## Why The Slope Is Too Interesting

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.

## Why Call The Intercept b

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.