 We can graph an equation by finding points, but is there something we can do that's a little bit more efficient? Well, let's take a closer look at that. While every equation corresponds to the graph of some curve, some equations are easier to use than others. So if I want to find the coordinates of points that satisfy this equation, I have to do a bit of work. On the other hand, if I want to find coordinates of a point to satisfy this equation, it's a lot easier. And so it's convenient if the equation is in a standard form. What's a standard form? Well, it's any form that we've defined as being convenient. And in this particular case, an equation in two variables is in standard form when it's in the form a times x plus b times y equals c for some real numbers a, b, and c. And so here the thing to notice is that all of our x and y terms are over on the left hand side and we have a number over on the right hand side. So let's try to put this equation in standard form. And again, the thing to notice is that standard form consists of having all of our variables on one side. So we'll write down our equation and we'll start moving our variables to one side. So I'll add eight y. I'll subtract seven x. And now I have my variables on the left and my constant on the right. And so this equation is now in standard form. Now we could say that ax plus by equals c is a first degree polynomial in x and in y. But we also have another name for it. We call it a linear equation. And that's because the graph of a linear equation is a straight line. Now before we continue we should point out something important. Fermat said that any equation in two variables corresponds to the graph of some curve. But this particular equation in two variables gives us a straight line. And what that means is that we have to consider a straight line to be a curve. Now a normal person doesn't typically regard a straight line as a curve, but mathematicians aren't normal people. The theorem that the graph of a linear equation is a straight line is a very important theorem for the following reasons. Since the graph will be a straight line, we can invoke an axiom of geometry. There is a unique straight line between two points. And this suggests the following approach to graphing this equation. First, we'll find two solutions x1, y1, x2, y2 to the linear equation. Since these correspond to points, we'll graph these points. And then because there is a unique straight line between the two points, we'll draw the line through the points. So just as a quick illustration, we already found three points on the graph of y equals 2x plus 7. Now, we might note that the equation y equals 2x plus 7 can be put into standard form. And so we know that the graph of y equals 2x plus 7 will be that of a straight line. Since it will be the graph of a straight line, we only need two of the points. So let's get rid of the one that involves fractions. So here's a useful thing to remember. The secret to graphing, graph first, then label. What do we mean by that? Well, let's start on our set of axes. Now, I have two points I want to graph. The first one is 0, 7. Now, you know this specifies a point that we get to by going to the right 0 units and then up 7 units. But what's important here is that we don't go to the right any distance at all and then we go up some distance. So rather than counting off our spaces 1, 2, 3, 4, 5, 6, 7, we'll just start at the origin, then go up some distance and simply declare that that distance is the correct amount. How do we make that declaration? We label the point. Similarly, the second point, 5, 9, tells us we should start at the origin and count out 1, 2, 3, 4, 5 units to the right and then go up 1, 2, 3, 4, 5, 6, 7, 8, 9. But again, we can just start at the origin. Go out some distance and then up some distance. Now, while we don't have to make our distance as exact, we should at least make them consistent. So for this first point, we went up by 7 units. Because we're now supposed to be going up by 9 units, we should make sure that we go up a little bit farther. And again, we declare that this is the correct place by labeling it. And since we know that this is the graph of a straight line, we'll draw the line and, guess what? We'll label it. We'll put down the equation of the line.