Write the value of k for which the system of equations x + ky = 0,

The given equations are

$x+k y=0$

$2 x-y=0$

$a_{1}=1, a_{2}=2, b_{1}=k, b_{2}=-1$

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

$\frac{b_{1}}{b_{2}}=\frac{k}{-1}$

For unique solution $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}$

$-1 \times 1 \neq 2 \times k$

$-1 \neq 2 k$

$\frac{-1}{2} \neq k$

For all real values of $k$, except $k=\frac{-1}{2}$ the equations have unique solutions.