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Solving Equations with Variable on Both Sides

y+73=1+3 y-25

Question

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

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Solution

We have: $\frac{y + 7}{3} = 1 + \frac{3 y - 2}{5}$
$\Rightarrow \frac{y + 7}{3} = \frac{(5 \times 1) + 3 y - 2}{5}$
$\Rightarrow 5 (y + 7) = 3 (3 + 3 y)$
$\Rightarrow 5 y + 35 = 9 + 9 y$
$\Rightarrow 9 y - 5 y = 35 - 9$
$\Rightarrow 4 y = 26$
$\Rightarrow y = \frac{13}{2}$
CHECK: Substituting $y = \frac{13}{2}$ in the given equation, we get
LHS $= \frac{y + 7}{3} = \frac{\frac{13}{2} + 7}{3} = \frac{13 + (2 \times 7)}{2 \times 3} = \frac{13 + 14}{6} = \frac{27}{6} = \frac{9}{2}$
RHS $= 1 + \frac{3 y - 2}{5} = 1 + \frac{(3 \times \frac{13}{2}) - 2}{5} = 1 + \frac{39 - (2 \times 2)}{2 \times 5} = 1 + \frac{35}{10} = \frac{45}{10} = \frac{9}{2}$
$∴$ LHS $=$ RHS
Hence, $y = \frac{13}{2}$ is a solution of the given equation.

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