Hi wonderful math people. We're going to solve systems of equations by graphing. One of the first things we want to look at is given 2 equations and a point, is that point located on both of the lines. So if we look at this example here, we're going to ask is the point 2/3 on X + 2 Y equal 8? Well, the way we're going to do this is we're going to substitute the two which is in the abscissa location in for the X. So 2 + 2. Now we're going to substitute the Y which is 3 in the given point or the ordinate in for the Y value. So the X value here which is 2 when in for the X and the Y value of three went in for the Y. Now the question is does this make a true statement? So 2 + 2 * 3 is 6/2 plus 6/8 equal 8. So the point is on the line. It made a true statement. Now, just because the point is on one line doesn't necessarily mean it's definitely on the other line. So we need to check the same point with the second line. We're going to do the same process. We're going to put the abscissa, the two in for the X and the ordinate, or the three in for the Y, and we're going to see if we have a true statement. 6 - 15 equal negative seven, 6 - 15 negative, 9 equal -7 not a true statement. So the point's not on the second line, not on the second line. Now, if the point had been true on both lines, the point would have been at their intersection, or possibly the lines might have been the same line. So we're going to look at three different possible cases here. 1 is if the two lines intersect and intersect in any old fashioned 2 lines that intersect. The lines are referred to as being consistent because there's at least one solution, and independent because one line doesn't depend upon where the other line is. If I knew the solution and only one of the given lines, there's actually infinitely many lines that could still pass through the given known point. The next possibility that we're interested in would be if the two lines were parallel. If they're parallel, it means there's no solutions, so they're inconsistent, and they're also independent of each other. The reason they're independent of each other is there are infinitely many parallel lines. Given these two lines, I have infinitely many parallel. So what happens is, if I only knew this one line, could you absolutely beyond a doubt tell me which of the other lines is the parallel one? And the answer is no, because there's infinitely many parallel to that given line. The last possibility is if the lines are the same same line. If they're the same line, they have infinitely many solutions. So they're called consistent and they're dependent. If you know one line and you know that they're consistent, do you absolutely, positively know where the other line is? And the answer's yes, the lines have to be the same line. If I know one line and I know it's consistent and it's the same line, then the other line has to be right on top. We're going to look at a couple examples now. The first one we're going to look at is oops. The first one we're going to look at is three X + y equal 1. So we have a couple different ways to graph this. One was to use X&Y intercepts. The other is to solve for Y. And this first one, let's just solve for Y. So Y equal negative 3X plus one. That means our Y intercept is the .01 and our slope is -3. Or we could think of it as -3 / 1. So if we go to 01 and now our slope being -3, we're going to go down 3 into the right one, down 3 into the right one, down 3 into the right one. We could also go back or up 3 into the left one, up 3 into the left, up 3 into the left. The more points, the more precise my line is. Hopefully. Yeah, pretend that's a little straight. OK, the other one, let's go ahead and do the X intercept and Y intercept in here. So if X = 0, then Y equal. Well, if X is 0, we'd get 0 + 2, Y equal 4, SO2Y equal 4, Y equal 2. So one point on this graph would be the .02 when X IS0Y is 2. Another point, if y = 0, then X is X + 2 * 0 equal 4, so X equal 4. Another point on the graph would be when X is 4, Y is 01234. When we connect those two points, you can see the two lines are going to intersect. Hence these two lines are consistent and independent because they intersect. There's a single solution, so they're consistent and independent. Our next example, 2X plus three y = 6 and Y equal -2 thirds X + 2. Here, if we do our X intercept and Y intercept, if I put zero in for X, I'm going to get Y equaling 2. So this is a point on my graph. The point X IS0Y is 2X is 0, Y is 2. If I do the other intercept, we're going to put zero in for the Y. So two X = 6 or X equal 3. So when X IS3Y is 0 so that's my other .30. These two points we can connect to get a line. If we look at this next graph Y equal or this next equation Y equal -2 thirds X + 2, this is already in slope intercept form. So our Y intercept here is 02 and our slope is -2 thirds. If we graph this one, we go to 02 and then the slope tells me down to and to the right 3. What do you notice? These two lines are the same line. If they're the same line, they're called consistent and dependent. Now we have special notation to show the solution. The special notation is the set of all X&Y values such that they're on the line 2X plus three y = 6. Or you could have actually used the other line also, so the set of all X&Y such that Y equal -2 thirds X + 2. Sometimes people get confused and they say it's all real numbers. Well it can't be all real numbers because if it was then this point out here would work and that point out there clearly isn't on the line. Our last example is going to be X + y equal 5 and Y equal negative negative X + 2. If I solve both of these for Y equal and graph them, we get AY intercept of 5012345 and a slope of -1, so Y intercept 05 slope -1. If we do the second one, we get AY intercept OF02A slope of -1. And if we look at these, these are parallel, so they're inconsistent and independent. Sometimes this is referred to as no solution. Oops, Dent. Sometimes we think of this as no solution because those two lines will never intersect. You've been great. Have a wonderful day.