The pair of linear equations $3 x + k y = 9$ and $6 x + 4 y = 18$ gives infinitely many solutions, then the value of $k$ is ___

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Solution

The given linear equations are: $3 x + k y = 9; 6 x + 4 y = 18$
Here, $a_{1} = 3$ , $b_{1} = k$ , $c_{1} = 9$ and $a_{2} = 6$ , $b_{2} = 4$ , $c_{2} = 18$
Given equations gives infinitely many solutions, if
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
$\Rightarrow \frac{3}{6} = \frac{k}{4} = \frac{9}{18}$
$\Rightarrow \frac{1}{2} = \frac{k}{4}$
$\Rightarrow k = 2$

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