Question

Find the value of k for which the following equations has a unique solution:
$$\begin{gathered} x-ky-2=0 \\ 3x+2y+5=0 \end{gathered}$$

Answer 1

The given system of equations are
$$\begin{gathered} x-ky-2=0.....\left(i\right) \\ 3x+2y+5=0.....\left(ii\right) \end{gathered}$$
This system of equations 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 $a_1=1,b_1=-k,c_1=-2\mathrm{and}{a}_2=3,b_2=2,c_2=5$
Now, for the given system of equations to have a unique solution, we must have
$$\begin{gathered} \frac{a_1}{a_2}\ne \frac{b_1}{b_2} \\ \Rightarrow \frac{1}{3}\ne \frac{-k}{2} \\ \Rightarrow k\ne -\frac{2}{3} \end{gathered}$$
Hence, $k\ne -\frac{2}{3}$.