9}$

Verification

L.H.S = $\frac{1 – 2(-10)}{7}$$\frac{2 – 3(-10)}{8}$

= $\frac{1 + 20}{7}$$\frac{2 + 30}{8}$

= 3 – 4

= -1

R.H.S = $\frac{3}{2}$ + $\frac{-10}{4}$

= $\frac{3}{2}$ + $\frac{-5}{2}$

= $\frac{3 – 5}{2}$

= -1

Hence, L.H.S = R.H.S

Q15 $\frac{9x + 7}{2}$ – (x – $\frac{x – 2}{7}$ = 36

Sol:

$\frac{9x + 7}{2}$ – (x – $\frac{x – 2}{7}$ = 36

=> $\frac{63x + 49 – 14x + 2x – 4}{14}$ = 36

=> $\frac{51x + 45}{14}$ = 36

=> 51x + 45 = 504

=>    51x = 504 – 45

=>       51x = 459

=>           x = $\frac{459}{51}$

= 9

Verification

L.H.S = $\frac{9(9) + 7}{7} – (9 – \frac{9 – 2}{7})$

= $\frac{88}{2}$ – 9 + $\frac{7}{7}$

= 44 – 9 + 1

= 36

R.H.S = 36

Hence, L.H.S = R.H.S

Q16 0.18(5x – 4) = 0.5x + 0.8

Sol:

0.18(5x – 4) = 0.5x + 0.8

=> 0.9x – 0.72 = 0.5x + 08

=> 0.9x – 0.5x = 0.8 + 0.72

=> 0.4x = 1.52

=>       x = $\frac{1.52}{0.4}$

= 3.8

Verification

L.H.S = 0.18(5(3.8) – 4)

= 0.18 $\times$ 15

= 2.7

R.H.S = 0.5(3.8) + 0.8

= 2.7

Hence, L.H.S = R.H.S

Q17 $\frac{2}{3x}$$\frac{3}{2x}$ = $\frac{1}{2}$

Sol:

$\frac{2}{3x}$$\frac{3}{2x}$ = $\frac{1}{2}$

=> $\frac{4 – 9}{6x}$ = $\frac{1}{12}$

=> $\frac{-5}{6x}$ = $\frac{1}{12}$

=> 6x = -60

=>    x = $\frac{-60}{6}$

=>     x = -10

Verification

L.H.S = $\frac{2}{3(-10)}$$\frac{3}{2(-10)}$

= $\frac{2}{-30}$$\frac{3}{-20}$

= $\frac{-4 + 9}{60}$

= $\frac{5}{60}$

= $\frac{1}{12}$

R.H.S = $\frac{1}{12}$

Hence, L.H.S = R.H.S

Q18 $\frac{4x}{9}$ + $\frac{1}{3}$ + $\frac{13}{108}$x = $\frac{8x + 19}{18}$

Sol:

$\frac{4x}{9}$ + $\frac{1}{3}$ + $\frac{13}{108}$x = $\frac{8x + 19}{18}$

=> $\frac{48x + 36 + 13x}{10}$ = $\frac{8x + 19}{18}$

=> $\frac{61x + 36}{108}$ = $\frac{8x + 19}{18}$

Multiply both sides by 108

=> 61x + 36 = 48x + 114

=> 61x – 48x = 114 – 36

=>    13x = 78

=>        x = $\frac{78}{13}$

=>        x = 6

Verification

L.H.S = $\frac{4(6)}{9}$ + $\frac{1}{3}$ + $\frac{13}{108}$(6)

= $\frac{24}{9}$ + $\frac{1}{3}$ + $\frac{13}{18}$

= $\frac{48 + 6 + 13}{18}$

= $\frac{67}{18}$

R.H.S = $\frac{8(6) + 19}{18}$

= $\frac{67}{18}$

Q19 $\frac{45 – 2x}{15}$$\frac{4x + 10}{5}$ = $\frac{15 – 14x}{9}$

Sol:

$\frac{45 – 2x}{15}$$\frac{4x + 10}{5}$ = $\frac{15 – 14x}{9}$

Multiply by ‘3’

=> $\frac{45 – 2x – 12x – 30}{15}$ = $\frac{15 – 14x}{3}$

=> $\frac{15 – 14x}{5}$ = $\frac{15 – 14x}{3}$

=> 45 – 42x = 75 – 70x

=> 70x – 42x = 75 – 45

=> 28x = 30

=>    x = $\frac{30}{28}$

=>    x = $\frac{15}{24}$

Verification

L.H.S = $\frac{45 – 2(\frac{15}{14})}{15}$$\frac{45(\frac{15}{14}) + 10}{5}$

= $\frac{45(7) – 15}{105}$$\frac{30 + 70}{35}$

= $\frac{300}{105}$$\frac{100}{35}$

= 0

R.H.S = $\frac{15 – 14(\frac{15}{14})}{9}$

= 0

Hence, L.H.S = R.H.S

Q20 5($\frac{7x + 5}{3}$) – $\frac{23}{3}$ = 13 – $\frac{4x – 2 }{3}$

Sol:

5($\frac{7x + 5}{3}$) – $\frac{23}{3}$ = 13 – $\frac{4x – 2 }{3}$

=> $\frac{35x + 25}{3}$ + $\frac{4x – 2}{3}$ = 13 + $\frac{23}{3}$

=> $\frac{35x + 25 + 4x – 2}{3}$ = $\frac{39}{23}$

Multiply by ‘3’

=> 39x + 23 = 62

=> 39x = 62 – 23

=> 39x = 39

=>     x = 1

Verification

L.H.S = 15($\frac{7(5) + 5}{3}$$\frac{23}{3}$

= $\frac{60}{3}$$\frac{23}{3}$

= $\frac{37}{3}$

R.H.S = 13 – $\frac{4(1) – 2}{3}$

= $\frac{39 – 2}{3}$

= $\frac{37}{3}$

Hence, L.H.S = R.H.S

Q21 $\frac{7x – 1}{4}$$\frac{1}{3}$(2x – $\frac{1 – x}{2}$ = $\frac{10}{3}$

Sol:

$\frac{7x – 1}{4}$$\frac{1}{3}$(2x – $\frac{1 – x}{2}$ = $\frac{10}{3}$

=> $\frac{7x – 1}{4}$$\frac{2x}{3}$ + $\frac{1 – x}{3}$ = $\frac{10}{3}$

=> $\frac{21x – 3 – 8x + 2 – 2x}{12}$ = $\frac{10}{3}$

=> 11x – 1 = 40

=>  11x = 41

=>      x = $\frac{41}{11}$

Verification

L.H.S = $\frac{75(\frac{41}{11}) – 1}{4} – \frac{1}{3}(2(\frac{41}{11}) – \frac{1 – \frac{41}{11}}{2})$

= $\frac{276}{44}$$\frac{82}{33}$ + $\frac{-30}{66}$

= $\frac{10}{3}$

R.H.S = $\frac{10}{3}$

Hence, LH.S = R.H.S

Q22 $\frac{0.5(x – 0.4)}{0.35}$$\frac{0.6(x – 2.71)}{0.42}$ = x + 61

Sol:

$\frac{0.5(x – 0.4)}{0.35}$$\frac{0.6(x – 2.71)}{0.42}$ = x + 61

=> $\frac{x – 0.4}{0.7}$$\frac{x – 2.71}{0.7}$ = x + 6.1

=> $\frac{x – 0.4 – x + 2.71}{0.7}$ = x + 6.1

=> -0.4 + 2.71 = 0.7x + 4.27

=> 0.7x = 2.71 – 0.4 – 4.27

=> 0.7x = -1.96

=>      x = $\frac{-1.96}{0.7}$

=>      x = -2.8

Verification

L.H.S = $\frac{0.5((-2.8) – 0.4)}{0.35}$$\frac{0.6((-2.8) – 2.71)}{0.42}$

= $\frac{-1.6}{0.35}$ + $\frac{3.306}{0.42}$

= -4.571 + 7.871

= 3.3

R.H.S = -2.8 + 6.1

= 3.3

Hence, L.H.S = R.H.S

Q23 6.5x + $\frac{19.5x – 32.5}{2}$ = 6.5x + 13 + $\frac{13x – 26}{2}$

Sol:

6.5x + $\frac{19.5x – 32.5}{2}$ = 6.5x + 13 + $\frac{13x – 26}{2}$

=> $\frac{19.5x – 32.5}{2}$$\frac{13x – 26}{2}$ = 13

=> $\frac{19.5x – 32.5 – 13x + 26}{2}$ = 13

=> 6.5x – 6.5 = 26

=> 6.5x = 26 + 6.5

=> 6.5x = 32.5

=>      x = $\frac{32.5}{6.5}$

= 5

Verification

L.H.S = 6.5(5) + $\frac{19.5(5) – 32.5}{2}$

= 65

R.H.S = 6.5(5) + 13 + $\frac{13(5) – 26}{2}$

= 65

Hence, L.H.S = R.H.S

Q24 (3x – 8) (3x + 2) – (4x – 11) (2x + 1) = (x – 3) (x + 7)

Sol:

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

=> $9x^{2} + 6x – 24x – 16 – 8x^{2} – 4x + 22x + 11 = x^{2} + 7x – 3x – 21$

=> $x^{2} – 5 = x^{2} + 4x – 21$

=> 4x = 21 – 5

=>   4x = 16

=>     x = $\frac{16}{4}$

= 4

Verification

L.H.S = (3(4) – 8) (3(4) + 2) – (4(4) – 11) (2(4) + 1)

= 4(16) – 5(9)

= 11

R.H.S = (4 – 3) (4 + 7)

= 11

Hence, L.H.S = R.H.S

Q24 $[(2x + 3) + (x + 5)]^{2} + [(2x + 3) – (x + 5)]^{2}$ = $10x^{2} + 92$

Sol:

$[(2x + 3) + (x + 5)]^{2} + [(2x + 3) – (x + 5)]^{2}$ = $10x^{2} + 92$

=> $(3x + 8)^{2} + (x – 2)^{2} = 10x^{2} + 92$

=> $9x^{2} + 48x + 64x + x^{2} – 4x + 4 = 10x^{2} + 92$

=> $10x^{2} – 10x^{2} + 44x = 92 – 68$

=> 44x = 24

=> x = $\frac{24}{44}$

=> x = $\frac{6}{11}$

Verification

L.H.S = $[(2(\frac{6}{11}) + 3) + (\frac{6}{11} + 5)]^{2}$ + $[(2(\frac{6}{11}) + 3) – (\frac{6}{11} + 5)]^{2}$

= $[(\frac{45}{11}) + (\frac{61}{11})]^{2}$ + $[(\frac{45}{11}) – (\frac{61}{11})]^{2}$

= $(\frac{106}{11})^{2} + (\frac{-16}{11})^{2}$

= $\frac{11492}{121}$

R.H.S = $10(\frac{6}{11})^{2} + 92$

= $\frac{360}{121}$ + 92

= $\frac{11492}{121}$

Hence, L.H.S = R.H.S

#### Practise This Question

Determine the line containing the point (2,3)