Slope-intercept form intro
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- [Voiceover] There's a lot of different ways that you could represent a linear equation. So for example, if you had the linear equation y is equal to 2x plus three, that's one way to represent it, but I could represent this in an infinite number of ways. I could, let's see, I could subtract 2x from both sides, I could write this as negative 2x plus y is equal to three. I could manipulate it in ways where I get it to, and I'm gonna do it right now, but this is another way of writing that same thing. y minus five is equal to two times x minus one. You could actually simplify this and you could get either this equation here or that equation up on top. These are all equivalent, you can get from one to the other with logical algebraic operations. So there's an infinite number of ways to represent a given linear equation, but I what I wanna focus on in this video is this representation in particular, because this one is a very useful representation of a linear equation and we'll see in future videos, this one and this one can also be useful, depending on what you are looking for, but we're gonna focus on this one, and this one right over here is often called slope-intercept form. Slope-intercept form. And hopefully in a few minutes, it will be obvious why it called slop-intercept form. And before I explain that to you, let's just try to graph this thing. I'm gonna try to graph it, I'm just gonna plot some points here, so x comma y, and I'm gonna pick some x values where it's easy to calculate the y values. So maybe the easiest is if x is equal to zero. If x is equal to zero, then two times zero is zero, that term goes away, and you're only left with this term right over here, y is equal to three. Y is equal to three. And so if we were to plot this. Actually let me start plotting it, so that is my y axis, and let me do the x axis, so that can be my x, oh that's not as straight as I would like it. So that looks pretty good, alright. That is my x axis and let me mark off some hash marks here, so this is x equals one, x equals two, x equals three, this is y equals one, y equals two, y equals three, and obviously I could keep going and keep going, this would be y is equal to negative one, this would be x is equal to negative one, negative two, negative three, so on and so forth. So this point right over here, zero comma three, this is x is zero, y is three. Well, the point that represents when x is equal to zero and y equals three, this is, we're right on the y axis. If they have a line going through it and this line contains this point, this is going to be the y- intercept. So one way to think about it, the reason why this is called slope-intercept form is it's very easy to calculate the y-intercept. The y-intercept here is going to happen when it's written in this form, it's going to happen when x is equal to zero and y is equal to three, it's gonna be this point right over here. So it's very easy to figure out the intercept, the y-intercept from this form. Now you might be saying, well it says slope-intercept form, it must also be easy to figure out the slope from this form. And if you made that conclusion, you would be correct! And we're about to see that in a few seconds. So let's plot some more points here and I'm just gonna keep increasing x by one. So if you increase x by one, so we could write that our delta x, our change in x, delta Greek letter, this triangle is a Greek letter, delta, represents change in. Change in x here is one. We just increased x by one, what's gonna be our corresponding change in y? What's going to be our change in y? So let's see, when x is equal to one, we have two times one, plus three is going to be five. So our change in y is going to be two. Let's do that again. Let's increase our x by one. Change in x is equal to one. So then if we're gonna increase by one, we're gonna go from x equals one to x equals two. Well what's our corresponding change in y? Well when x is equal to two, two times two is four, plus three is seven. Well our change in y, our change in y is equal to two. Went from five- when x went from one to two, y went from five to seven. So for every one that we increase x, y is increasing by two. So for this linear equation, our change in y over change in x is always going to be, our change in y is two when our change in x is one, or it's equal to two, or we could say that our slope is equal to two. Well let's just graph this to make sure that we understand this. So when x equals one, y is equal to five. And actually we're gonna have to graph five up here. So when x is equal to one, y is equal to, and actually this is a little bit higher, this, let me clean this up a little bit. So this one would be, erase that a little bit. Just like that. So that's y is equal to four, and this is y is equal to five. So when x is one, y is equal to five, so it's that point right over there. So our line is going to look- you only need two points to define a line, our line is going to like, let me do this in this color right over here. Our line is going to look like, is going to look, is going to look something like, is going to look, let me see if I can, I didn't draw it completely at scale, but it's going to look something like this. This is the line, this is the line, y is equal to 2x plus three. But we already figured out that its slope is equal to two, when our change in x is one, when our change in x is one, our change in y is two. If our change in x was negative one, if our change in x was negative one, our change in y is negative two. And you can see that, if from zero we went, we went down one, if we went to negative one, then what's our y going to be? Two times negative one is negative two plus three is one. So we see that, the point negative one comma one is on the line as well. So the slope here, our change in y over change in x, if we're going from between any two points on this line, is always going to be two. But where do you see two in this original equation? Well you see the two right over here. And when you write something in slop-intercept form, where you explicitly solve for y, y is equal to some constant times x to the first power plus some other constant, the second one is going to be your intercept, your y-intercept, or it's going to be a way to figure out the y-intercept, the intercept itself is this point, the point at which the line intercepts the y axis, and then this two is going to represent your slope. And that makes sense because every time you increase x by one, you're gonna multiply that by two, so you're gonna increase y by two. So this is just a, kinda of a get your feet wet with the idea of slope-intercept form, but you'll see, at least for me, this is the easiest form for me to think about what the graph of something looks like, because if you were given another, if you were given another linear equation, let's say y is equal to negative x, negative x plus two. Well immediately you say, okay look, my yintercept is going to be the point zero comma two, so I'm gonna intersect the y axis right at that point, and then I have a slope of, the coefficient here is really just negative one, so I have a slope of negative one. So as we increase x by one, we're gonna decrease y by one. Increase x by one, you're gonna decrease y by one. If you increase x by two, you're gonna decrease y by two. And so our line is gonna look something like this. Let me see if I can draw it relatively neatly. It's going to look something, I think I can do a little bit better than that. It's 'cus my graph paper is hand drawn. It's not ideal, but I think you get, you get the point. It's gonna look something like that. So from slope-intercept form, very easy to figure out what the y-intercept is, and very easy to figure out the slope. The slope here, slope here is negative one. That's this negative one right over here, and the y-intercept, y-intercept is the point zero comma two, very easy to figure out 'cus essentially that gave you the information right there.