Write the value of $k$ for which the system of equations $3 x - 2 y = 0$ and $k x + 5 y = 0$ has infinitely many solutions.

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Solution

The given equations are
$3 x - 2 y = 0$
$k x + 5 y = 0$
For the equations to have an infinite number of solutions, $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
Therefore, $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}}$
$\Rightarrow \frac{3}{k} = \frac{- 2}{5}$ [ $∵ \frac{a_{1}}{a_{2}} = \frac{3}{k}, \frac{b_{1}}{b_{2}} = \frac{- 2}{5}$ ]
By cross multiplication, we have
$\Rightarrow 3 \times 5 = - 2 \times k$
$\Rightarrow 15 = - 2 k$
$\Rightarrow k = \frac{15}{- 2} = - \frac{15}{2}$
Hence, the value of k for the system of equation $3 x - 2 y = 0$ and $k x - 2 y = 0$ is $\frac{- 15}{2}$ .

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