 One of the most useful things in mathematics is the transition between an algebraic representation of something and a geometric representation of something. In other words, it is the difference between a formula or an equation and a picture. This shows up in the topic of graphing equations. So when we graph an equation, if we have an equation in two variables x and y, we can form the graph of the equation by identifying all points whose x and y coordinates make the equation a true state. And in other words, if I find a point with coordinates x and y that make the equation true, this is going to be a point on the graph of the equation. And any point that makes the equation false should not be a point on the graph. For example, let's consider this, I have a couple of possible points and I have this graph of whatever this looks like. And so I want to know whether these points are on the graph of this equation. And so that's an easy determination. I check if I take a look at the point at zero, zero, my x-coordinate is zero, my y-coordinate is zero. So if I substitute these into the equation and get a true statement, I have a point of the graph, otherwise I don't. So there's my equation. And what I want to know, because I don't know for certain, is whether x equals zero, y equals zero makes the statement true. So I'll let x equals zero, I'll substitute that in, y is equal to zero, so I'll substitute that in, x is zero. So I'll substitute that in, y, zero, substitute that in. And I'll do a little bit of arithmetic. That's going to be zero from the left, zero on the right. And it is in fact true that zero equals zero. So substituting these values into my equation gives me a true statement. And so the point is on the graph. Well, what about that other point, this point one, negative one? Well, my x-coordinate is one, my y-coordinate is negative one. So I want to substitute in x equals one, y equals negative one into the equation and see if I get a true statement. So again, I have my equation. Now x is one, so I'll substitute that in, y is negative one. So I'll substitute that in. And so I'll have my new expressions here. Left-hand side, one squared plus one squared is one plus one is two squared is two. And I get four. And the big question is, is four equal to two? And no. So this is a false statement. So my point one, negative one is not a point on the graph of the equation. Well, I might actually want to graph the equation rather than checking to see whether a particular point is on the graph of the equation. So here's a general procedure for that. Rather than trying to find random points, it's easier to actually locate points on the graph by the following method. What we're going to do is we're going to choose a value for one of the variables. Either x or y, doesn't matter. We'll substitute this into the equation, giving us an equation in one variable. And we'll do a little bit of algebra and we'll solve the equation for the other variable. And this will give us a set of coordinates for a point. So for example, let's consider the equation 3x minus 4y equals 36. And the two points that I might want to find are the x and the y intercepts. Now those are the points where the graph crosses either the x or the y axis. So the x intercept is going to be where the graph crosses the x axis. And if I'm on the x axis, then any point on the x axis will have y coordinate zero. So because any point on the x axis has y equals zero, I'll let y equals zero in my equation and solve for x. So there's my equation. I'm going to allow y to be zero. And I get the equation 3x equals 36. 4 times zero is zero. 3x equals 36. And I'll solve for x by dividing both sides by three. And I'll get x equals 12. And what this tells me is that y equals zero x equals 12 makes this a true statement, then the point where y equals zero x equals 12 is a point on the graph. Now remember we specify the coordinates by giving the horizontal distance the x coordinate first. So this is going to be the point x coordinate 12, y coordinate zero, and the point 12 zero is the x intercept. Now how about the y intercept? Well any point on the y axis will have x equal to zero. So I'll substitute x equals zero into the equation and solve for y. So there's my equation. Once again I'll let x be zero. I'll do a little bit of arithmetic cleanup. Negative four y equals 36. Divide by negative four y equals negative nine. And now I know that the point where x is equal to zero, y equal to negative nine, that's going to make the equation a true statement. So the point where x equals zero, y equals negative nine will also be on the graph. And again when I give the coordinates of a point, I want to identify the x coordinate first, the horizontal distance first, the y coordinate next, x equals zero, y equals negative nine. And my point is going to be zero negative nine, and that's going to be my y intercept. So I found the x and y intercepts. Now we can graph the line between the two of them. So I need two points, which I have, and so I'll go ahead and graph them. I'll set down the origin. I'll set down my principal direction. And it's convenient to also have a perpendicular direction there. Now I'll emphasize that perpendicular direction is mainly for convenience, mainly to make sure we're going in the right direction when we move off the principal direction. So let's see. So my first point is at twelve zero. Now there is an art to graphing, and the art is this. The thing we should focus on is sine and direction, and we should worry less about magnitude. So here this point, twelve zero, as a set of directions, that says positive first coordinate, I'm going to go to the right some distance, but I'm not going to go any place vertically. So I'm going to be on the principal axis. So I'll go right some distance, and there's my point, and the first point, twelve zero. Since we want to make sure that nobody can understand what we've done, well actually we want to make sure that our work is as clear as possible. So what we should always do, anytime we put down a point, we always want to put down the coordinates of that point. This is going to be the point twelve zero. Likewise my second point, zero negative nine, and again sine and direction are going to be the important things. We'll worry less about magnitude. Zero says I go right zero from the origin. I don't go any place right or left. Negative nine says I'm going to go down by nine units. So don't go any place right or left, but go down nine units someplace around here. There's our point, and because we want to make our work very clear, we're going to label that point with the coordinates zero negative nine. Since we want to graph the line between the two points, then we have our two points. We'll draw the line in between those two points, and again you never want to put anything down on the graph without labeling what it is. This is the line between the two points. This is going to be labeled by our equation three x minus four y equals thirty six, and there's our graph. Let's take a look at another example. So I want to find two different points on the graph of y equals two thirds x plus seven, and then graph them. So we're not looking for specific points like the x or y intercept, so we can just pick values for variables and solve for the other one. And we can read this equation as a formula for y once you know the value of x. So it's easiest if we pick a value for x here and then calculate what the value of y is. So since we get to pick whatever value we want, we should pick values like y equals three thousand nine hundred eight three point one three nine eight seven one four three, which are very hard to work. Well, I get to pick that value of x, so maybe I'll pick something easy. How about x equals zero? So I'll let x equals zero, and my formula says y equals two thirds x plus seven, y equals two thirds times zero plus seven, zero plus seven, y equals seven. So the point where x equals zero y equals seven is on the graph of the equation, and I could pick a different value for x, like oh I don't know x equals pi plus the fifth root of 11. Let's pick something easy. I'm multiplying by two thirds, so what I might want to do is I might want to pick a value of x that I can multiply easily by two thirds. So for example if x equals three, two thirds times three is an easy computation. So y equals two thirds x plus seven, y equals two thirds times three plus seven, two thirds times three is easy to calculate. That three will cancel. I'll be left with two plus seven, y equals nine, and I have a second point where x equals three, y equals nine. So there's a second point on the graph of the equation. And I'll graph my two points. So again my origin and principal directions look something like that. My first point zero seven, follow the directions to the point, zero units to the right, and then seven units up. Don't go right or left, but go up someplace. My point is up here someplace. Again plot the point and label. My second point is three nine, go over three and nine, and here you might want to get at least a little bit of the magnitude correct. Here I went up seven to the point. If I'm going to go up nine, I should go a little farther up. And so my point over three up nine, over three up nine should be a little higher than this point. I'll be up there someplace and label it. And finally I'm going to draw the line through the points and label it with the equation. So there's my line and there's my equation x equals two thirds x plus seven.