Answer
We have to choose $$k=-\frac{2}{3}.$$
Work Step by Step
Step 1: Make the right sides of the equations to be equal. To do this multiply the first equation by $-3$:
\begin{align*}
-3kx-3y=&-12\\
2x-3y=&-12
\end{align*}
Step 2: Find such $k$ for which the left sides of the two equations will be completely identical (same coefficients multiplying $x$ and $y$). In this way we will have only one independent equation and that means infinitely many solutions. We need that $-3k = 2$ which means that
$$k = -\frac{2}{3}.$$
Now we really have
\begin{align*}
2x-3y=&-12\\
2x-3y=&-12
\end{align*}
which means we can drop one equation and the system will have infinitely many solutions.
