Strategies for eliminating variables in a system examples

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. Pause this video. Would adding the equations eliminate a variable in this system? All right, now let's do it together.

If we add these equations, we have on the left-hand side, we have 5x plus 5x, which is going to be 10x. Then you have negative 3y plus 4y, which is just a positive 1y or just plus y, is equal to negative 3 plus 6, which is just going to be equal to positive 3. We haven't eliminated any variables, so choice A I could rule out; that did not eliminate a variable. Let me cross it out.

Now check it. Subtract the bottom equation from the top. Well, when we subtract the bottom from the top, 5x minus 5x, that's going to be zero x's, so I won't even write it down. We've already seen we've eliminated an x, so I'm already feeling good about choice B.

But then we can see negative 3y minus 4y is negative 7y, negative 3 minus 6 is going to be negative 9. So choice B does successfully eliminate the x's, so I will select that choice.

Choice C: Multiply the top equation by 2, then add the equations. Pause the video. Does that eliminate a variable? Well, we're going to multiply the top equation by 2, so it's going to become 10x minus 6y is equal to negative 6. You could already see if you then add the equations 10x plus 5x, you're going to have 15x; that's not going to get eliminated. Negative 6y plus 4y is negative 2y; that's not going to be eliminated, so we can rule that out as well.

Let's do another example. 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 2, we are going to get—if we multiply it by 2, we're going to get 2x minus 4y is equal to 10. And then, if we were to add the equations, 4x plus 2x is 6x, so that doesn't get eliminated. Positive 4y plus negative 4y is equal to 0, so the y's actually do get eliminated when you add 4y to negative 4y. So I like choice A, and I'm going to delete this so I have space to work on the other choices.

So I like this one. What about choice B? Pause the video. Does that work? Multiply the bottom equation by 4, then subtract the bottom equation from the top equation.

All right, let's multiply the bottom equation by 4. What do we get? We're going to get 4x minus 8y is equal to 20. Yep, we multiplied it by 4. Then, subtract the bottom equation from the top. So we would subtract 4x from 4x; well, that's looking good; that would eliminate the x's. So I'm feeling good about choice B.

Then we could see if we subtract negative 8y from 4y. Well, subtracting a negative is the same thing as adding a positive, so that would actually get us to 12y. If we're subtracting negative 8y from 4y, and then if we subtract 20 from negative 2, we get to negative 22. But we see that 4x minus 4x 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 one-half, then add the equations. Let's try that out. Pause the video. All right, let's just multiply times one-half.

So the left-hand side times one-half, we distribute the one-half. Is one way to think about it: 4x times one-half is going to be 2x plus 4y times one-half is 2y, is equal to negative 2 times one-half is equal to negative 1. Now, and then they say add the equations. So 2x plus x is going to be 3x; so that's not going to eliminate the x's. 2y plus negative 2y—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 because 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 is subtract the bottom equation from the top equation. Pause the video. Does that work?

Well, if we subtract the bottom from the top, so if you subtract a negative 2x, that's the same thing as adding 2x. You're adding 2x to 3x; that's 5x. The x's don't get eliminated. Subtracting 4y from negative 3y is going to get us to negative 7y. The y's don't get eliminated, so I would rule this one out. Nothing's getting eliminated there.

Multiply the top equation by 3, multiply the bottom equation by 2, then add the equations. Pause the video. Does that work? All right, so if I multiply the top equation by 3, I'm going to get 9x minus 9y is equal to 21.

Then if I multiply the bottom by 2—so this is times 2—I'm going to get 2 times negative 2 is negative 4x plus 8y is equal to 14. And then they say add the equations. Well, if I add 9x to negative 4x, that doesn't eliminate the x's; that gets me to positive 5x. If I add negative 9y to positive 8y, 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 2, multiply the bottom equation by 3, then add the equations. 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 2, and we're going to multiply the bottom equation by 3. So the top equation times 2 is going to be 6x minus 6y is equal to 14. And then with this bottom equation, when you multiply it by 3, both sides—that's the only way to ensure that the equation is saying the same thing—if you do the same thing to both sides; that's really the heart of algebra.

So negative 2 times 3 is negative 6x, and I already like where this is going because when I add these two, they're going to get eliminated. Plus 4y times 3 is going to be plus 12y, 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.