Why A Line Doesn’t Have An Equation

[ To resume after some interruptions — it’s been quite a busy few weeks — the linear interpolations that I had been talking about, I will need equations describing a line. ]

To say something is the equation representing a line is to lie in the article. It’s little one, of the same order as pretending there’s just one answer to the question, “Who are you?” Who you are depends on context: you’re the person with this first-middle-last name combination. You’re the person with this first name. You’re the person with this nickname. You’re the third person in the phone queue for tech support. You’re the person with this taxpayer identification number. You’re the world’s fourth-leading expert on the Marvel “New Universe” line of comic books, and sorry for that. You’re the person who ordered two large-size fries at Five Guys Burgers And Fries and will soon learn you’ll never live long enough to eat them all. You’re the person who knows how to get the sink in the break room at work to stop dripping. These may all be correct, but depending on the context some of these answers are irrelevant, and maybe one or two of them is useful, or at least convenient. So it is with equations for a line: there are many possible equations. Some of them are just more useful, or even convenient.

The idea of an equation for a line follows right after we insist we can draw a line, and put some coordinate system onto the surface we used to draw that line. The coordinate system matches up points of space with combinations, ordered sets, of numbers. We might, in theory, describe a line as the list of all the combinations of coordinates which describe points on the line. It’s probably not surprising to say there’s an infinite number of these groups, which right away warns us we shouldn’t be trying to do that. More, a lot of these combinations are going to look an awful lot like each other, with only tiny changes. That’s another warning to us not to do that: enormous lists of nearly identical items are hard to pay attention to, and we’re not likely to do good work on enormous lists of things hard to pay attention to.

Rather than pay attention to these enormous lists of things we might try seeing whether there’s any kind of relationship among the coordinates. If there are, we can reduce the number of things we have to pay attention to, and capture them as a couple of easy-to-grasp ideas. Happily, if we’re using a non-weird coordinate system like the Cartesian scheme — that’s the one we might as well default to, the one that looks like grid paper drawn across the universe — lines can be represented by awfully nice-looking, easy-to-work-with relationships.

Let’s suppose we draw the line in a two-dimensional system, so that we have two coordinates for each point. When we want to talk about the first coordinate in each of these pairs we need some name; using x as the variable, the stand-in for this number whose value we may or may not want right now, is extremely popular. For the second coordinate y is a quite popular choice. There could be other choices made. Which variables we use isn’t actually important; we could use t instead of x or use z instead of y and it wouldn’t make the math any different; we’d just use different symbols. (There could be value in this. If we think one of the coordinates represents time, t may be a very good variable choice, because the sight of the t reminds us of what we mean to represent. But that’s a mnemonic device, not a mathematic one.)

We say that an equation describes a line (or another curve) if whenever we have a combination of the variables x, y (or whatever we call them) which makes the equation true, then we also have a set of coordinates which falls on the line (or other curve). If we’re being a little loose, we might say that’s the equation for the line. But it’s too big a claim to say there’s just the one. Any equation can be rewritten many ways over, all with the same solutions, all describing the same line. Different aspects will be more convenient in some representations than others, is all. That’s a pattern to be seen in many mathematical fields; there are many possible ways to describe something, and we stick with a few that are particularly convenient, to the point we might forget there are alternatives.