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8th grade (Eureka Math/EngageNY)

Unit 4: Lesson 4

Topic D: Systems of linear equations and their solutions

Systems of equations with substitution: potato chips

To solve a system of equations using substitution...

1. Isolate one of the variables in one of the equations, e.g. rewrite 2x+y=3 as y=3-2x.
2. You can now express the isolated variable using the other one. *Substitute* that expression into the second equation, e.g. rewrite x+2y=5 as x+2(3-2x)=5.
3. Now you have an equation with one variable! Solve it, and use what you got to find the other variable.

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Created by Sal Khan.

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  • duskpin tree style avatar for user Seneth
    What does he mean by the last video?
    (36 votes)
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  • duskpin tree style avatar for user Vladimir
    Does Sal draw these pictures? if he does, I'm very impressed.
    (25 votes)
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  • leaf red style avatar for user Jack McClelland
    So, which method is faster/easier/recommended: elimination or substitution?
    (13 votes)
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  • sneak peak green style avatar for user BUGATTI VEYRON
    I don't know why my teacher put me into this class because i really don't know any of this. Every time i look at this question i get over whelmed because i dont get the problems he is doing.
    😵😠🤯
    (7 votes)
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    • male robot hal style avatar for user johnnyman242
      Let's use this example:
      {7x+10y=36
      {2x-y=-9
      First, label the equations. 7x+10y=36 would be Equation 1, and 2x-y=-9 would be Equation 2. Next, multiply Equation 2 by an integer so that either the x values (2x and 7x) or the y values (-y and 10y) are equal. You could multiply Equation 2 by -10, for example, so the y values would be equal, since -y • -10 = 10y. Label the new equation, -20x+10y=90, Equation 3. Next, subtract Equation 3 from Equation 1. You should get 27x = -54. Divide both sides of the equation by 27, and you should get x=-2, so you've solved for x. Now, substitute x=-2 into Equation 2. You should get (-4)-y=(-9) (it is okay if you don't have the parentheses, I just put them there to emphasize the fact that -4 and -9 are negative). Add 4 to both sides of the equation to get - y=-5. Divide both sides of the equation by -1 to get y=5. You are done! (x,y)=(-2,5)!

      My explanation is a little complicated, you might have to reread this. Hope it helps!
      (6 votes)
  • old spice man green style avatar for user Benaiah Stone
    the value for m = -4w+ 11 @ essentially the equation for the line which you could then use to find the answer to any combination/ variable to the problem
    (10 votes)
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  • marcimus pink style avatar for user arho626
    Why does and how does this substitution work?
    (4 votes)
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  • blobby green style avatar for user 150738
    This does not make any sense to me at all, could you explain this in easier terms?
    (5 votes)
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    • leaf green style avatar for user Katriana
      I can try, though I'm not sure if this is easier or not. A system of equations is where you have more than one equation with the same variables and you need to find out what values of the variables will work for all the equations.
      Here is an example: 2x+3y=12; 5x+y=17
      Substitution is one way to solve it.
      First, we can rearrange one of the equations in order to isolate one of the variables:
      5x+y=17
      y=17-5x
      We now have a way to express y in terms of x, so we can put it into the other equation instead of y in order to solve for x:
      2x+3y=12
      2x+3(17-5x)=12
      2x+51-15x=12
      51-13x=12
      -13x=-39
      x=3
      Now we know what value x needs to be to satisfy both equations, so we can use that in place of x to solve for y.
      5x+y=17
      5(3)+y=17
      15+y=17
      y=2
      We now know the values for both x and y, and in order to be quite sure, we can check them by putting them both into the first equation again:
      2(3)+3(2)?=12
      6+6?=12
      12=12
      This system of equations could also be solved in several other ways, such as elimination or graphing. In graphing, the point at which all equations in the system intersect is the solution.
      (2 votes)
  • starky sapling style avatar for user Rhylie Templeman
    Im still confused on everything..
    (4 votes)
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  • leaf green style avatar for user Merouane Merad
    How do we solve y intercept?
    (4 votes)
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  • blobby green style avatar for user Kevin  Schoonover
    He starts off by referring to "the last video" that features a problem with chips. That video is on elimination, but substitution comes before it in the current playlist. Did that originally come first? Which should I learn first?
    (3 votes)
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    • mr pink green style avatar for user David Severin
      It does not matter which is learned first, so learning them in any order or even in parallel will work. The key is that when you have learned different methods, you hopefully can start choosing the best method based on what you see. You are correct that Sal moves things around quite often.
      (2 votes)

Video transcript

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.