Solving systems with substitution
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We're given a system of equations here, and we're told to solve for x and y. Now, the easiest thing to do here, since in both equations they're explicitly solved for y, is say, well, if y is equal to that, and y also has to equal this second equation, then why don't we just set them equal to each other? Or another way to think about it is, if y is equal to this whole thing right over here-- that's what that first equation is telling us-- and if we have to find an x and a y that satisfy both of these equations, if y is equal to that, why can't I just substitute that right here for y? And if we do that, the left-hand side of this bottom equation becomes negative 1/4x plus 100. And then that is going to be equal to this right-hand side-- and I'll do it in the same color-- is equal to negative 1/4x plus 120. Now, the first thing we might want to do is maybe get all of our x terms onto the left- or the right-hand side of the equation. And if we wanted to get rid of these x terms from the right-hand side, get them on the left-hand side, the best thing to do is to add 1/4x to both sides of this equation. So let me do that. So we're going to add 1/4x here, add 1/4x here, and you might already be sensing that something shady is going on. So let's do it. So negative 1/4x plus 1/4x. They cancel out. You get 0x. So the left side of the equation is just 100. And then the right side of the equation, same thing. Negative 1/4x plus 1/4x. They cancel out. No x's. And you're just left with is equal to 120. Which we know is definitely not the case. 100 is not equal to 120. We got this nonsensical equation here, that 100 equals 120. So this type of system has no solution. You know it has no solution because in order for it to have any solution, these two numbers would have to be equal to each other, and they are not equal to each other. And if you look at the original equations, it might jump out at you why they have no solutions. Both of these lines, or both of these equations, if you view them as lines, have the exact same slope. But they have different y-intercepts. So if I just were to do a really quick graph here. That's my y-axis, that is my x-axis, so it's y and x. This first graph over here, its y-intercept is 100. Let me do it a little bit lower. Its y-intercept-- let's say that that is 100, so it intersects right there. And there's a slope of negative 1/4. So maybe it looks something like this. That's that first line. This second line-- I'll do it in pink right here-- y is equal to negative 1/4x plus 120, its y-intercept might be right here at 120. But it has the same slope, negative 1/4, so its slope, the line would look something like this. So you see that there are no x and y points that satisfy both of these equations. Another way to think about it. If y-- you take an x. This first equation says, OK, you take your x, multiply it by negative 1/4, and add 100, and that's going to give you y. Now, here we say, well, you take that same x, and you multiply it by negative 1/4 and add 120, and that has to be equal to y. Well, the only way that that would ever be true is if 100 and 120 were the same number, and they're not the same number. So you're never going to have a solution of this system. These two lines are never going to intersect, and that's because they have the exact same slope.