EXERCISE 2.2 — Step by Step Solutions

1. \( \dfrac{x}{2} - \dfrac{1}{5} = \dfrac{x}{3} + \dfrac{1}{4} \)
Solution:
Transpose \( \dfrac{x}{3} \) to LHS and \( \dfrac{1}{5} \) to RHS:

\[ \dfrac{x}{2} - \dfrac{x}{3} = \dfrac{1}{4} + \dfrac{1}{5} \]
By Cross Multiplication:

LHS: \(\dfrac{3 \times x - 2 \times x}{2 \times 3}\) = \(\dfrac{3x - 2x}{6}\) = \(\dfrac{x}{6}\)

RHS: \(\dfrac{5 \times 1 + 4 \times 1}{4 \times 5}\) = \(\dfrac{5 + 4}{20}\) = \(\dfrac{9}{20}\)
Equation becomes:
\[ \dfrac{x}{6} = \dfrac{9}{20} \]
Transpose 6 to RHS:
\[ x = \dfrac{9}{20} \times 6 \]
Simplify numbers:
\[ x = \dfrac{9 \times 6}{20} = \dfrac{54}{20} \]
Reduce numerator and denominator by 2:
\[ \dfrac{54^{(2 \times 27)}}{20^{(2 \times 10)}} = \dfrac{27}{10} \]
Final Answer: \(x = \dfrac{27}{10}\).
2. \( \dfrac{n}{2} - \dfrac{3n}{4} + \dfrac{5n}{6} = 21 \)
LCM of 2, 4, 6 = 12:
So Equation becomes:
\[ \dfrac{6n}{12} - \dfrac{9n}{12} + \dfrac{10n}{12} = 21 \]
Combine terms:
\[ \dfrac{6n - 9n + 10n}{12} = 21 \]
\[ \dfrac{7n}{12} = 21 \]
\[ 7n = 21 \times 12\]
\[ n = \dfrac{21 \times 12}{7}\]
\[ n = \dfrac{\cancel{21}^{7 \times 3} \times 12}{\cancel{7}_{1}}\]
\[ n = 3 \times 12\]
Final Answer: \(n = 36\).
3. \( x+7-\dfrac{8x}{3} = \dfrac{17}{6}-\dfrac{5x}{2} \)
Transpose terms with x to LHS and constants to RHS:
\[ x - \dfrac{8x}{3} + \dfrac{5x}{2} = \dfrac{17}{6} - 7 \]
LCM of 3, 2 = 6. Convert all terms to denominator 6:
\[ LHS: \dfrac{6x}{6} - \dfrac{16x}{6} + \dfrac{15x}{6} \]
\[ = \dfrac{6x-16x+15x}{6} = \dfrac{5x}{6} \]
\[ RHS: \dfrac{17}{6} - 7 = \dfrac{17}{6} - \dfrac{42}{6} \]
\[ = \dfrac{17-42}{6} = -\dfrac{25}{6} \]
Equation becomes:
\[ \dfrac{5x}{6} = -\dfrac{25}{6} \]
Multiply both sides by 6:
\[ 6 \times \dfrac{5x}{6} = 6 \times -\dfrac{25}{6} \]
Cancel:
\[ \cancel{6} \times \dfrac{5x}{\cancel{6}} = \cancel 6 \times -\dfrac{25}{\cancel 6} \]
Simplify:
\[ 5x = -25 \]
\[ \implies x = -5 \]
Final Answer: \(x = -5\).
4. \( \dfrac{x-5}{3} = \dfrac{x-3}{5} \)
Cross multiply:
\[ 5(x-5) = 3(x-3) \]
Expand both sides:
\[ 5x - 25 = 3x - 9 \]
Transpose terms:
\[ 5x - 3x = -9 + 25 \]
Simplify:
\[ 2x = 16 \]
Divide both sides by 2:
\[ x = \dfrac{16}{2} = 8 \]
Final Answer: \(x = 8\).
5. \( \dfrac{3t-2}{4} - \dfrac{2t+3}{3} = \dfrac{2}{3} - t \)
Bring all terms to one side:
\[ \dfrac{3t-2}{4} - \dfrac{2t+3}{3} + t - \dfrac{2}{3} = 0 \]
Take denominator 12:
\[ \dfrac{9t-6}{12} - \dfrac{8t+12}{12} + \dfrac{12t}{12} - \dfrac{8}{12} \]
Combine numerators:
\[ \dfrac{9t-6 - 8t-12 + 12t - 8}{12} = \dfrac{13t - 26}{12} \]
Equation:
\[ \dfrac{13t-26}{12} = 0 \]
Multiply both sides by 12:
\[ 13t - 26 = 0 \]
Simplify:
\[ 13t = 26 \implies t = \dfrac{26}{13} = 2 \]
Final Answer: \(t = 2\).
6. \( m - \dfrac{m-1}{2} = 1 - \dfrac{m-2}{3} \)
Transpose:
\[ m - \dfrac{m-1}{2} + \dfrac{m-2}{3} = 1 \]
Convert to denominator 6:
\[ \dfrac{6m}{6} - \dfrac{3m-3}{6} + \dfrac{2m-4}{6} \]
Simplify numerator:
\[ \dfrac{6m - 3m + 3 + 2m - 4}{6} = \dfrac{5m - 1}{6} \]
Equation:
\[ \dfrac{5m-1}{6} = 1 \]
Multiply both sides by 6:
\[ 5m - 1 = 6 \]
Simplify:
\[ 5m = 7 \implies m = \dfrac{7}{5} \]
Final Answer: \(m = \dfrac{7}{5}\).
7. \( 3(t-3) = 5(2t+1) \)
Expand both sides:
\[ 3t - 9 = 10t + 5 \]
Transpose:
\[ 3t - 10t = 5 + 9 \]
Simplify:
\[ -7t = 14 \]
Divide both sides by -7:
\[ t = \dfrac{14}{-7} = -2 \]
Final Answer: \(t = -2\).
8. \( 15(y-4) - 2(y-9) + 5(y+6) = 0 \)
Expand:
\[ 15y - 60 - 2y + 18 + 5y + 30 = 0 \]
Combine like terms:
\[ 18y - 12 = 0 \]
Simplify:
\[ 18y = 12 \implies y = \dfrac{12}{18} = \dfrac{2}{3} \]
Final Answer: \(y = \dfrac{2}{3}\).
9. \( 3(5z-7) - 2(9z-11) = 4(8z-13) - 17 \)
Expand both sides:
\[ 15z - 21 - 18z + 22 = 32z - 52 - 17 \]
Simplify LHS and RHS:
\[ -3z + 1 = 32z - 69 \]
Transpose terms:
\[ -3z - 32z = -69 - 1 \]
Simplify:
\[ -35z = -70 \implies z = \dfrac{-70}{-35} = 2 \]
Final Answer: \(z = 2\).
10. \( 0.25(4f-3) = 0.05(10f-9) \)
Expand both sides:
\[ 1f - 0.75 = 0.5f - 0.45 \]
Transpose:
\[ 1f - 0.5f = -0.45 + 0.75 \]
Simplify:
\[ 0.5f = 0.3 \]
Divide both sides by 0.5:
\[ f = \dfrac{0.3}{0.5} \]
Simplify fraction:
\[ f = \dfrac{3}{5} = 0.6 \]
Final Answer: \(f = 0.6\).