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Intro to linear equation standard form

The standard form for linear equations in two variables is Ax+By=C. For example, 2x+3y=5 is a linear equation in standard form. When an equation is given in this form, it's pretty easy to find both intercepts (x and y). This form is also very useful when solving systems of two linear equations.

Video transcript

- [Voiceover] We've already looked at several ways of writing linear equations. You could write it in slope-intercept form, where it would be of the form of Y is equal to MX plus B, where M and B are constants. M is the coefficient on this MX term right over here and M would represent the slope. And then from B you're able to figure out the y-intercept. The Y, you're able to figure out the y-intercept from this. Literally the graph that represents the XY pairs that satisfy this equation, it would intersect the y-axis at the point X equals zero, Y is equal to B. And it's slope would be M. We've already seen that multiple times. We've also seen that you can also express things in point-slope form. So let me make it clear. This is slope-intercept. Slope- intercept. And these are just different ways of writing the same equations. You can algebraically manipulate from one to the other. Another way is point-slope. Point-slope form. And in point-slope form, if you know that some, if you know that there's an equation where the line that represents the solutions of that equation has a slope M. Slope is equal to M. And if you know that X equals, X equals A, Y equals B, satisfies that equation, then in point-slope form you can express the equation as Y minus B is equal to M times X minus A. This is point-slope form and we do videos on that. But what I really want to get into in this video is another form. And it's a form that you might have already seen. And that is standard form. Standard. Standard form. And standard form takes the shape of AX plus BY is equal to C, where A, B, and C are integers. And what I want to do in this video, like we've done in the ones on point-slope and slope-intercept is get an appreciation for what is standard form good at and what is standard form less good at? So let's give a tangible example here. So let's say I have the linear equation, it's in standard form, 9X plus 16Y is equal to 72. And we wanted to graph this. So the thing that standard form is really good for is figuring out, not just the y-intercept, y-intercept is pretty good if you're using slope-intercept form, but we can find out the y-intercept pretty clearly from standard form and the x-intercept. The x-intercept isn't so easy to figure out from these other forms right over here. So how do we do that? Well to figure out the x and y-intercepts, let's just set up a little table here, X comma Y, and so the x-intercept is going to happen when Y is equal to zero. And the y-intercept is going to happen when X is equal to zero. So when Y is zero, what is X? So when Y is zero, 16 times zero is zero, that term disappears, and you're left with 9X is equal to 72. So if nine times X is 72, 72 divided by nine is eight. So X would be equal to eight. So once again, that was pretty easy to figure out. This term goes away and you just have to say hey, nine times X is 72, X would be eight. When Y is equal to zero, X is eight. So the point, let's see, Y is zero, X is one, two, three, four, five, six, seven, eight. That's this point, that right over here. This point right over here is the x-intercept. When we talk about x-intercepts we're referring to the point where the line actually intersects the x-axis. Now what about the y-intercept? Well, we said X equals zero, this disappears. And we're left with 16Y is equal to 72. And so we could solve, we could solve that. So we could say, alright 16Y is equal to 72. And then divide both sides by 16. We get Y is equal to 72 over 16. And let's see, what is that equal to? That is equal to, let's see, they're both divisible by eight, so that's nine over two. Or we could say it's 4.5. So when X is zero, Y is 4.5. And so, we could plot that point as well. X is zero, Y is one, two, three, 4.5. And just with these two points, two points are enough to graph a line, we can now graph it. So let's do that. So let me, oops, though I was using the tool that would draw a straight line. Let me see if I can... So the line will look something like that. There you have it. I've just graphed, I've just graphed, this is the line that represents all the X and Y pairs that satisfy the equation 9X plus 16Y is equal to 72. Now, I mentioned standard form's good at certain things and the good thing that standard form is, where it's maybe somewhat unique relative to the other forms we looked at, is it's very easy to figure out the x-intercept. It was very easy to figure out the x-intercept from standard form. And it wasn't too hard to figure out the y-intercept either. If we looked at slope-intercept form, the y-intercept just kinda jumps out at you. At point-slope form, neither the x nor the y-intercept kind of jump out at you. The place where slope-intercept or point-slope form are frankly better is that it's pretty easy to pick out the slope here, while in standard form you would have to do a little bit of work. You could use these two points, you could use the x and y-intercepts as two points and figure out the slope from there. So you can literally say, "Okay, if I'm going from "this point to this point, my change in X "to go from eight to zero is negative eight. "And to go from zero to 4.5," I wrote that little delta there unnecessarily. Let me. So when you go from eight to zero, your change in X is equal to negative eight. And to go from zero to 4.5, your change in Y is going to be 4.5. So your slope, once you've figured this out, you could say, "Okay, this is going to be "change in Y, 4.5, over change in X, "over negative 8." And since I, at least I don't like a decimal up here, let's multiply the numerator and the denominator by two. You get negative nine over 16. Now once again, we had to do a little bit of work here. We either use these two points, it didn't just jump immediately out of this, although you might see a little bit of a pattern of what's going on here. But you still have to think about is it negative? Is it positive? You have to do a little bit of algebraic manipulation. Or, what I typically do if I'm looking for the slope, I actually might put this into, into one of the other forms. Especially slope-intercept form. But standard form by itself, great for figuring out both the x and y-intercepts and it's frankly not that hard to convert it to slope-intercept form. Let's do that just to make it clear. So if you start with 9X, let me do that in yellow. If we start with 9X plus 16Y is equal to 72 and we want to put it in slope-intercept form, we can subtract 9X from both sides. You get 16Y is equal to negative 9X, plus 72. And then divide both sides by 16. So divide everthing by 16. And you'll be left with Y is equal to negative 9/16X, that's the slope, you see it right there, plus 72 over 16, we already figured out that's 9/2 or 4.5. So I could write, oh I'll just write that as 4.5. And this form over here, much easier to figure out the slope and, actually, the y-intercept jumps out at you. But the x-intercept isn't as obvious.