 This video is going to be on how to graph to solve systems. Now I'm going to look at some systems that aren't graphed. We have to graph them. And how would we do that? Well, there are a couple ways that we could do this. One way we could do it is just to pick points. So I'm going to take care of this first one here, 2x plus y. I'm going to pick some points. In fact, I'm going to let x be 0. If x is 0, then I have 2 times 0 plus y. And that's equal to negative 3. So I know that y is negative 3. And if I let y be 0, then I have 2 times x plus 0, which is my y, equal to negative 3. And so that's 2x equal negative 3 divided by 2. And x is equal to negative 3 over 2. Or some of you may prefer to say that that's negative 1.5. It might be easier to graph if we think of it as negative 1.5. And two points would give us a line. So 0, negative 3, 1, 2, 3. Notice our scale is 2 on our graph. But each dotted line is 1. And then negative 1.5, so here's negative 1. And negative 1.5 and 0 is my x-intercept. And I can kind of draw my line. And it won't be perfect here because I don't have a straight edge since I'm doing it on my thing. Well, let's try the same thing for the other equation. You got your x, you got your y. We're going to let x be 0. In fact, I should do this in another color. Let x be 0. So now we have 0 minus y is supposed to be equal to negative 3. So that tells us that negative y is equal to negative 3. And y will be equal to 3 when we divide by negative 1. So 0, 3. And then if we let y be 0, we'll have x minus 0 is equal to negative 3. So the x-intercept is negative 3, 0. So 0, 3 is the y-intercept. And negative 3, 0 is the x-intercept. And if I draw my line through there, it looks like I have an intersection point right here. And that point looks like it is negative 2 and 1. And you can double check if you want to. 2 times negative 2 plus y is supposed to be equal to negative 3. And this is negative 4 plus y, y was 1, my fault. Plus 1 is equal to negative 3, and that's true. And if we do the other equation, negative 2 is our x minus 1, which is our y. And that's supposed to be equal to negative 3. Negative 2 minus 1 equal negative 3. So it satisfies both. We know we did it right. All right. So how else could we have done this? Well, another way that we could have done it was to solve for y. So if I take the top equation, I have x minus 4y is equal to 4. And I can subtract the x from both sides. So I have negative 4y is equal to 4 minus x. And then I can divide by negative 4, everything. And that gives me y equal to 4 divided by negative 4 will be negative 1. And negative x divided by negative 4 will be plus 1 over 4x. Now I know that this value here, the negative 1, that's my y-intercept or my b. And this 1 fourth right here, that's my slope. So I should be able to graph it using this y-intercept and the slope. Y-intercept is negative 1. And then I go up 1 and over 4, 1, 2, 3, 4. And then I go up 1 and over 4, 1, 2, 3, 4. Or remember that this could be and could also be the opposite of 1, which is negative 1. And the opposite of 4, which is negative 4. So I could go down 1 and to the left 4. And I'm going to start from my y-intercept so that I can get to the left part of my graph. Down 1 to the left 4. And again, down 1 to the left 4. And if I try to graph my line, that's not going to be the easiest thing to do. But I can try to get them in there. And if you have a straight edge on your paper, it's much easier to do. Well, I need to do the same thing then for my second equation. And I think I'll do that under the graph. We have 3x minus 4y equal to negative 4. So we subtract our 3x. And we have negative 4y equal to negative 4 minus 3x. And we divide everything by negative 4. And we find out that y would be equal to positive 1, negative 4 divided by negative 4. And then a negative times a negative again is going to be plus 3 over 4x. Well, this is my b. So b is 1, 0, 1. So I'll graph that. And then this is my slope. So it means I'm going to go up 3, 1, 2, 3, and over 4. 1, 2, 3, 4. Well, if you look at that, if I do it again up 3, 1, 2, 3, and then over 4, 1, 2, 3, 4. I can see that I'm getting farther and farther away from my line. So I probably need to come back here and try m is equal to the opposite of 3, which is negative 3. And the opposite of 4, which is negative 4. So that tells me to go down 3, 1, 2, 3, and over 4. 1, 2, 3, 4. And down 3, 1, 2, 3, and over 4. 1, 2, 3, 4. Now, you probably realized that the point I graphed before that was actually the point of intersection. But I wanted to be able to draw the line. But there's our point of intersection. And that point of intersection looks like negative 4 in the x direction and negative 2 in the y direction. I find that if I set it equal to y, it's a lot easier to graph and it's a little more accurate because I can use my slope to help me get the points. So we're going to use this one. x plus y equal 8. Solve it for y. So x plus y equal 8. I have to subtract the x from both sides. And then I have my y equal to 8 minus x. So if I graph this then, I have 8. And let's go up 8. 1, 2, 3, 4, 5, 6, 7, 8. And then I'm going to go down 1 and to the right 1 because that's what the negatives and positives tell me. So down 1 to the right 1, down 1 to the right 1, down 1 to the right. And so on. I need to get several of these in here. Oh, I'm not drawing my line very well, but we can guesstimate when we get this second equation in. It says that x is equal to 5. Remember with one variable equation, you either have a vertical line or a horizontal line. This says that x is always 5. So if I come to x equal 5 and then find points where x is always 5, it's going to be this vertical line at 5. And then if I look closely, I can see that here's my ordered pair and that would be 5 in the x. And if I go across, I've gone up 3. So it's 5 3 is the ordered pair and we can double check. 5 plus 3 is 8. And of course, x is equal to 5. Alright, now let's look at this example. If I subtract x from both sides here, I'm going to have y is equal to 2 minus x. That was a pretty easy one to do. And let's go ahead and graph that one. You've got 1 2 is our y intercept. And then we're going to go down 1 and over 1. This is a negative slope again. Down 1, over 1, down 1, over 1. Or up 1 to the left 1, up 1 to the left 1. And I get my line. And my line looks something like this. Now, if I had to take this equation, I would subtract 3x from both sides. So I have 3y equal to negative 9 minus that 3x. I'm going to divide everything by 3. And I end up with y is equal to negative 3 minus x. So let's graph that one in red. 1, 2, negative 3. And then I need to go up to get to my graph. So I'm going to write my slope as positive 1 over negative 1. That's still a negative 1. Up 1 to the left 1, up 1 to the left 1, up 1 to the left 1. And I could use my slope from my y intercept. So down 1 to the right 1, down 1 to the right 1. And if you look at these two lines, what do you notice? These two lines are parallel. And if you look at them, parallel lines have the same slope. That means they're going at the same rate. So they're never going to cross. So there is no solution here. These lines will never have anything in common. This is also called inconsistent. It's inconsistent with the solution. When we did our other examples, they were consistent with the solution. And these are inconsistent. Let's look at our last example then. Taking the top equation, solving for y, I would have to subtract 3x from both sides. And that will give me 3y on this side. And I'll have the sixth, and then I subtract the 3x from it. So minus 3x. Now we divide everything by 3. And we find out that y is equal to 2 minus x. So we do 2, and then down 1 over 1, down 1 over 1, or up 1 to the left 1, up 1 to the left 1. And we get a nice little line here. And then if we try the second equation, if we add 2x to both sides, we're going to have negative 2y equal to negative 4 plus 2x. And if we divide everything by negative 2, then y is going to be equal to negative 4 divided by negative 2 is positive 2. And positive 2 divided by negative 2 is going to be minus 1x. So we start at 2, and we go down 1 over 1, and oh, look at that. Every point on this line satisfies the same ones on the other line. They are the same line. So these two are the same line, which means that every single point for one equation is going to satisfy the other equation. So they have infinite solutions, or the same line, you could answer it either way. And this one is consistent with the solution. So there's two types that are consistent. We have this type, which is consistent, but they're also called dependent. They have a solution, but they're dependent on each other because they're the same line. If we go back and look at some of our other examples where we had one solution like this one, this one is a consistent because it has a solution, but it's a special one called independent because the two lines are independent of each other. So you can have consistent and independent, consistent and dependent, whether the same line like this one, or you can have two parallel lines, which are inconsistent.