Intro to point-slope form

So what I've drawn here in yellow is a line. And let's say we know two things about this line. We know that it has a slope of m, and we know that the point a, b is on this line. And so the question that we're going to try to answer is, can we easily come up with an equation for this line using this information? Well, let's try it out. So any point on this line, or any x, y on this line, would have to satisfy the condition that the slope between that point-- so let's say that this is some point x, y. It's an arbitrary point on the line-- the fact that it's on the line tells us that the slope between a, b and x, y must be equal to m. So let's use that knowledge to actually construct an equation. So what is the slope between a, b and x, y? Well, our change in y-- remember slope is just change in y over change in x. Let me write that. Slope is equal to change in y over change in x. This little triangle character, that's the Greek letter Delta, shorthand for change in. Our change in y-- well let's see. If we start at y is equal to b, and if we end up at y equals this arbitrary y right over here, this change in y right over here is going to be y minus b. Let me write it in those same colors. So this is going to be y minus my little orange b. And that's going to be over our change in x. And the exact same logic-- we start at x equals a. We finish at x equals this arbitrary x, whatever x we happen to be at. So that change in x is going to be that ending point minus our starting point-- minus a. And we know this is the slope between these two points. That's the slope between any two points on this line. And that's going to be equal to m. So this is going to be equal to m. And so what we've already done here is actually create an equation that describes this line. It might not be in any form that you're used to seeing, but this is an equation that describes any x, y that satisfies this equation right over here will be on the line because any x, y that satisfies this, the slope between that x, y and this point right over here, between the point a, b, is going to be equal to m. So let's actually now convert this into forms that we might recognize more easily. So let me paste that. So to simplify this expression a little bit, or at least to get rid of the x minus a in the denominator, let's multiply both sides by x minus a. So if we multiply both sides by x minus a-- so x minus a on the left-hand side and x minus a on the right. Let me put some parentheses around it. So we're going to multiply both sides by x minus a. The whole point of that is you have x minus a divided by x minus a, which is just going to be equal to 1. And then on the right-hand side, you just have m times x minus a. So this whole thing has simplified to y minus b is equal to m times x minus a. And right here, this is a form that people, that mathematicians, have categorized as point-slope form. So this right over here is the point-slope form of the equation that describes this line. Now, why is it called point-slope form? Well, it's very easy to inspect this and say, OK. Well look, this is the slope of the line in green. That's the slope of the line. And I can put the two points in. If the point a, b is on this line, I'll have the slope times x minus a is equal to y minus b. Now, let's see why this is useful or why people like to use this type of thing. Let's not use just a, b and a slope of m anymore. Let's make this a little bit more concrete. Let's say that someone tells you that I'm dealing with some line where the slope is equal to 2, and let's say it goes through the point negative 7, 5. So very quickly, you could use this information and your knowledge of point-slope form to write this in this form. You would just say, well, an equation that contains this point and has this slope would be y minus b, which is 5-- y minus the y-coordinate of the point that this line contains-- is equal to my slope times x minus the x-coordinate that this line contains. So x minus negative 7. And just like that, we have written an equation that has a slope of 2 and that contains this point right over here. And if we don't like the x minus negative 7 right over here, we could obviously rewrite that as x plus 7. But this is kind of the purest point-slope form. If you want to simplify it a little bit, you could write it as y minus 5 is equal to 2 times x plus 7. And if you want to see that this is just one way of expressing the equation of this line-- there are many others, and the one that we're most familiar with is y-intercept form-- this can easily be converted to y-intercept form. To do that, we just have to distribute this 2. So we get y minus 5 is equal to 2 times x plus 2 times 7, so that's equal to 14. And then we can get rid of this negative 5 on the left by adding 5 to both sides of this equation. And then we are left with, on the left-hand side, y and, on the right-hand side, 2x plus 19. So this right over here is slope-intercept form. You have your slope and your y-intercept. So this is slope-intercept form. And this right up here is point-slope form.