Question

Find k for which the system kx − y = 2 and 6x − 2y = 3 has a unique solution.

Answer 1

The given system is
$$\begin{gathered} kx-y-2=0.....\left(i\right) \\ 6x-2y-3=0.....\left(ii\right) \end{gathered}$$
Here, $a_1=k,b_1=-1,c_1=-2.a_2=6,b_2=-2\mathrm{and}{c}_2=-3$.
For the system, to have a unique solution, we must have
$$\begin{gathered} \frac{a_1}{a_2}\ne \frac{b_1}{b_2} \\ \Rightarrow \frac{k}{6}\ne \frac{-1}{-2}=\frac{1}{2} \\ \Rightarrow k\ne 3 \end{gathered}$$
Hence, $k\ne 3$.