 An important property of straight lines is known as the slope. And this comes from the following idea. Given two lines, we could say that the first line is steeper than the second. Equivalently, we could say that the second line is shallower than the first. We'd like to be able to assign a numerical value to steepness. And here's a useful strategy we'll find many opportunities to use. Look for the right triangle. We might approach the problem as follows. If we follow the line, the steepness can be measured by how far up or down we move over a given distance. So suppose we move some distance to the right. We'll call that about the run. Then we might move up or down some distance. We'll call that the rise. And then this suggests that what we think about as the steepness of the line, our slope, is the ratio of rise over run. So let's try to find the slope of the line between 0, negative 3, and 8, 0. Now, slope is a geometric concept. So the hardest way to solve this problem is to not draw a picture. And that's great for building character. But it's not so great for learning math, so let's draw a picture. We'll graph the points and the line. So again, a useful strategy is to look for the right triangle. So we can use our two points as two of the vertices, and we'll extend sides horizontally and vertically from those points. So let's take a little run. We'll move horizontally from our starting point until we're right below our ending point. So how far have we gone? Since we're moving horizontally, let's take a look at our horizontal coordinates. We started at x equals 0 and ended at x equals 8. So a useful thing to remember is how far you've gone is always the end point minus the beginning point. So since we started at x equals 0 and ended at x equals 8, our run was 8 minus 0 or 8 units. How about our rise? Well, now we can move vertically to the point. And to determine how far we've gone, we'll take a look at our vertical coordinates. We started at y equals negative 3 and ended at y equals 0. How far we've gone is end minus beginning, so our rise was 0 minus negative 3 or 3 units. Our slope will be the ratio of rise over run. So the slope will be 3 over 8. Now it's helpful if you always move from left to right, but that means that sometimes our height will increase and sometimes it will decrease. And so as with the coordinates themselves, we should consider rise as having a sign. If the height increases, the rise will be positive. If the height decreases, the rise will be negative. And if the height doesn't change, the rise will be 0. Now as long as you keep in mind how far you've gone is always end minus beginning, the signs will take care of themselves. So for example, let's say we want to find the slope of the line between the points 3, 4, and 1, 9. So again, we'll do this the hardest way possible and not graph. Well, evidently, you want to do this the easy way, so let's graph our points. Remember, the secret to graphing is graph first, then label. So our point 3, 4 is out some distance, up some distance, so we'll plot and label. The other point 1, 9 is also out and up. But because our x-coordinate is less, we don't go out so far, and because our y-coordinate is greater, we'll go up further. So that point might be here, labeled. And we'll draw our line. And the first thing to recognize here is if we do want to go from left to right, we should view our line as running from 1, 9 to 3, 4. So the run, we go from x equals 1 to x equals 3, that's 3 minus 1, that's 2 units to the right. The rise, we go from y equals 9 to y equals 4. And so our rise, end minus beginning, 4 minus 9, that's negative 5 units. And the slope will be the ratio rise over run, negative 5 over 2. Now if we put all of these ideas together, we get to a formula for calculating the slope. And remember, one of the worst ways to learn mathematics is to memorize a bunch of formulas. You should understand the concept. So if you remember what slope is, if you remember this concept of a ratio of the rise over the run, you don't actually need to remember the formula. In fact, if you remember what slope is, if you keep in mind this concept of rise over run, you can reproduce it or generate the formula. So let's do that. So the one thing to remember is that how far you've gone is always end minus beginning. So let's say I run between the points x1, y1, and x2, y2. If I want to find the rise, well, I start at a height of y1 and I end at a height of y2. So the rise end minus beginning is going to be y2 minus y1. On the other hand, the run is going to be x2 minus x1. And now I can find the ratio, rise over run, y2 minus y1, over x2 minus x1. So let's go ahead and calculate that, the slope of the line between 3, 4, and 5, 9, and the slope of the line between 3, 4, and 1, 9. So we'll pull in our slope formula for reference, but we don't really need it. But since this is the first time we're calculating slope from the formula, let's go ahead and keep it out for reference. So we want to go from the point 3, 4, to 5, 9. So we want to find the difference in the y values, 9 minus 4, divided by the difference in the x values, 5 minus 3. And so we can simplify this and get our slope, 5 halves. If we want to go between 3, 4, and 1, 9, we can still find the difference in the y values, divided by the difference in the x values, and that gives us a slope of 5 over negative 2, or negative 5 halves.