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Point-slope & slope-intercept equations

Point-slope is the general form y-y₁=m(x-x₁) for linear equations. It emphasizes the slope of the line and a point on the line (that is not the y-intercept). We can rewrite an equation in point-slope form to be in slope-intercept form y=mx+b, to highlight the same line's slope and y-intercept.

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Video transcript

- [Voiceover] Let's say we have a linear equation and we know when X is equal to four that Y is equal to nine and we've plotted that point here on our XY plane. Actually, I forgot to label the x-axis right over there. Now let's say we also know, we also know that when X is equal to six Y is equal to one. And we've plotted that point there. And so this green line represents all of the solutions to this linear equation. Now what I want to do in this video is I want to say, well can we find that linear equation and can we express it in both point-slope form and in slope-intercept form. And I encourage you, like always, pause the video and see if you can do it. So let's first think about point-slope form. Point, point-slope form. And point-slope form is very easy to generate if you know a point on the line, or if you know a point that satisfies, where the X and Y coordinates satisfy the linear equation, and if you were to know the slope of the line that represents the solution set of that linear equation. Now for sure we actually were given two points that are solutions, that represent solutions to the linear equation. To fully apply point-slope, or to apply point-slope easily, we just have to figure out the slope. And what we could do is, we could just evaluate well what's the slope between the two points that we know? And we just have to remind ourselves that slope, slope, is equal to, slope is equal to change in y over change in x. Sometimes people say rise over run. And what's that going to be? Well, if we say that this second point right over here, if we say this is kind of our, if we're starting at this point and we go to that point, then our change in Y, going from this point to that point is going to be, it's going to be equal to one minus, one minus nine. One minus nine. This point right over here is the point six comma one. So we started at Y equals nine, we finish at Y equals one, our change in Y is going to be one minus nine. We have a negative eight change in Y, which makes sense. We've gone down eight. So this is going to be equal to, this is going to be equal to negative eight. That's our change in Y. And what's our change in X? Well we go from X equals four to X equals six. So we end up at X equals six and we started at X equals four. We started at X equals four, so our change in X is six minus four, which is equal to two. Which is equal to two. And you could have even done it visually. To go from this point to this point your change in Y, your change in Y is you went down eight. So your change in, let me write this. So your change in Y is equal to negative eight. And what was your change in X? Again to get to this point? Well, your change in X is positive two. So your change in X is equal to two. And so what's your slope? Change in Y over change in X. Negative eight over two is equal to negative four. So now that we have a, now that we know the slope and we know a point, we know a, we actually know two points on the line, we can express this in point-slope form. And so let's do that. And the way I like to it is I always like to just take it straight from the definition of what slope is. We know that the slope between any two points on this line is going to be negative four. So if we take an arbitrary Y that sits on this line and if we find the difference between that Y and, let's focus on this point up here. So if we find the difference between that Y and this Y, and nine, and it's over the difference between some X on the line and this X, and four. This is going to be the slope between any XY on this line and this point right over here. And the slope between any two points on a line are going to have to be constant. So this is going to be equal to the slope of the line. It's going to be equal to negative four. And we're not in point-slope form or classic point-slope form just yet. To do that, we just multiply both sides times X minus four. So we get Y minus 9, we get Y minus nine is equal to our slope, negative four times X minus four. Time X minus four. And this right over here is our classic, this right over here is our classic point-slope form. We have the point, sometimes they even put parenthesis like this, but we could figure out the point from this point-slope form. The point that sits on this line with things that make both sides of this equation equal to zero. So it would be X equals four, Y equals nine, which we have right up there, and then the slope is right over here, it's negative four. Now from this can we now express this linear equation in y-intercept form? And y-intercept form, just as a bit of a reminder, it's Y is equal to MX plus B. Where this coefficient is our slope and this constant right over here allows us to figure out our y-intercept. And to get this in this form we just have to simplify a little bit of this algebra. So you have Y minus nine. Y minus nine is equal to, well let's distribute this negative four. And I'll just switch some colors. Let's distribute this negative four. Negative four times X is negative four X. Negative four time negative four is plus 16. And now, if we just want to isolate the Y on the left hand side, we can add nine to both sides. So let's do that. Let's add nine, let's add nine to both sides. Let's add nine to both sides. On the left-hand side we're just left with Y. And on the right-hand side we're left with negative four X and then 16 plus nine is plus 25. And there you have it. We have the same linear equation, but it's now represented in slope-intercept form. Once again, we see the slope right over here and now we can figure out what the y-intercept is. The y-intercept when X is equal to zero, Y is going to be equal to 25. My axis right here, I haven't drawn it high enough, but if I made it even taller and taller and you see this line is gonna intersect the y-axis when Y is equal to, Y is equal to 25. So there you go, we wrote it in point-slope form, that is that right over there, and we wrote it in Y, sorry, we wrote it in slope, we wrote it in slope-intercept form. Point-slope and slope-intercept. Hopefully you enjoyed that.