Question-9

Consider a system of equations:

\[ \begin{aligned} x_{1}-3x_{2} & =4\\ 3x_{1}+kx_{2} & =-12 \end{aligned} \]

where \(k\in\mathbb{R}\). For what value of \(k\) does this system have:

For this system to have a unique solution, the slopes of the two equations should be different:

\[ \cfrac{1}{3}\neq\cfrac{-3}{k}\implies k\neq-9 \]

For the system to have infinitely many solutions, the two lines should be identical:

\[ \cfrac{1}{3}=\cfrac{-3}{k}=\cfrac{-1}{3} \]

This is not possible, so the system will never have infinitely solutions. For no solution, we need the slopes to be identical and the intercepts to be different:

\[ \cfrac{1}{3}=\cfrac{-3}{k}\neq\cfrac{-1}{3} \]

We have \(k=-9\).