 So, let's take a look at a few more examples of graphing equations in standard form. So, maybe I'm going to be kind, or maybe you'll be lucky and the universe gives you an equation in standard form. Don't count on it. All right, enough comments from the Peanut Gallery. If we have this equation in standard form, we know it's going to graph as a line, so we need to find two points on the line. So, we'll pick a value for one of the variables and solve for the other. So, if we let x equals 0, substituting and solving gives us y equals negative 3. And if we let y equals 0, then substituting and solving gives us x equals 8. And so, we know two points on this graph are 0, negative 3, and 8, 0. So again, the secret to graphing is graph first, then label. So, this first point, 0, negative 3 says go down three units. So, we'll start at the origin and go down some distance and declare that it's the correct distance by labeling it. This second point says go right eight units. So, we'll start at the origin, go right some distance and declare it's the correct amount by labeling it. And since we know this is the graph of a straight line, we'll draw the line between the two points and we'll declare that it's correct by writing down the equation. Let's fly around with this idea a little bit. So, let's start off by writing the equation x equals 5 in standard form. And then we'll find two points on the line and graph it. So, in order to write this in standard form, we need values a, b, and c where our equation can be written in the form ax plus by equals c. So, let's remember a couple important rules of algebra. First of all, we can always include or omit a coefficient of 1. So, I need a coefficient of x, so I can put in a 1 and now I have 1x equals 5. Now, standard form also requires us to have a y term, which we don't have, but more generally, standard form requires us to have a sum of two things. So, we want to rewrite this as a sum of two things. So, another rule that we can remember from algebra is that we can always add 0. So, let's add 0 and you might say, well, that isn't standard form, there's no y there. And that's true, but remember, it doesn't matter what you write down first, fix things as you go along. So, here the fix is to remember that we can multiply 0 by anything we want and it doesn't change anything. So, I'll multiply 0 by y and now I have my equation in standard form 1x plus 0y equals 5. So, now let's look for ordered pairs x, y that satisfy this equation. And so, we might try to let x equals 0 equals means replaceable. So, we'll replace x with 0 in our equation and we find this is false. And so, no value of y will solve this equation. Well, that means we can't choose values for x and solve for y and so, we should choose values for y instead. So, let's let y equals 0 equals means replaceable. So, every time we see a y, we'll replace it with 0. And then we can solve this equation for x and so, x equals 5, y equals 0, 5, 0 is a point on the graph. Well, let's let y equals 1. So, equals means replaceable. So, we'll replace every occurrence of y with 1, which gives us an equation which we can then solve. And so, that tells us the point where x equals 5, y equals 1, 5, 1 is a point on the graph. So, now let's graph. We'll put down our coordinate axes. This first point, 5, 0, says we'll start at the origin and go to the right sum distance, but don't go up or down and label. This second point says we go to the right the same distance we went before, but this time we'll go up one unit. We'll label, draw the line, and label that too. So, it's worth emphasizing we can always go through this procedure of finding points on the line, but it may be worth noting that this equation x equals 5 gave us a vertical line, and this is in fact an example of a much more general result. Namely, the graph of x equals h is a vertical line through the point hk. What if we have the line y equals one-third? So, we'll go through the same process, we'll write it in standard form, then find two points in graph. So, y equals one-third, we can always include a coefficient of 1, we can always add 0, we can always multiply 0 by anything, and that gives us our equation in standard form. Now, we do have this fraction here, and you should be able to work with fractions. They are inevitable. But if you want to, you can multiply everything by the denominator 3 and eliminate the fractions. So, if we take that last step, we'll end up with another form of our equation. If we let x equals 0, our equation becomes, which we solve. If x equals 5, our equation will be, which we solve, and that gives us two points on the graph, x equals 0, y equals one-third, 0, one-third, x equals 5, y equals one-third, 5, one-third. So, we'll graph these two points. So, 0, one-third, don't go right, but do go up some amount, 5, one-thirds, go to the right a bunch, and then up the same amount that you went the first time. And notice this is a horizontal line through all points whose y coordinate is one-third, and so this suggests the following theorem. The graph of y equals k is a horizontal line through hk.