 Eventually we want to graph equations, but in order to do that we'll need to find points on the graph, and so a good place to start is with what are known as the x and y intercepts. These come about as follows. Graphing only requires an origin and two principal directions. You need to know where you're starting and what directions you're going in. But it's sometimes convenient if we set down an axis for each of our principal directions to serve as a reminder of the directions that we should be traveling in. Now an important idea is that we're not actually traveling along the axes. Rather the axes tell us which directions we should be moving in. So in rectangular coordinates we have a horizontal axis that tells us we should be moving left and right, and we also have a vertical axis that tells us we should be moving up and down. Or because our horizontal coordinate is also known as our x-coordinate, we call the horizontal axis the x-axis, and the vertical axis is known as the y-axis. Now typically we draw these axes so they cross at the origin, and because they are our representations of our principal directions, we don't always draw the principal directions. And we can also omit the origin. However it's vitally important to understand that like the principal directions, the axes have to be drawn in some place, but they don't actually exist in a particular location. They represent directions. This leads to the following idea. If two curves cross, that is a point of intersection. So we might have these two things crossing at these two points, and these are points of intersection. Now you might look at this and say, wait a minute, that's not a curve, that's a straight line. And that's true, it is a straight line, but we also call it a curve. We'll explain why in a little bit. Now if we think about these curves as living on the coordinate plane, then someplace there's an x-axis, and our curve can cross this x or horizontal axis at an x or horizontal intercept. And again, how you speak influences how you think. This is an intercept, not an intersection. And the reason for the distinction is that the x-axis in some sense doesn't really exist. It's a reference that shows one of our principal directions, but it can be drawn any place. Similarly, someplace floating around there's a y-axis, and we can talk about the y-intercepts. And again, because the x and y-axes don't really exist, we call these intercepts and not intersections. Now geometrically, finding the x and y-intercepts is easy. Suppose the graph crosses the x-axis. If we have the graph, we just look for the intercepts. So we cross the x-axis here and here, and we cross the y-axis here. The problem is figuring out the exact locations of these points. But finding an exact location is something that algebra is good for. So let's translate this into an algebraic problem. So let's take a look at our x-intercepts first. So if I want to get to the x-intercept, what I have to do is start at the origin and go out some distance h. Then go vertically a distance of k equal to 0. And that means the coordinates of any x-intercept will be h0. So for example, let's say I want to find the x-intercept of the graph 3x plus 5y equals 15. So remember the x-intercept will have coordinates h0. And that means we need to find a point where the x-coordinate is, I don't know, but our y-coordinate is 0. Well, let's let our y-coordinate be 0. If we do that, then our equation becomes, and we can solve this for x, so x equals 5. So if y equals 0, x equals 5. So what's the x-intercept? And it's important to remember that the x-intercept is a point, which means that both coordinates must be given. So that means that the point on the graph is 5, 0. What if our graph crosses the y-axis? To get to the y-intercept, start at the origin, go some distance k vertically, but go distance h equals 0 horizontally. Don't move left, don't move right. And this takes us to the point 0k. So let's try to find the y-intercept of our graph of 3x plus 5y equals 15. So the y-intercept will have coordinates 0k. So we're looking for a point on the graph with coordinates 0k. So if we let x equals 0, our equation becomes, and solving this gives us y equals 3. So if x equals 0, then y equals 3. And remember, the y-intercept is a point, and both coordinates must be given. So the y-intercept is 0, 3.