 We're going to be looking at graphing a system of equations to find a solution. We start off with this example, and I've already given you the graph, and these are the equations. So we won't really need to calculate it for this one. We just want to see how it works, and then we'll see how we can use a calculator to find the answers as well. So when we had these two equations, remember that this one right here is going to be y is equal to negative one-half x plus six, and this equation right here is going to be y is equal to one-half x minus four. Every point on here satisfies this equation, and let's circle this in green so that I can say that every point on here satisfies this equation. Again, every point on this line satisfies this equation. So when we're looking for a solution, we want to know what satisfies both equations. And you can see that they have one point in common right here at the intersection point. So it's an ordered pair, and the ordered pair would be x is ten, and y looks to be one. Now let's see how we could actually do that in our calculator. So y is equal to, so this first one will say negative point five x, because point five is the same thing as one-half, plus six, and then enter. And my second equation, instead, I'm going to put one divided by two times x. But when I end out, put a fraction in parentheses, and then minus four. Okay, you can write it either way. And we want to look at the graph. However, if you look at our graph, it's ten one. If we, I'm in a standard window, I believe, when I graph. Okay, to find that intersection point, you do second, and then trace. Put it there so you can come back to it. Second, trace, and then you choose five, because it says intersect, and that's what we want to know. Where do they intersect? So you can choose five, and then you're just going to press enter three times. So press enter, enter, enter, and I'm going to get an error. And the reason why is right on the edge of my screen, and it doesn't like it have intersections right at the edge of your screen. So I can just come in here and change a window. It didn't work, so I'm going to choose the window. And I didn't, I was over at ten, and yes, that's my answers at ten, but it doesn't, again, want it to be right on the edge. So I just need to go a little further. I'm going to let my x max be fifteen instead of ten. Now when I graph, I should be able to see my intersection not right on the edge. And then again, second, trace, option five, and then enter, enter, enter. And the intersection it tells me is ten, one, just like we had here. And so you have another example. In this example, we're going to want to use our calculators because we don't want to have to graph it by hand. This one's great for the calculator. In fact, I can come back in here and put that one in here right away. Clear up my equation, and I can put p is equal to two m, in our case, and the calculator is going to say two x minus twelve. But the second equation, enter and clear before I forget, the second equation isn't p equal. So I have to solve for p. I have to solve for the same variable. So I've got to solve for p, which means I'm going to add three m to both sides. So I'm going to have negative seventeen plus three m. And that's what I want to put my calculator. So negative seventeen plus three x in our case. Now we can look at our graph and see if we can find the intersection point. There's one line, there's the second line, and I can see my intersection point. So I'm ready to do second, trace, five, enter, enter, enter. And you would want to make a sketch of your graph. So here's my graph, and it looks something like this and something like this. And this point right here, then, is my solution, which says five negative two. Okay, what do you think we're looking for in our table? Remember they have to be the exact same point for both equations. So this is the x for both equations. And I need my y's to be exactly the same. So here you see it, y, x is five, y is negative two here, and y is negative two here. And we know that that is going to be our solution. Y's are the same, four, x equal five. Here I have a y equal, so that's great to put in my calculator. This one's not so great, but I want to show you another little trick. Let's solve for y. So we're going to have negative three y equal to negative twelve. And I have to subtract the x, so minus x. And the quickest, easiest way to get this in your calculator is just to divide everything by negative three. But you've got to remember to put all of this in parentheses. So we're going to come in here, put our equation in, and one divided by three. This time x is going to take one and divide it by three and then multiply by x, so I don't have to have those parentheses. Plus the four, and that's my top equation. Now this time I have to have the parentheses. And it would do our division first before we did the addition or subtraction. But we want it to add and subtract first. So negative twelve minus x, close the parentheses, and then divided by negative three. So it's going to do the top part and then divide by negative three. We can look at our graph and see if it fits the standard window, and it does. And you're looking at that and you're probably saying, wait, it only showed us one line. Well, let's go look at our table for a second. And see if we really do have one line or not. Quick way to tell if we have one line. When you look at a table, notice all my y ones and y twos are exactly the same. So this is the same line. If I go back and look at my graph, so I can sketch it, it looks something like this, okay? If I had taken negative twelve and divided by negative three, that would have been positive four. And if I'd taken my negative x and divided by negative three, that would have been one-third x, which you'll notice is the same as the top equation. All right, let's look at another one set. These are nice equations. There's a point I want to make here, so I made them nice equations. Three x plus seven, enter, three x minus three, enter, and I'm going to graph them. But these are parallel. We will find out they're never going to cross, so they have no solution. And if you look in the table, you will notice that this is seven to negative three, ten to zero, there's always a difference of ten. Thirteen and three, there's a difference of ten. Sixteen and six, a difference of ten, they're never getting any closer, they're always the same distance away. So the y's are the same distance from each other. So let's make a summary. We can have a graph that has one point in common. We can have a graph that has infinite solutions or all points because they're the same line. And we can have a system that has no solution because they are parallel.