Find the value of k for which the system of equations
$5 x - 3 y = 0, 2 x + k y = 0$
has a nonzero solution.

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Solution

The given system of equations:
5x − 3y = 0 ....(i)
2x + ky = 0 ....(ii)
These equations are of the following form:
a₁x + b₁y + c₁= 0, a₂x + b₂y + c₂ = 0
Here, a₁ = 5, b₁= −3, c₁ = 0 and a₂ = 2, b₂= k, c₂ = 0
For a non-zero solution, we must have:
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}}$
$\Rightarrow \frac{5}{2} = \frac{- 3}{k}$
$\Rightarrow 5 k = - 6 \Rightarrow k = \frac{- 6}{5}$
Hence, the required value of k is $\frac{- 6}{5}$ .

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