 In two dimensions, the intersection of two curves is all points x and y that make both equations true. In three dimensions, the intersection of two surfaces is all points x, y, and z that make both equations true. And so what if we looked at the intersection of the graph and the three coordinate planes? So for example, let's sketch the intersection of the graph of 2x plus 3y plus 5c equals 30 and the three coordinate planes. So the portion of the graph on the xy plane, well remember the xy plane has equation z equal to 0, equals means replaceable, and so our equation becomes, so we can find points by letting the variables take on different values. For example, if x equals 0, we find, and if y equals 0, we find. So this will be the line going to the point x equals 0, y equals 10, and x equals 15, y equals 0. Well, not quite. Remember we are trying to graph in three dimensions, so there is a third coordinate, and we need to know what z is. And that's a tremendous problem. How are we ever going to know what z is in this case, if only there was some way we had of knowing what the value of z was. Oh wait, here it is. And it's useful to keep in mind, if it's not written down, it didn't happen. Part of the reason that it's useful to write things down, like the portion of the graph on the xy plane will have z equal to 0, is that at some point we may actually want to remember that we're letting z equals 0. So let's try to graph this. We're looking at the xy plane. And in some sense, this is the easiest because this is just our familiar xy coordinate plane, and we know what that line looks like. Now similarly, we can look at the portion of the graph on the yz plane. Well remember, that's where x is equal to 0, so equals means replaceable. And again, we can find points by picking values for the coordinates of sum, and then solving for the last one. So if y is equal to 0, we get, and if z equals 0, we get, and so we go through the points x equals 0, y equals 0, z equals 6, and x equals 0, y equals 10, z equals 0. Now if we want to graph this, it's helpful to keep in mind what our perspective is. So right now we're staring at the xy plane, which means that we're also looking along the z axis so we don't see it. If we turn into the first octant so we can see the yz plane, then, so the line we just drew is in the same plane as our line of sight, and the important thing to recognize here is our y axis runs horizontally and a z axis runs vertically. And so we can graph our points, keeping in mind the values of y and z. And we'll get a straight line. Third times the charm, the portion of the graph on the xz plane will have y equal to 0, and so we'll let y equals 0. And we can pick values for the remaining variables and solve for the last. And we don't have to get too fancy. We can either let x equals 0 or z equals 0 and solve. And so this equation will correspond to the line going through 0, 0, 6 and 15, 0, 0. Now something odd happens when we try to graph this. So remember we want to graph from the perspective of the first octant, and right now we're looking at the yz plane. If we want to look at the xz plane, we're going to have to rotate around a little bit. And here's the important thing to notice. The positive x axis is actually running to the left. So when we graph this point 15, 0, 0, it's actually to the left of the origin. And our line will look like. And let's see what this looks like in three dimensions. And we can do this for more complicated graphs as well. So let's consider the graph of z equals x squared plus y squared and the coordinate planes. So the points on the graph and the xy plane will have z equal to 0. And this has the solution 0, 0 only, which is just a single point at our origin. If I want to sketch the points on the graph and the yz plane will have x equal to 0, and that gives us. And we should recognize this as the equation of a parabola opening upward. So again, we'll turn to face the yz plane and sketch our parabola. And if I want to look at the points on the graph and the xz plane, well that will have y equals 0. And this will look like a parabola opening upward. So again, moving around to face the xz plane, graphing our parabola. And again, if we want to look at this in three dimensions, now we don't have to limit ourselves to the coordinate planes. We can find the intersection of any curve and any plane. So for example, we want to find and describe the intersection of z squared equals x squared plus y squared and the plane 2x plus 3y plus 5z equals 30. And that will be a system of two equations and three unknowns. And we have a nonlinear system, so that's going to be a little bit challenging. So let's focus on planes of the form x equals a or y equals b or z equals c. So let's find and describe the intersection of z squared equals x squared plus y squared and z equals 4 and also the intersection with x equals 3. E equals means replaceable, since z equals 4, we can replace in our equation and we see that this is a circle centered at the origin with a radius of 4. And if I go a little bit further, since z is equal to 4, then this circle is 4 units above the xy plane. And so we might say that the intersection with z equals 4 is a circle centered at the origin with radius 4, that is 4 units above the xy plane. With x equals 3, e equals means replaceable. Now it's useful to think about our perspective, our z axis is going to be the vertical axis, the y axis is going to be the horizontal axis. And so this is going to be a hyperbola opening upward and because x equals 3, that means we're actually going to be 3 units away from the yz plane.