How to Find the Slope of a Line

[On screen is a graph with two lines with their points labeled, (-2, -2) and (5, 3)]

Female Narrator: Welcome! In this video we’re going to go over how to find the slope of a line, given any two points on that line. Here’s a line segment with two points: (-2, -2) and (5, 3). Let's suppose that this line represents data that a company has collected and they want to know the rate of change that the data describes. The slope is the rate of change. So we’ve done this before, and we know that to find the slope, we just need to create a stair step, So we’re finding the slope of this step. From this point up to this new point. So let's first calculate the rise, because we need to find the slope m, which equals the rise over the run. [Writes, m=rise/run.] So let's calculate the rise, which is the length of this leg of our triangle.

We’ve created our triangle here, and we need to know the leg of this triangle. Well, we know that our greatest Y value is 3 and our lowest Y value is negative one. What is the distance between 3 and negative one. So it's pretty easy to see, in this scenario that the length of this side of the triangle is four units, because we are going from three down to two, one, zero, negative one. [Draws curves to count the points, resulting in four curves.] But if we didn’t know, if we couldn’t see it so easily in that way, how would we go about doing it? Well if we subtracted three minus negative one, so three minus a negative one, subtracting  a negative is the same as adding. So we would have three plus one, equals four! That would be in the rise. So subtracting the Y value from this point, from the Y value of the other point, we were able to get the height of this side of the triangle. Now we need to figure out the run. So the run will be in blue. So our greatest X value is five and our lowest X value is negative two. If we subtract, again, five minus a negative two, it should give us the distance between these two points. We can look at it here, and see if we are right. Because here we will have five steps, or units, plus another 2 units gives us 7. It should equally seven, just like this one, [refers to the right side of the triangle.] because it's one, two, three, 4, 5, 6, 7, steps or units long. So let's see; if we take five minus a negative two, [writing it out.] five minus negative two equals, five minus negative two is the same as 5 plus 2, which does equal seven.

So by taking the X value of this point, and subtracting off the X value of the other point, we were able to get the length of this other side of the triangle. So what can we conclude from this about the slope? Well we know that the slope is the rise over the run. And we calculated that to be four over seven. SO our slope, M is equal to four sevenths. Now suppose these were very different numbers, suppose that we weren’t able to just immediately see the lengths of these legs. What did we do here that made it possible for us to find the slope. Well as we said before the slope is equal to the rise over the run. So what did we do, in this case, to find our slope. Well given two points, let's call this point number two, [The point (5, 3) on the graph] and we’ll say that this is point number one [The point (-2, -1) on the graph]. What did we do in order to calculate this slope. Well first of all, let’s figure out what we did in the rise. So i just moved this over, so that when we write, this line won’t get in the way.

So M, our slope is equal to our rise over our run. We took the Y value from point 2, the Y value, from point two, I’m going to denote that by putting a little two next to it, so this is Y of two. And this is x of point 2. We’ll put comma there. The five is the C value of point number two, and the three is the Y value of point number two. So we took the Y value of point number two, and we subtracted off the Y value of point number one. Let's do the same thing for this point. So negative 2 is the Xz value of point number one and negative one is the Y value of point number one. So what we did is we took the Y value of point number two, which is three and we subtracted off the Y value of point number one, which is a negative one. We took Y two minus Y one. Then to find the run, We did a similar thing. We took the X value, number five, and we subtracted off the negative 2. So we took the X value of point two, so X of two, minus the negative 2, which is the X value of point number one, X one. So what we have right here is the equation, which allows us to find the slope of any line given two points.

OS the slope is equal to Y two minus Y one divided by X two minus X one. Now one important thing to always keep in mind, is to always keep your points straight. It doesn't matter which point you name as number one or number two, but just remember to keep them in order. So if you start with this point as point number two, make sure that it's X value and it's Y value stay in the right spot. And if this is point number one, make sure it's X and Y values stay in the right spot, in the equation.

[End of video.]