 We're going to be talking about slope again, but this time we're going to be using the slope formula and then we're going to talk about how to use that formula to be able to graph a line. But let's remind ourselves what we did. This is the very same graph that we saw in the last video and we started at the y-intercept here and we went down to the x-intercept. And we said that we went down four, which was our rise, and we went to the right two, which is a positive two in the run direction. So rise is negative four over run, which is two, and we said our slope was negative two. Now we're going to prove to ourselves that this formula really works. So there's a couple fundamentals we need to know. First of all, we need two points. This point we're going to call up here x-one and y-one, and we know that that's the y-intercept and it was zero-four. And then down here we're going to call this point x-two, y-two, and we know that that point is two-zero because it was the x-intercept. The slope formula says y-two minus y-one over x-two minus x-one. So I need to go find my y-two, that's this one right here, and it happens to be zero. And then I need to go find my y-one and subtract that, y-one happens to be four, so I'm going to say zero minus four. And then I have x-two. I have to start with the same point, okay? So this is the point I'm always going to start with. Well, I start with the y, I have to start with that x. So two minus y-one, which over here is zero, and we have then negative four over two, or again, negative two slope. Now we want to find two other points just to prove to ourselves that this really does work. So we're going to pick a point that looks nice on this graph on the graph, on our line. This point right here looks really nice. And that one, we're going to come out here, we're going to call that one. It looks like we went over one in the x-direction and up two in the y-direction. And then this point down here looks like a really nice point. And it looks like we went over three in the x-direction and down two in the y-direction. So those are our two points that we're going to use. And let's call this one x-one, y-one. It doesn't really matter which one you choose. You just have to decide that one's going to be x-one, y-one's going to be x-two, y-two. And let's use our slope formula again. So I'm going to, y-two happens to be negative two minus my y-one, remember the formula. So y-one happens to be two over, remember we have to start with this point. So x-two is going to be three minus x-one, which is one. And when I subtract, I get negative four on the top over two on the bottom, which again is negative two. So three times now, I've used rise over run, I've used the x and y intercepts, I've used just two random points on my line and every time I found the slope to be negative two. So talking about equations of lines, we're at this point right here, just so you can follow along where I am. It says the equation of this line is actually y equal to negative two x plus four. And it says, what did you find the slope to be above? And we found three times that to be negative two. Is that value anywhere in the equation? Think about it for a second. Here's your equation. Where do you see the value of negative two? Well you see the value of negative two as the coefficient on x, okay? What did you find the y intercept to be for this point? Remember we found that to be zero four. And do we see that value of four anywhere in that equation? And here it is right here as the constant. So we can say that the coefficient on x is equal to our slope and the constant would be equal to our y intercept. Now given the equation y equal one-third x plus two, what's the slope of the line? Well the slope again is the coefficient on x, so it would be one-third. And does it have a positive or negative slope? It's a positive slope because it's a positive one-third that we're looking at. And what is the y intercept? Well remember that that is going to be our constant and our constant is two. So it would be the point zero two. Okay so now we want to use that line that we just found the slope in the y intercept for and use it to be able to graph the line. So remember here that we said the slope was equal to one-third and this is rise over run and the y intercept is the point zero two. So let's plot zero two. So start with the y intercept, alright? So zero two would be this point right here. And then we're going to use the rise to be the second thing we're going to do. So from that point we rise one. So we go up one because it was positive we're going to go up. And then third thing we're going to do is do the run. So from there it's kind of like going to a stop sign and then turning. So we're going to turn at this stop line and go three to the right because it's a positive three and we would then have our new point. Alright, let's try that again. Step one in this case would be start with this point. Step two would be go up one. Step three would be to go over three. One, two, three and if you see we're off our graph so that's not the best. So what happens if we get off our graph? We really haven't even found a second point that fits our graph. Well there's another thing we can use. Thinking about the fact that it's one over three is our slope. But we can say that that's the same thing as negative one over negative three because a negative divided by a negative is a positive. So I can change the signs. And this time I'm going to go down one and to the left three. So I go, let's start here first and see if we can get back to where we were. So we go down one and then one, two, three. Oh, there we are at our point. Try it again. Go down one and then to the left three and there we are back at our y intercept. And then to find another point we can go down one and over three. And now we have three points that lie nicely on my graph and actually a fourth one. If I could draw a straight line we'd have our line. So again, find a point. The y intercept is in the equation so that's a good place to start. Then go your rise and over and then your run. One last thing about this changing signs. If this had been a negative one third, then if we wanted to change that one to go the other direction we would say it was a positive over negative three. It's still equivalent. The negative can be in either part of the fraction. So if we have both positives we can change into both negatives. But if you have a negative you make it a positive and if you have a positive you make it a negative and then you can go the other direction.