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.
For all these forms I’m going to use x and y as the variables, and I’ll be looking for equations which are true for exactly the same pairs of values of x and y that are also coordinates of some point on the line we’re interested in.
For the slope-intercept form we need just what the name suggests: the slope and an intercept. Here, particularly, the intercept is the y-intercept, the y-coordinate of the point where the line we’re interested in crosses the y-axis. Another way of putting that, without changing the meaning any, is to say it’s the y-coordinate of whatever point on the line has an x-coordinate of zero. We should expect only the one point, since we don’t deal with those annoying curved lines that go back on themselves, not at this level.
This coordinate is some number, possibly a whole number, possibly something rational, possibly irrational. But I might not know which number it is; for example, in this problem of interpolating Charlotte, North Carolina’s population in 1975, I have no idea what the number should be. I’ll be able to find it out, though, with effort. I need to write something down in the equation, though, and so I fall back on using a variable, a letter which stands for the number I might at some later point find out.
It’s not actually required by the laws of the universe that we use the letter b for this y-intercept, but it might as well be, at least in English-written mathematics. I’m told other languages use different letters, but the only other language I can read is Math Book French, and the Math Books that I have read in French haven’t been ones that would talk about common forms of equations of lines. (Also, I read Math Book French very, very badly, but can recognize π fairly well.)
That’s a convention, just as arbitrary but just as understood as using x for an unknown-but-to-be-determined number is, or in a programming language using the letter
i to represent an index. You could use a different letter if you liked, but, if you use b other people looking at your work will start with a pretty good idea what it means. Not much reason to give that up.
Though since it is an arbitrary choice which letter to use, the natural question is: why b? Why not d or k or any of the other good letters of the Roman alphabet, or the handful of usable other symbols from the Greek alphabet? Heck, why not a? An early letter in the alphabet makes sense: Rene Descartes gave us the pretty good convention that letters near the start of the alphabet should be for known quantities — and, looking at a line on a piece of graph paper, we know, or at least could easily find by looking at (this is known as “solving by inspection”), the intercept — and letters near the end for quantities we don’t know, like, “some arbitrary point somewhere in the universe that happens to be on this line.”
So why b and not a? For that matter, why b and not c, which has that nice suggestion of “constant number” to it? I can’t give any compelling reason. It seems to be one of those things mathematicians just kind of drifted into doing, although there’s some reason behind it. The letter a is often used as the coordinate for the x-intercept, describing where a line crosses the x-axis, and the parallelism between a and x-intercept with b and y-intercept seems like a good reason to leave things like they are. It’s just not a fully satisfactory explanation for why things are like they are. References on the origins of mathematics symbols will point out that the Irish mathematician George Salmon, 1819 – 1904, published in 1848 A Treatise On Conic Sections which used a and b for the intercepts, and showed several convenient forms of lines which are perfectly recognizable and which don’t look funny to the modern eye. On the other hand, one can find perfectly good texts written a century later which were still picking oddball letters like h or k for the y-intercept.