myhelper: RS Aggarwal solution class 8 chapter 8 Linear Equations Exercise 8A

Exercise 8A

Page-107

Q1 | Ex-8A | Class 8 | RS AGGARWAL | Linear Equations Solution | myhelper

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Question 1:

Solve:
8x + 3 = 27 + 2x

Answer 1:

$8 x + 3 = 27 + 2 x$

$\Rightarrow 8 x - 2 x = 27 - 3$

$\Rightarrow 6 x = 24$

$\Rightarrow x = \frac{24}{6} = 4$

$∴ x = 4$

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Question 2:

Solve:
5x + 7 = 2x − 8

Answer 2:

$5 x + 7 = 2 x - 8$

$\Rightarrow 5 x - 2 x = - 8 - 7$

$\Rightarrow 3 x = - 15$

$\Rightarrow x = \frac{- 15}{3} = - 5$

$∴ x = - 5$

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Question 3:

Solve:
2z − 1 = 14 − z

Answer 3:

$2 z - 1 = 14 - z$

$\Rightarrow 2 z + z = 14 + 1$

$\Rightarrow 3 z = 15$

$\Rightarrow z = \frac{15}{3} = 5$

$∴ z = 5$

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Question 4:

Solve:
9x + 5 = 4(x − 2) + 8

Answer 4:

$9 x + 5 = 4 (x - 2) + 8$

$\Rightarrow 9 x + 5 = 4 x - 8 + 8$

$\Rightarrow 9 x + 5 = 4 x$

$\Rightarrow 9 x - 4 x = - 5$

$\Rightarrow 5 x = - 5$

$\Rightarrow x = \frac{- 5}{5} = - 1$

$∴ x = - 1$

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Question 5:

Solve:
$\frac{7 y}{5} = y - 4$

Answer 5:

$\frac{7 y}{5} = y - 4$

$By cross multiplication :$

$\Rightarrow 7 y = 5 (y - 4)$

$\Rightarrow 7 y = 5 y - 20$

$\Rightarrow 7 y - 5 y = - 20$

$\Rightarrow 2 y = - 20$

$\Rightarrow y = \frac{- 20}{2} = - 10$

$∴ y = - 10$

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Question 6:

Solve:
$3 x + \frac{2}{3} = 2 x + 1$

Answer 6:

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

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

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

$(L . C . M . of 1 and 3 is 3)$

$\Rightarrow x = \frac{3 - 2}{3}$

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

$∴ x = \frac{1}{3}$

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Question 7:

Solve:
15(y − 4) − 2(y − 9) + 5(y + 6) = 0

Answer 7:

$15 (y - 4) - 2 (y - 9) + 5 (y + 6) = 0$

$\Rightarrow 15 y - 60 - 2 y + 18 + 5 y + 30 = 0$

$\Rightarrow 15 y - 2 y + 5 y - 60 + 18 + 30 = 0$

$\Rightarrow 18 y - 12 = 0$

$\Rightarrow 18 y = 12$

$\Rightarrow y = \frac{12}{18} = \frac{2}{3}$

$∴ y = \frac{2}{3}$

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Question 8:

Solve:
3(5x − 7) − 2(9x − 11) = 4(8x − 13) − 17

Answer 8:

$3 (5 x - 7) - 2 (9 x - 11) = 4 (8 x - 13) - 17$

$\Rightarrow 15 x - 21 - 18 x + 22 = 32 x - 52 - 17$

$\Rightarrow 15 x - 18 x - 21 + 22 = 32 x - 69$

$\Rightarrow - 3 x + 1 = 32 x - 69$

$\Rightarrow 1 + 69 = 32 x + 3 x$

$\Rightarrow 70 = 35 x$

$\Rightarrow 35 x = 70$

$(b y transposition)$

$\Rightarrow x = \frac{70}{35} = 2$

$∴ x = 2$

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Question 9:

Solve:
$\frac{x - 5}{2} - \frac{x - 3}{5} = \frac{1}{2}$

Answer 9:

$\frac{x - 5}{2} - \frac{x - 3}{5} = \frac{1}{2}$

$\Rightarrow 10 (\frac{x - 5}{2}) - 10 (\frac{x - 3}{5}) = 10 (\frac{1}{2})$

$(multiplying throughout by 10, which is the L . C . M . of 2, 2 and 5)$

$\Rightarrow 5 (x - 5) - 2 (x - 3) = 5$

$\Rightarrow 5 x - 25 - 2 x + 6 = 5$

$\Rightarrow 5 x - 2 x - 25 + 6 = 5$

$\Rightarrow 3 x - 19 = 5$

$\Rightarrow 3 x = 5 + 19$

$\Rightarrow 3 x = 24$

$\Rightarrow x = \frac{24}{3} = 8$

$∴ x = 8$

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Question 10:

Solve:
$\frac{3 t - 2}{4} - \frac{2 t + 3}{3} = \frac{2}{3} - t$

Answer 10:

$\frac{3 t - 2}{4} - \frac{2 t + 3}{3} = \frac{2}{3} - t$

$\Rightarrow \frac{3 t - 2}{4} - \frac{2 t + 3}{3} = \frac{2 - 3 t}{3}$

$(3 is the L . C . M . of 1 and 3)$

$\Rightarrow 12 (\frac{3 t - 2}{4}) - 12 (\frac{2 t + 3}{3}) = 12 (\frac{2 - 3 t}{3})$

$(multiplying throughout by 12, which is the L . C . M . of 4, 3 and 3)$

$\Rightarrow 3 (3 t - 2) - 4 (2 t + 3) = 4 (2 - 3 t)$

$\Rightarrow 9 t - 6 - 8 t - 12 = 8 - 12 t$

$\Rightarrow 9 t - 8 t - 6 - 12 = 8 - 12 t$

$\Rightarrow t - 18 = 8 - 12 t$

$\Rightarrow t + 12 t = 18 + 8$

$\Rightarrow 13 t = 26$

$\Rightarrow t = \frac{26}{13} = 2$

$∴ t = 2$

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Question 11:

Solve:
$\frac{2 x + 7}{5} - \frac{3 x + 11}{2} = \frac{2 x + 8}{3} - 5$

Answer 11:

$\frac{2 x + 7}{5} - \frac{3 x + 11}{2} = \frac{2 x + 8}{3} - 5$

$\Rightarrow \frac{2 x + 7}{5} - \frac{3 x + 11}{2} = \frac{2 x + 8 - 15}{3}$

$(L . C . M . of 3 and 1 is 3)$

$\Rightarrow 30 (\frac{2 x + 7}{5}) - 30 (\frac{3 x + 11}{2}) = 30 (\frac{2 x + 8 - 15}{3})$

$(multiplyin g throughout by 30, which is the L . C . M . of 5, 2 and 3)$

$\Rightarrow 6 (2 x + 7) - 15 (3 x + 11) = 10 (2 x + 8 - 15)$

$\Rightarrow 12 x + 42 - 45 x - 165 = 20 x - 70$

$\Rightarrow 12 x - 45 x + 42 - 165 = 20 x - 70$

$\Rightarrow - 33 x - 123 = 20 x - 70$

$\Rightarrow - 33 x - 20 x = 123 - 70$

$\Rightarrow - 53 x = 53$

$\Rightarrow x = \frac{53}{- 53}$

$\Rightarrow x = - 1 ∴ x = - 1$

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Question 12:

Solve:
$\frac{5 x - 4}{6} = 4 x + 1 - \frac{3 x + 10}{2}$

Answer 12:

$\frac{5 x - 4}{6} = 4 x + 1 - \frac{3 x + 10}{2}$

$\Rightarrow \frac{5 x - 4}{6} = \frac{2 (4 x + 1) - 3 x - 10}{2}$

$(L . C . M . of 1 and 2 is 2)$

$\Rightarrow \frac{5 x - 4}{6} = \frac{8 x + 2 - 3 x - 10}{2}$

$\Rightarrow \frac{5 x - 4}{6} = \frac{8 x - 3 x + 2 - 10}{2}$

$\Rightarrow \frac{5 x - 4}{6} = \frac{5 x - 8}{2}$

$\Rightarrow 2 (5 x - 4) = 6 (5 x - 8)$

$\Rightarrow 10 x - 8 = 30 x - 48$

$\Rightarrow 10 x - 30 x = - 48 + 8$

$\Rightarrow - 20 x = - 40$

$\Rightarrow x = \frac{- 40}{- 20} = 2 ∴ x = 2$

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Question 13:

Solve:
$5 x - \frac{1}{3} (x + 1) = 6 (x + \frac{1}{30})$

Answer 13:

$5 x - \frac{1}{3} (x + 1) = 6 (x + \frac{1}{30})$

$\Rightarrow 5 x - \frac{1 (x + 1)}{3} = 6 (\frac{30 x + 1}{30})$

$(L . C . M . of 1 and 30 is 30)$

$\Rightarrow 5 x - \frac{(x + 1)}{3} = \frac{30 x + 1}{5}$

$\Rightarrow \frac{15 x - x - 1}{3} = \frac{30 x + 1}{5}$

$(L . C . M . of 1 and 3 is 3)$

$\Rightarrow \frac{14 x - 1}{3} = \frac{30 x + 1}{5}$

$\Rightarrow 5 (14 x - 1) = 3 (30 x + 1)$

$(by cross multiplication)$

$\Rightarrow 70 x - 5 = 90 x + 3$

$\Rightarrow 70 x - 90 x = 3 + 5$

$\Rightarrow - 20 x = 8$

$\Rightarrow x = \frac{8}{- 20} = \frac{- 2}{5} ∴ x = - \frac{2}{5}$

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Question 14:

Solve:
$4 - \frac{2 (z - 4}{3} = \frac{1}{2} (2 z + 5)$

Answer 14:

$4 - \frac{2 (z - 4)}{3} = \frac{1}{2} (2 z + 5)$

$\Rightarrow \frac{12 - 2 (z - 4)}{3} = \frac{1 (2 z + 5)}{2}$

$(L . C . M . of 1 and 3 is 3)$

$\Rightarrow \frac{12 - 2 z + 8}{3} = \frac{2 z + 5}{2}$

$\Rightarrow \frac{20 - 2 z}{3} = \frac{2 z + 5}{2}$

$\Rightarrow 2 (20- 2 z) = 3 (2 z + 5)$

$(by cross multiplication)$

$\Rightarrow 40 - 4 z = 6 z + 15$

$\Rightarrow 40 - 15 = 6 z + 4 z \Rightarrow 25 = 10 z$

$\Rightarrow 10 z = 25 (by transposition)$

$\Rightarrow z = \frac{25}{10} = \frac{5}{2}$

$∴ z = \frac{5}{2}$

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Question 15:

Solve:
$\frac{3 (y - 5)}{4} - 4 y = 3 - \frac{(y - 3)}{2}$

Answer 15:

$\frac{3 (y - 5)}{4} - 4 y = 3 - \frac{(y - 3)}{2}$

$\Rightarrow \frac{3 y - 15}{4} - 4 y = 3 - \frac{y - 3}{2}$

$\Rightarrow \frac{3 y - 15 - 16 y}{4} = 3 - \frac{y - 3}{2}$

$(L . C . M . of 4 and 1 is 4)$

$\Rightarrow \frac{- 13 y - 15}{4} = \frac{6 - y + 3}{2}$

$\Rightarrow \frac{- 13 y - 15}{4} = \frac{9 - y}{2}$

$\Rightarrow 2 (- 13 y - 15) = 4 (9 - y)$

$\Rightarrow - 26 y - 30 = 36 - 4 y$

$\Rightarrow - 26 y + 4 y = 36 + 30$

$\Rightarrow - 22 y = 66$

$\Rightarrow 22 y = - 66$

$(multiplying both the sides with a - ve sign)$

$\Rightarrow y = - \frac{66}{22} = - 3$

$∴ y = - 3$

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Question 16:

Solve:
$\frac{8 x - 3}{3 x} = 2$

Answer 16:

$\frac{8 x - 3}{3 x} = 2$

$\Rightarrow 8 x - 3 = 2 (3 x)$

$(by cross multiplication)$

$\Rightarrow 8 x - 3 = 6 x$

$\Rightarrow 8 x - 6 x = 3$

$\Rightarrow 2 x = 3$

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

$∴ x = \frac{3}{2}$

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Question 17:

Solve:
$\frac{9 x}{7 - 6 x} = 15$

Answer 17:

$\frac{9 x}{7 - 6 x} = 15$

$\Rightarrow \frac{9 x}{7 - 6 x} = \frac{15}{1}$

$\Rightarrow 1 (9 x) = 15 (7 - 6 x)$

$(by cross multiplication)$

$\Rightarrow 9 x = 105 - 90 x$

$\Rightarrow 9 x + 90 x = 105$

$\Rightarrow 99 x = 105$

$\Rightarrow x = \frac{105}{99} = \frac{35}{33}$

$∴ x = \frac{35}{33}$

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Question 18:

Solve:
$\frac{3 x}{5 x + 2} = - 4$

Answer 18:

$\frac{3 x}{5 x + 2} = - 4$

$\Rightarrow \frac{3 x}{5 x + 2} = \frac{- 4}{1}$

$\Rightarrow 1 (3 x) = - 4 (5 x + 2)$

$(by cross multiplication)$

$\Rightarrow 3 x = - 20 x - 8$

$\Rightarrow 3 x + 20 x = - 8$

$\Rightarrow 23 x = - 8$

$\Rightarrow x = \frac{- 8}{23}$

$∴ x = \frac{- 8}{23}$

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Question 19:

Solve:
$\frac{6 y - 5}{2 y} = \frac{7}{9}$

Answer 19:

$\frac{6 y - 5}{2 y} = \frac{7}{9}$

$\Rightarrow 9 (6 y - 5) = 7 (2 y)$

$(by cross multiplication)$

$\Rightarrow 54 y - 45 = 14 y$

$\Rightarrow 54 y - 14 y = 45$

$\Rightarrow 40 y = 45$

$\Rightarrow y = \frac{45}{40} = \frac{9}{8}$

$∴ y = \frac{9}{8}$

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Question 20:

Solve:
$\frac{2 - 9 z}{17 - 4 z} = \frac{4}{5}$

Answer 20:

$\frac{2 - 9 z}{17 - 4 z} = \frac{4}{5}$

$\Rightarrow 5 (2 - 9 z) = 4 (17 - 4 z)$

$(by cross multiplication)$

$\Rightarrow 10 - 45 z = 68 - 16 z$

$\Rightarrow 10 - 68 = 45 z - 16 z$

$\Rightarrow - 58 = 29 z$

$\Rightarrow 29 z = - 58 (by transposition)$

$\Rightarrow z = \frac{- 58}{29} = - 2$

$∴ z = - 2$

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Question 21:

Solve:
$\frac{4 x + 7}{9 - 3 x} = \frac{1}{4}$

Answer 21:

$\frac{4 x + 7}{9 - 3 x} = \frac{1}{4}$

$\Rightarrow 4 (4 x + 7) = 1 (9 - 3 x)$

$(by cross multiplication)$

$\Rightarrow 16 x + 28 = 9 - 3 x$

$\Rightarrow 16 x + 3 x = 9 - 28$

$\Rightarrow 19 x = - 19$

$\Rightarrow x = \frac{- 19}{19} = - 1$

$∴ x = - 1$

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Question 22:

Solve:
$\frac{7 y + 4}{y + 2} = \frac{- 4}{3}$

Answer 22:

$\frac{7 y + 4}{y + 2} = \frac{- 4}{3}$

$\Rightarrow 3 (7 y + 4) = - 4 (y + 2)$

$(by cross multiplication)$

$\Rightarrow 21 y + 12 = - 4 y - 8$

$\Rightarrow 21 y + 4 y = - 8 - 12$

$\Rightarrow 25 y = - 20$

$\Rightarrow y = \frac{- 20}{25} = \frac{- 4}{5}$

$∴ y = \frac{- 4}{5}$

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Question 23:

Solve:
$\frac{15 (2 - y) - 5 (y + 6)}{1 - 3 y} = 10$

Answer 23:

$\frac{15 (2 - y) - 5 (y + 6)}{1 - 3 y} = 10$

$\Rightarrow \frac{30 - 15 y - 5 y - 30}{1 - 3 y} = 10$

$\Rightarrow \frac{- 20 y}{1 - 3 y} = 10$

$\Rightarrow 1 (- 20 y) = 10 (1 - 3 y)$

$(by cross multiplication)$

$\Rightarrow - 20 y = 10 - 30 y$

$\Rightarrow - 20 y + 30 y = 10$

$\Rightarrow 10 y = 10$

$\Rightarrow y = \frac{10}{10} = 1$

$∴ y = 1$

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Question 24:

Solve:
$\frac{2 x - (7 - 5 x)}{9 x - (3 + 4 x)} = \frac{7}{6}$

Answer 24:

$\frac{2 x - (7 - 5 x)}{9 x - (3 + 4 x)} = \frac{7}{6}$

$\Rightarrow \frac{2 x - 7 + 5 x}{9 x - 3 - 4 x} = \frac{7}{6}$

$\Rightarrow \frac{7 x - 7}{5 x - 3} = \frac{7}{6} \Rightarrow 6 (7 x - 7) = 7 (5 x - 3)$

$(by cross multiplication)$

$\Rightarrow 42 x - 42 = 35 x - 21$

$\Rightarrow 42 x - 35 x = 42 - 21$

$\Rightarrow 7 x = 21$

$\Rightarrow x = \frac{21}{7} = 3$

$∴ x = 3$

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Question 25:

Solve:
$m - \frac{(m - 1)}{2} = 1 - \frac{(m - 2)}{3}$

Answer 25:

$m - \frac{(m - 1)}{2} = 1 - \frac{(m - 2)}{3}$

$\Rightarrow \frac{2 m - m + 1}{2} = 1 - \frac{(m - 2)}{3}$

$(L . C . M . of 1 and 2 is 2)$

$\Rightarrow \frac{m + 1}{2} = \frac{3 - m + 2}{3}$

$(L . C . M . of 1 and 3 is 3)$

$\Rightarrow \frac{m + 1}{2} = \frac{5 - m}{3}$

$\Rightarrow 3 (m + 1) = 2 (5 - m)$

$(by cross multiplication)$

$\Rightarrow 3 m + 3 = 10 - 2 m$

$\Rightarrow 3 m + 2 m = 10 - 3$

$\Rightarrow 5 m = 7$

$\Rightarrow m = \frac{7}{5}$

$∴ m = \frac{7}{5}$

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Question 26:

Solve:
$\frac{3 x + 5}{4 x + 2} = \frac{3 x + 4}{4 x + 7}$

Answer 26:

$\frac{3 x + 5}{4 x + 2} = \frac{3 x + 4}{4 x + 7}$

$\Rightarrow (4 x + 7) (3 x + 5) = (4 x + 2) (3 x + 4)$

$(by cross multiplication)$

$\Rightarrow 12 x^{2} + 20 x + 21 x + 35 = 12 x^{2} + 16 x + 6 x + 8$

$\Rightarrow 12 x^{2} + 41 x + 35 = 12 x^{2} + 22 x + 8$

$\Rightarrow 12 x^{2} - 12 x^{2} + 41 x - 22 x = 8 - 35$

$\Rightarrow 19 x = - 27$

$\Rightarrow x = \frac{- 27}{19}$

$∴ x = \frac{- 27}{19}$

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Question 27:

Solve:
$\frac{9 x - 7}{3 x + 5} = \frac{3 x - 4}{x + 6}$

Answer 27:

$\frac{9 x - 7}{3 x + 5} = \frac{3 x - 4}{x + 6}$

$\Rightarrow (x + 6) (9 x - 7) = (3 x + 5) (3 x - 4)$

$(by cross multiplication)$

$\Rightarrow 9 x^{2} - 7 x + 54 x - 42 = 9 x^{2} - 12 x + 15 x - 20$

$\Rightarrow 9 x^{2} + 47 x - 42 = 9 x^{2} + 3 x - 20$

$\Rightarrow 9 x^{2} - 9 x^{2} + 47 x - 3 x = - 20 + 42$

$\Rightarrow 44 x = 22$

$\Rightarrow x = \frac{22}{44} = \frac{1}{2} ∴ x = \frac{1}{2}$

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Question 28:

Solve:
$\frac{2 - 7 x}{1 - 5 x} = \frac{3 + 7 x}{4 + 5 x}$

Answer 28:

$\frac{2 - 7 x}{1 - 5 x} = \frac{3 + 7 x}{4 + 5 x}$

$\Rightarrow (4 + 5 x) (2 - 7 x) = (1 - 5 x) (3 + 7 x)$

$(by cross multiplication)$

$\Rightarrow 8 - 28 x + 10 x - 35 x^{2} = 3 + 7 x - 15 x - 35 x^{2}$

$\Rightarrow - 35 x^{2} - 18 x + 8 = - 35 x^{2} - 8 x + 3$

$\Rightarrow - 35 x^{2} + 35 x^{2} - 18 x + 8 x = - 8 + 3 \Rightarrow - 10 x = - 5$

$\Rightarrow x = \frac{- 5}{- 10} = \frac{1}{2}$

$∴ x = \frac{1}{2}$