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

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

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Solution

$\frac{3 a - 2}{3} + \frac{2 a + 3}{2} = a + \frac{7}{6}$
$\frac{3 a - 2}{3} + \frac{2 a + 3}{2} = a + \frac{7}{6}$
$\frac{2 (3 a - 2) + 3 (2 a + 3) = 6 a + 7}{6}$ (L.C.M. of 3,2,6=6)
$\Rightarrow 6 a - 4 + 6 a + 9 = 6 a + 7$
$\Rightarrow 12 a + 5 = 6 a + 7$
$\Rightarrow 12 a - 6 a = 7 - 5$ (By transposition)
$\Rightarrow 6 a = 2 \Rightarrow a = \frac{2}{6} = \frac{1}{3}$
$∴ a = \frac{1}{3}$
Verification:
$L . H . S . = \frac{3 a - 2}{3} + \frac{2 a + 3}{2} = \frac{3 \times \frac{1}{3} - 2}{3} + \frac{2 \times \frac{1}{3} + 3}{2} = \frac{1 - 2}{3} + \frac{\frac{2}{3} + 3}{2} = \frac{- 1}{3} + \frac{11}{3 \times 2} = \frac{- 1}{3} + \frac{11}{6} = \frac{- 2 + 11}{6} = \frac{9}{6} = \frac{3}{2} R . H . S . = a + \frac{7}{6} = \frac{1}{3} + \frac{7}{6} = \frac{2 + 7}{6} = \frac{9}{6} = \frac{3}{2} ∴ L . H . S . = R . H . S .$

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