Find the value of $k$ , if the lines $3 x + 2 k y = 2$ and $2 x + 5 y = - 1$ has infinite solutions.

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Solution

$3 x + 2 k y - 2 = 0$ ; $2 x + 5 y + 1 = 0$
$a_{1} = 3$ , $a_{2} = 2$ , $b_{1} = 2 k$ , $b_{2} = 5$ , $c_{1} = - 2$ , $c_{2} = 1$
For parallel lines
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} + \frac{c_{1}}{c_{2}}$
$\frac{3}{2} = \frac{2 k}{5} + \frac{- 2}{1}$
$\frac{15}{4} = k$
$k = \frac{15}{4}$ .

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