## Elementary Linear Algebra 7th Edition

We have to choose $$k=-\frac{2}{3}.$$
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.