Representing line intersections as a system of linear equations

(Note: this article was originally written in $\LaTeX$ and transcribed to WordPress, so forgive the equation alignment. Get the original).

In a previous post, I outlined an analytical solution for intersecting lines and
ellipses. In this post I’m doing much the same thing but rather with lines on lines. I’ll point out why the normal slope-intercept form for a line is a poor representation, and what we can do about that.