Video Transcript: Substitution Method

Systems of equations with substitution: potato chips 

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. 

Systems of equations with substitution: -3x-4y=-2 & y=2x-5 

So that it's less likely that we get shown up by talking birds in the future, we've set a little bit of  exercise for solving systems of equations with substitution. And so this is the first exercise or  the first problem that they give us. -3x-4y=-2 and y=2x-5 So let me get out my little scratch pad  and let me rewrite the problem. So this is -3x-4y=-2 and then they tell us y=2x-5. So, what's  neat about this is that they've already solved the second equation. They've already made it  explicitly solved for y which makes it very easy to substitute for. We can take this constraint, the constraint on y in terms of x and substitute it for y in this first blue equation and then solve for  x. So let's try it out. So this first blue equation would then become -3x-4 but instead of putting a y there the second constraint tells us that y needs to be equal to 2x-5. So it's 4(2x-5) and all of  that is going to be equal to -2. So now we get just one equation with one unknown. and now we just have to solve for x. So, let's see if we can do that. So, it's -3x and then this part right over  here we have a -4, be careful, we have a -4 we want to distribute. We are going to multiply  -4*2x which is -8x and -4*-5 is positive 20 and thats going to equal -2. And now we can  combine all the x terms so -3x-8x, that's going to be -11x and then we have -11x+20=-2. Now to  solve for x, we'll subtract 20 from both sides to get rid of the 20 on the left hand side. On the  left hand side, we're just left with the -11x and then on the right hand side we are left with -22.  Now we can divide both sides by -11. And we are left with x is equal to 22 divided by 11 is 2, and  the negatives cancel out. x = 2. So we are not quite done yet. We've done, I guess you can say  the hard part, we have solved for x but now we have to solve for y. We could take this x value to  either one of these equations and solve for y. But this second one has already explicitly solved  for y so let's use that one, so it says y = 2 times and instead of x, we now know that the x value  where these two intersect, you could view it that way is going to be equal to 2, so 2 * 2 - 5 let's  figure out the corresponding y value. So you get y=2(2)-5 and y = 4 - 5 so y = -1. And you can  verify that it'll work in this top equation If y = -1 and x=2, this top equation becomes -3(2) which  is -6-4(-1) which would be plus 4. And -6+4 is indeed -2. So it satisfies both of these equations  and now we can type it in to verify that we got it right, although, we know that we did, so x=2  and y=-1. So, let's type it in... x=2 and y=-1. Excellent, now we're much less likely to be  embarassed by talking birds.