After we have the intercept, the other thing we need is the slope. This is a very easy thing to start calculating and it’s extremely testable, but the idea weaves its way deep into all mathematics. It’s got an obvious physical interpretation. Imagine the x-coordinates are how far we are from some reference point in a horizontal direction, and the y-coordinates are how far we are from some reference point in the vertical direction. Then the slope is just the grade of the line: how much we move up or down for a given movement forward or back. It’s easy to calculate, it’s kind of obvious, so here’s what’s neat about it.
Just because there are in principle uncountably many possible equations for any line doesn’t mean we ever actually see any of them. Actually, we just about always pick one of a handful of representations. They’re just the convenient ones. I’m going to say there’s four patterns that actually get used, because I can only think of three that turn up, as long as we’re sticking to Cartesian coordinate systems and aren’t doing something weird like parametric descriptions, and I want to leave some hedge room for when I realize I overlooked the obvious. The first one — that I want to talk about, anyway, and just about the first one anyone encounters — is called the slope-intercept form, and it’s probably what someone means if they do talk about “the” equation for a line.