Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College solving systems of equations by graphing. So given 2 equations and a point, we want to know if that point is on both the lines, IE is that point where those two lines intersect. So what we do is we're actually going to put in the point for the variables. Remember, we do it in alphabetical order. So in this case, the two is going to go in for the X and the three is going to go in for the Y. And we want to know if it's a true statement or not. Well, 2 + 6 really does equal 8. So yes, the point is on this first line, but the question is, is the point the intersection point of the two lines? So we actually have to check it in the other line also. So does 3 * 2 - 5 * 3 equal -7? Well, 6 - 15 that doesn't equal -7 So the point is not on both lines, so it's not the intersection point. We're going to have three different cases about two lines. If the two lines intersect, they're called consistent because there's at least one equation, one solution to the equations, and independent. Pictorially, what this looks like is you have two lines, Here's one line, and the other line could be in any location as long as there's a point in common. So the fact that they're independent means that it could be this line here going through the given point, or this line, or this line, or this line. So just because I know one line and the solution doesn't necessarily mean I know the other line. The fact that there's a solution means it's consistent. The fact that just because I know one line in the solution, I don't know the other makes them independent. If it's parallel, they're going to be inconsistent because there's not going to be any solutions, but they're also going to be independent. So pictorially, if I have one line and I know that's the solution and there is no solution with the other line, there's actually infinitely many lines. That other line might be if they're the same line, there's going to be infinitely many solutions. They're called consistent and they're going to be dependent this time. If I know one line and I know that it's consistent, I know the other line has to be right on top of it. Now, this one has very special notation for solution sets. We're going to use bracey, parenthesis, whatever variables we're using. Let's call it X&Y in this case, such that a big long line Y equal MX plus B or whatever that given line was. If they're in different formats, you could use either of the two formats here. It absolutely must look like that for the solution when they're the same line. So let's look at a couple examples. If we have three X + y equal 5 and X + - 2 Y equal 4, I'm going to solve for Y in the first one. So I get Y equal negative three X + 5. That tells me we're going to go up five because we have AY intercept of the point. Remember, it's always a .05 and a slope of -3 that -3 We could think of as down 3 and to the right one, or we could think of as up three and to the left one. So down 3 into the right one, down 3 into the right one. The more points I use, the more accurate my line's going to be. So oops, thought I was going to draw. Ah, not straight at all. There we go, better. So the second equation of the line is going to be X. Oh, let's solve for Y -2 Y equal negative X + 4 divide. So we get Y equal 1 half X -, 2 Y intercept zero -2 the slopes 1/2. So zero -2 that's going to put me down here. Up one and to the right two, up one and to the right two, up one and to the right two, or down one and to the left two. Because remember 1 / 2 we could think of as -1 / -2 if we wanted. Then when we connect those points, oh need to probably move that one up to being where the points are. We can see that the point of intersection for this is going to be over 2 and down one. So the point of intersection is going to be two one. And that makes these intersecting lines. So they're consistent because there's at least one solution, but they're independent. Just because I know one line doesn't mean I know the other line. Let's look at one more example. If we have Y equal 2X plus three y = 6, solving for Y again, we're going to get 3 Y equaling negative two X + 6 / 3 Y equal -2 thirds X + 2 my Y intercept the .02 my slope -2 / 3. Or we could think of that as 2 / -3. So 02 down to to the right three down to to the right three. Connecting those points. If we solve the other equation, it's actually already in Y equal form, so Y equal -2 thirds X + 2, we can see 02 negative 2/3 of the slope. It's going to be the same line. So this is going to be written very important to write it correctly X, Y such that now I could actually state either of the two lines because they're the same line. Frequently we use the Y equal 1 because that's our slope intercept form Y equal -2 thirds X + 2 because they're the same line. There's infinitely many solutions, hence it's consistent. And if I know one line, I absolutely know the other one because it's the same line, thus they're dependent. Let's do one more example. I know I said one more minute ago, but let's do this one also. Y equal negative X + 5 S 05 Y intercept slope -1, so 0512345 slope -1. I could think of that as -1 / 1 down 1 / 1, down 1 / 1, and then the other line is already in Y equal negative X + 2 Y intercept form SO02 and the slope is -1. Remember that could be thought of as -1 / 1. So we go to 02, we go down 1 / 1, down 1 / 1, down 1 / 1. When I connect that, I can see that these are going to be parallel lines because they have the same slope. And so the fact that they're parallel lines means that there is no intersection. So if there is no intersection, they're inconsistent, no solution. And they're also in dependent. So parallel lines here, parallel lines, inconsistent, independent. You try the next one. Thank you and have a wonderful day.