Answer
We have to choose $k=-2$.
Work Step by Step
Step 1: Make the right sides of the equations equal. This can be done by multiplying the second equation by $-2$:
\begin{align*}
4x+ky=&6\\
-2kx-2y=&6
\end{align*}
Step 2: Now since we want infinitely many solutions we want that the left sides of the equations be identical (same coefficients multiplying $x$ and $y$) so that we really have only one independent equation. If we choose $k=-2$ this will be the case because then we will have:
\begin{align*}
4x-2y=&6\\
4x-2y=&6
\end{align*}
which is what we wanted.
