– [ Voiceover ] There’s a distribute of different ways that you could represent a linear equation. then for example, if you had the linear equality yttrium is equal to 2x plus three, that ‘s one way to represent it, but I could represent this in an space number of ways. I could, let ‘s see, I could subtract 2x from both sides, I could write this as negative 2x plus yttrium is equal to three. I could manipulate it in ways where I get it to, and I ‘m gon na do it correctly nowadays, but this is another means of writing that same thing. y minus five is adequate to two times ten 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 phone number of ways to represent a given linear equation, but I what I wan sodium focus on in this video recording is this representation in detail, because this one is a identical useful representation of a analogue equation and we ‘ll see in future television, this one and this one can besides be utilitarian, depending on what you are looking for, but we ‘re gon na 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 precisely try to graph this thing. I ‘m gon na try to graph it, I ‘m just gon na plot some points here, sol ten comma y, and I’m gon na pick some ten values where it ‘s easily to calculate the yttrium values. then possibly the easiest is if x is equal to zero. If adam is peer to zero, then two times zero is zero, that term goes away, and you ‘re only left with this term correct 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 bloc, so that can be my adam, oh that ‘s not a straight as I would like it. So that looks reasonably commodity, all right. That is my ten axis and let me mark off some hashish marks here, so this is x equals one, x equals two, x equals three, this is yttrium equals one, y equals two, yttrium 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 adequate to negative one, negative two, damaging three, so on and so forth. indeed this degree right over here, zero comma three, this is adam is zero, yttrium is three. Well, the period that represents when x is equal to zero and y equals three, this is, we ‘re right on the yttrium axis. If they have a production line going through it and this note contains this point, this is going to be the y- wiretap. So one way to think about it, the cause why this is called slope-intercept class is it ‘s very easy to calculate the y-intercept. The y-intercept here is going to happen when it ‘s written in this class, it ‘s going to happen when ten is equal to zero and y is equal to three, it ‘s gon sodium be this steer 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 besides be easily to figure out the slope from this form. And if you made that conclusion, you would be decline ! And we ‘re about to see that in a few seconds. So let ‘s plot some more points hera and I ‘m just gon na 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. exchange in ten hera is one. We merely increased x by one, what ‘s gon sodium be our corresponding variety in y ? What ‘s going to be our change in y ? So get ‘s see, when ten is adequate to one, we have two times one, plus three is going to be five. so our change in yttrium is going to be two. Let ‘s do that again. Let ‘s increase our x by one. change in ten is adequate to one. therefore then if we ‘re gon na increase by one, we ‘re gon na go from ten equals one to x equals two. Well what ‘s our correspond change in y ? well when adam is equal to two, two times two is four, plus three is seven. Well our change in y, our transfer in yttrium is peer to two. become 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. then for this linear equation, our exchange in yttrium over deepen in x is constantly going to be, our change in y is two when our deepen in adam is one, or it ‘s equal to two, or we could say that our slope is peer to two. Well let ‘s just graph this to make indisputable that we understand this. indeed when ten equals one, yttrium is peer to five. And actually we ‘re gon na 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 peer to four, and this is yttrium is equal to five. so when ten is one, yttrium is adequate to five, so it ‘s that sharpen right over there. So our line is going to look- you only need two points to define a line, our agate line is going to like, let me do this in this discolor mighty over hera. Our lineage 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 did n’t draw it wholly at scale, but it ‘s going to look something like this. This is the note, this is the argumentation, yttrium is equal to 2x plus three. But we already figured out that its slope is equal to two, when our change in adam is one, when our change in adam is one, our exchange in y is two. If our change in x was damaging one, if our change in ten was damaging one, our change in yttrium 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 yttrium going to be ? Two times minus one is negative two plus three is one. So we see that, the point negative one comma one is on the line vitamin a well. So the slope here, our change in y over change in ten, if we ‘re going from between any two points on this line, is constantly going to be two. But where do you see two in this original equation ? Well you see the two veracious over here. And when you write something in slop-intercept mannequin, where you explicitly solve for yttrium, y is equal to some constant times x to the beginning exponent plus some early 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 yttrium axis, and then this two is going to represent your gradient. And that makes sense because every time you increase x by one, you ‘re gon na multiply that by two, so you ‘re gon na increase y by two. So this is fair a, rather of a catch your feet wet with the idea of slope-intercept shape, 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 equality, let ‘s say y is equal to negative ten, negative adam plus two. well immediately you say, okay look, my yintercept is going to be the period zero comma two, so I’m gon na intersect the yttrium axis correct at that point, and then I have a slope of, the coefficient here is very merely negative one, so I have a slope of damaging one. therefore as we increase x by one, we ‘re gon na decrease yttrium by one. Increase x by one, you’re gon na decrease y by one. If you increase x by two, you ‘re gon na decrease y by two. And so our line is gon na 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 moment better than that. It ‘s ‘cus my graph newspaper is hand draw. It ‘s not ideal, but I think you get, you get the point. It ‘s gon na look something like that. thus from slope-intercept form, very easy to figure out what the y-intercept is, and very easy to figure out the slope. The gradient hera, gradient here is negative one. That ‘s this negative one mighty over here, and the y-intercept, y-intercept is the compass point zero comma two, very easy to figure out ‘cus basically that gave you the information right there.
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