Substitution method for systems of equations
Just as you were solving the potato chip conundrum in the last video, the king's favorite magical bird comes flying along and starts whispering into the king's ear. And this makes you a little bit self-conscious, a little bit insecure, so you tell the king, what is the bird talking about. And the king says, well, the bird says that he thinks that there's another way to do the problem. And you're not used to taking advice from birds. And so being a little bit defensive, you say, well, if the bird thinks he knows so much, let him do this problem. And so the bird whispers a little bit more in the king's ear and says, OK, well I'll have to do the writing because the bird does not have any hands, or at least can't manipulate chalk. And so the bird continues to whisper in the king's ear. And the king translates and says, well, the bird says, let's use one of these equations to solve for a variable. So let's say, let's us this blue equation right over here to solve for a variable. And that's essentially going to be a constraint of one variable in terms of another. So let's see if we can do that. So here, if we want to solve for m, we could subtract 400 w from both sides. And we would have 100 m. If we subtract 400w from the left, this 400w goes away. If we subtract 400w from the right, we have is equal to negative 400w plus 1,100. So what got us from here to here is just subtracting 400w from both sides. And then if we want to solve for m, we just divide both sides by 100. So we just divide all of the terms by 100. And then we get m is equal to negative 400 divided by 100, is negative 4w. 1,100 divided by 100 is 11. Plus 11. So now we've constrained m in terms of w. This is what the bird is saying, using the king as his translator. Why don't we take this constraint and substitute it back for m in the first equation? And then we will have one equation with one unknown. And so the king starts to write at the bird's direction. 200, so he's looking at that first equation now, he says 200. Instead of putting an m there, the bird says well, by the second constraint, m is equal to negative 4w plus 11. So instead of writing an m, we substitute for m the expression negative 4w plus 11. And then we have the rest of it, plus 300w, is equal to 1,200. So just to be clear, everywhere we saw an m, we replaced it with this right over here, in that first equation. So the first thing, you start to scratch your head. And you say, is this a legitimate thing to do. Will I get the same answer as I got when I solved the same problem with elimination? And I want you to sit and think about that for a second. But then the bird starts whispering in the king's ear. And the king just progresses to just work through the algebra. This is one equation with one unknown now. So the first step would be to distribute the 200. So 200 times negative 4w is negative 800w. 200 times 11 is 2,200. Plus 2,200. And then we have the plus 300w. Plus 300w is equal to positive 1,200. Now we just need to solve for w. We first might want to group this negative 800w with this 300w. Negative 800 of something plus 300 of something is going to be negative 500w. And then we still have this plus 2,200 is equal to 1,200. Now to solve for w, we'd want to subtract 2,200 from both sides. So subtract 2,200, subtract 2,200. On the left-hand side, you're left just with the negative 500w. And on the right-hand side, you are left with negative 1,000. This is starting to look interesting, because if we divide both sides by negative 500, we get w is equal to 2, which is the exact same answer that we got when we tried to figure out how many bags of chips each woman would eat on average. When we tried to solve it with elimination, we got the exact right answer. So at least for this example, it seems like the substitution method that this bird came up with worked just as well as the elimination method that you had originally done the first time you wanted to figure out the potato chip conundrum. And if now, you actually wanted to figure out how many chips the men would eat, well, you could do exactly the same thing you did the last time. You know one of the variables. You can substitute it back into one of the equations and then solve for m. And you could try that yourself to verify that you actually will get the same value for m as well. And in fact, this would probably be the easiest equation to substitute into, because it explicitly solves for m already.