Simplify and solve the following linear equations.

1. $\dfrac{x}{2} - \dfrac{1}{5} = \dfrac{x}{3} + \dfrac{1}{4}$

$\dfrac{x}{2} - \dfrac{1}{5} = \dfrac{x}{3} + \dfrac{1}{4}$
$\Rightarrow \dfrac{x}{2} - \dfrac{x}{3} = \dfrac{1}{4} + \dfrac{1}{5}$
$\Rightarrow \dfrac{3x - 2x}{6} = \dfrac{5 + 4}{20}$
$\Rightarrow x = \dfrac{9}{20} \times 6$
$\Rightarrow x = \dfrac{54}{20}$
$\Rightarrow x = \dfrac{27}{10}$

2. $\dfrac{n}{2} - \dfrac{3n}{4} + \dfrac{5n}{6} = 21$

$\dfrac{6n - 9n + 10n}{12} = 21$
$\Rightarrow \dfrac{7n}{12} = 21$
$\Rightarrow 7n = 21 \times 12$
$\Rightarrow n = \dfrac{252}{7}$
$\Rightarrow n = 36$

3. $x + 7 - \dfrac{8x}{3} = \dfrac{17}{6} - \dfrac{5x}{2}$

$x - \dfrac{8x}{3} + \dfrac{5x}{2} = \dfrac{17}{6} - 7$
$\Rightarrow \dfrac{6x - 16x + 15x}{6} = \dfrac{17 - 42}{6}$
$\Rightarrow \dfrac{5x}{6} = -\dfrac{25}{6}$
$\Rightarrow 5x = -25$
$\Rightarrow x = -5$

4. $\dfrac{x-5}{3} = \dfrac{x-3}{5}$

$5(x-5) = 3(x-3)$
$\Rightarrow 5x - 25 = 3x - 9$
$\Rightarrow 5x - 3x = -9 + 25$
$\Rightarrow 2x = 16$
$\Rightarrow x = 8$

5. $\dfrac{3t - 2}{4} - \dfrac{2t + 3}{3} = \dfrac{2}{3} - t$

$9t - 6 - 8t - 12 = 8 - 12t$
$\Rightarrow t - 18 = 8 - 12t$
$\Rightarrow 13t = 26$
$\Rightarrow t = 2$

6. $m - \dfrac{m-1}{2} = 1 - \dfrac{m-2}{3}$

$m - \dfrac{m}{2} + \dfrac{1}{2} = 1 - \dfrac{m}{3} + \dfrac{2}{3}$
$\Rightarrow \dfrac{m}{2} + \dfrac{m}{3} = \dfrac{1}{2} + \dfrac{2}{3}$
$\Rightarrow \dfrac{3m + 2m}{6} = \dfrac{3+4}{6}$
$\Rightarrow \dfrac{5m}{6} = \dfrac{7}{6}$
$\Rightarrow m = \dfrac{7}{5}$

7. $3(t-3) = 5(2t+1)$

$3t - 9 = 10t + 5$
$\Rightarrow -7t = 14$
$\Rightarrow t = -2$

8. $15(y-4) - 2(y-9) + 5(y+6) = 0$

$15y - 60 - 2y + 18 + 5y + 30 = 0$
$\Rightarrow 18y - 12 = 0$
$\Rightarrow 18y = 12$
$\Rightarrow y = \dfrac{2}{3}$

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

$15z - 21 - 18z + 22 = 32z - 52 - 17$
$\Rightarrow -21z + 1 = 32z - 69$
$\Rightarrow -35z = -70$
$\Rightarrow z = 2$

10. $0.25(4f - 3) = 0.05(10f - 9)$

$f - 0.75 = 0.5f - 0.45$
$\Rightarrow 0.5f = 0.30$
$\Rightarrow f = \dfrac{0.30}{0.5}$
$\Rightarrow f = \dfrac{3}{5}$
$\Rightarrow f = 0.6$

1. $\dfrac{x}{2} - \dfrac{1}{5} = \dfrac{x}{3} + \dfrac{1}{4}$

$\dfrac{x}{2} - \dfrac{1}{5} = \dfrac{x}{3} + \dfrac{1}{4}$
$\Rightarrow \dfrac{x}{2} - \dfrac{x}{3} = \dfrac{1}{4} + \dfrac{1}{5}$
$\Rightarrow \dfrac{3x - 2x}{6} = \dfrac{5 + 4}{20}$
$\Rightarrow x = \dfrac{9}{20} \times 6$
$\Rightarrow x = \dfrac{54}{20}$
$\Rightarrow x = \dfrac{27}{10}$

2. $\dfrac{n}{2} - \dfrac{3n}{4} + \dfrac{5n}{6} = 21$

$\dfrac{6n - 9n + 10n}{12} = 21$
$\Rightarrow \dfrac{7n}{12} = 21$
$\Rightarrow 7n = 21 \times 12$
$\Rightarrow n = \dfrac{252}{7}$
$\Rightarrow n = 36$

3. $x + 7 - \dfrac{8x}{3} = \dfrac{17}{6} - \dfrac{5x}{2}$

$x - \dfrac{8x}{3} + \dfrac{5x}{2} = \dfrac{17}{6} - 7$
$\Rightarrow \dfrac{6x - 16x + 15x}{6} = \dfrac{17 - 42}{6}$
$\Rightarrow \dfrac{5x}{6} = -\dfrac{25}{6}$
$\Rightarrow 5x = -25$
$\Rightarrow x = -5$

4. $\dfrac{x-5}{3} = \dfrac{x-3}{5}$

$5(x-5) = 3(x-3)$
$\Rightarrow 5x - 25 = 3x - 9$
$\Rightarrow 5x - 3x = -9 + 25$
$\Rightarrow 2x = 16$
$\Rightarrow x = 8$

5. $\dfrac{3t - 2}{4} - \dfrac{2t + 3}{3} = \dfrac{2}{3} - t$

$9t - 6 - 8t - 12 = 8 - 12t$
$\Rightarrow t - 18 = 8 - 12t$
$\Rightarrow 13t = 26$
$\Rightarrow t = 2$

6. $m - \dfrac{m-1}{2} = 1 - \dfrac{m-2}{3}$

$m - \dfrac{m}{2} + \dfrac{1}{2} = 1 - \dfrac{m}{3} + \dfrac{2}{3}$
$\Rightarrow \dfrac{m}{2} + \dfrac{m}{3} = \dfrac{1}{2} + \dfrac{2}{3}$
$\Rightarrow \dfrac{3m + 2m}{6} = \dfrac{3+4}{6}$
$\Rightarrow \dfrac{5m}{6} = \dfrac{7}{6}$
$\Rightarrow m = \dfrac{7}{5}$

7. $3(t-3) = 5(2t+1)$

$3t - 9 = 10t + 5$
$\Rightarrow -7t = 14$
$\Rightarrow t = -2$

8. $15(y-4) - 2(y-9) + 5(y+6) = 0$

$15y - 60 - 2y + 18 + 5y + 30 = 0$
$\Rightarrow 18y - 12 = 0$
$\Rightarrow 18y = 12$
$\Rightarrow y = \dfrac{2}{3}$

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

$15z - 21 - 18z + 22 = 32z - 52 - 17$
$\Rightarrow -21z + 1 = 32z - 69$
$\Rightarrow -35z = -70$
$\Rightarrow z = 2$

10. $0.25(4f - 3) = 0.05(10f - 9)$

$f - 0.75 = 0.5f - 0.45$
$\Rightarrow 0.5f = 0.30$
$\Rightarrow f = \dfrac{0.30}{0.5}$
$\Rightarrow f = \dfrac{3}{5}$
$\Rightarrow f = 0.6$