Question

Write the value of k for which the system of equations has a unique solution.

Answer 1

The given pair of linear equations is
$$\begin{gathered} 3x+ky=0.....\left(i\right) \\ 2x-y=0.....\left(ii\right) \end{gathered}$$
Which is of the form $a_{1}x+b_{1}y+c_1=0\mathrm{and}{a}_{2}x+b_{2}y+c_2=0$, where $a_1=3,b_1=k,c_1=0,a_2=2,b_2=-1\mathrm{and}{c}_2=0$
For the system to have a unique solution, we must have
$$\begin{gathered} \frac{a_1}{a_2}=\frac{b_1}{b_2} \\ \Rightarrow \frac{3}{2}\ne \frac{k}{-1} \\ \Rightarrow k\ne -\frac{3}{2} \end{gathered}$$
Hence, $k\ne -\frac{3}{2}$.