**Question:**

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

**Solution:**

The given equations are

$3 x-2 y=0$

$k x+5 y=0$

$\frac{a_{1}}{a_{2}}=\frac{3}{k}, \frac{b_{1}}{b_{2}}=\frac{-2}{5}, \frac{c_{1}}{c_{2}}=\frac{0}{0}$

For the equations to have infinite number of solutions, $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$

Therefore,

$\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}$

$\frac{3}{k}=\frac{-2}{5}$

By cross multiplication we have

$3 \times 5=-2 \times k$

$15=-2 k$

$\frac{15}{-2}=k$

Hence, the value of $k$ for the system of equation $3 \times-2 y=0$ and $k x+5 y=0$ is $\frac{-15}{2}$.

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