In this problem, I understand that my solution set is defined by the line with direction vector <1,1,3>. To arrive at this solution, I know that I need two equations, specifically planes, but I am unsure of how to come about them. I believe that I should set a 2x3 matrix (which can be interpreted as squishing space from $\mathbb R^3$ to $\mathbb R^2$ because the solution set is a line, I but am not quite sure where to go from there.
On
Pick two nonzero vectors orthogonal to $\mathbf v$. An easy way to generate them is to remember that for any nonzero vector $(a,b,c)^T$, at least two of $(c,-b,0)^T$, $(-c,0,a)^T$ and $(b,-a,0)^T$ are nonzero and all are orthogonal to it. Each of these two vectors can be used as the normal of a distinct plane that contains $L$. I trust that you can construct their equations from these normals.
Guide:
We need two independent equations.
By looking at the first two coordinates, we can set $$x_1-x_2=0$$
Try to write down another equation by looking at the first and third coordinate.