Elimination strategies

- [Instructor] We're asked which of these strategies would eliminate a variable in the system of equations? Choose all answers that apply. So this first one says add the equations. So pause this video. Would adding the equations eliminate a variable in this system? All right, now let's do it together. So if we add these equations, we have, on the left hand side, we have five x plus five x, which is going to be 10 x, and then you have negative three y plus four y which is just a positive one y or just plus y is equal to negative three plus six, which is just going to be equal to positive three. We haven't eliminated any variables, so choice A, I could rule out. That did not eliminate a variable. Let me cross it out and not check it. Subtract the bottom equation from the top. Well when we subtract the bottom from the top, five x minus five x, that's going to be zero x's, so I won't even write it down, and we've already seen we've eliminated an x so I'm already feeling good about choice B, but then we can see negative three y minus four y is negative seven y. Negative three minus six is going to be negative nine and so choice B does successfully eliminate the x's. So I will select that. Choice C, multiply the top equation by two, then add the equations. Pause the video. Does that eliminate a variable? Well we're gonna multiply the top equation by two so it's going to become 10 x minus six y is equal to negative six, and you could already see if you then add the equations, 10 x plus five x, you're gonna have 15 x, that's not gonna get eliminated. Negative six y plus four y is negative two y. That's not going to be eliminated, so we can rule that out, as well. Let's do another example. One, they're asking us the same question. Which of these strategies would eliminate a variable in the system of equations? The first choice says multiply the bottom equation by two, then add the equations. Pause this video, does that work? All right, so if we multiply the bottom equation by two, we are going to get, if we multiply it by two, we're gonna get two x minus two minus four y, I should say. Two x, I'm just multiplying everything by two, minus four y is equal to 10. And then if we were to add the equations. Four x plus two x is six x, so that doesn't get eliminated. Positive four y plus negative four y is equal to zero y, so the y's actually do get eliminated when you add four y to negative four y. So I like choice A and I'm gonna delete this so I have space to work on the other choices, so I like one. What about choice B? Pause the video, does that work? Multiply the bottom equation by four, then subtract the bottom equation from the top equation. All right, let's multiply the bottom equation by four. What do we get? We're going to get four x minus eight y is equal to 20, yup. We multiplied it by four and then subtract the bottom equation from the top. So we would subtract four x from four x. Well that's looking good. That would eliminate the x's, so I'm feeling good about choice B. And then we could see if we subtract negative eight y from four y, well, subtracting a negative's the same thing as adding a positive, so that would actually get us to 12 y if we're subtracting negative eight y from four y. And then if we subtract 20 from negative two, we get to negative 22, but we see that four x minus four x is going to eliminate our x's, so that does definitely eliminate a variable, so I like choice B. Now what about choice C? Multiply the top equation by 1/2, then add the equations. Let's try that out, pause the video. All right, let's just multiply times 1/2, so the left hand side times 1/2, we distribute the one half is one way to think about it. Four x times 1/2 is going to be two x plus four y times 1/2 is two y is equal to negative two times 1/2 is equal to negative one. Now and then they say add the equations. So two x plus x is going to be three x, so that's not going to eliminate the x's. Two y plus negative two y, well that's going to be no y's. So that actually will eliminate the y's, so I like this choice, as well. So actually, all three of these strategies would eliminate a variable in the system of equations. This is useful to see 'cause you can see there's multiple ways to approach solving a system like this through elimination. Let's do another example. Which of these strategies would eliminate a variable in the system of equations? Same question again. So the first one, they suggest to subtract the bottom equation from the top equation. Pause this video, does that work? Well if we subtract the bottom from the top, so if you subtract a negative two x, that's the same thing as adding two x, 'cause you're adding two x to three x, that's five x. The x's don't get eliminated. Subtracting four y from negative three y's just gonna get us to negative seven y. The y's don't get eliminated, so I would rule this one out. Nothing's getting eliminated there. Multiply the top equation by three, multiply the bottom equation by two, then add the equations. Pause the video, does that work? All right, so if I multiply the top equation by three, I'm going to get nine x minus nine y is equal to 21, and then if I multiply the bottom by two, so this is times two, I'm going to get two times negative two is negative four x plus eight y is equal to 14, and then they say add the equations. Well if I add nine x to negative four x, that doesn't eliminate the x's. That gets me to positive five x, and if I add negative nine y to a positive eight y, that also doesn't eliminate the y's. That gets me to a negative y, so choice B, I can also rule out. Once again deleting all of this so I have space to try to figure out choice C. Multiply the top equation by two, multiply the bottom equation by three, then add the equation. So they're telling us to do it the other way around. Pause the video, does this work? All right, so we multiply the top equation by two and we're gonna multiply the bottom equation by three. So the top equation times two is going to be six x minus six y is equal to, is equal to 14. And then with this bottom equation, when you multiply it by three, both sides, that's the only way to ensure that the equation is saying the same thing is if you do the same thing to both sides. That's really the heart of algebra. So negative two times three is negative six x, and I already like where this is going, 'cause when I add these two, they're going to get eliminated, plus four y times three is gonna be plus 12 y is going to be equal to 21. And then they say add the equations, well, you immediately see when you add the x terms on the left hand side, they are going to cancel out. So I like choice C.