Question

A student is trying to solve the system of two equations given below:

Equation P: y + z = 6
Equation Q: 5y + 9z = 1

Which of the following is a possible step used in eliminating the y-term?

(y + z = 6) ⋅ 9
(y + z = 6) ⋅ −5
(5y + 9z = 1) ⋅ 9
(5y + 9z = 1) ⋅ 5

Answer 1

Well, this has to do with systems of linear equations and solving them with the substitution and addition methods. Do you know what those are?

Answer 2

A little but I know what your talking about

Answer 3

Well if we want to eliminate the y-term, we need to either make the y term above it equal to it and then subtract (substitution method) or make the number above it equal to it's negative and then add (addition method).

Answer 4

So how would I do that

Answer 5

Let's say we have the equations\[2x+y=4\] \[4x+3y=7\] If we want to get the y alone, we would mutiply the entire top by 2 and then subtract. \[2(2x+y=4)\to4x+2y=8\] With this we would subtract the bottom equation by the top. So\[4x+2y=8\] \[-(4x+3y=7)\] to get\[-y=1\] In this example I was eliminating the x variable with the substitution method. Understand how the process works?

Answer 6

I think so

Answer 7

So if I wanted to get rid of the 4x by the addition method, I would have multiplied by -2 instead of 2 and then added the equations instead of subtracting.

Answer 8

So now that we know how the methods work, let's look at the options for the problem. First thing we should notice is that both 3 and 4 are multiplying the side with the bigger variables. After looking at my example above, do we want to be multiplying the smaller variables or the bigger variables?

Answer 9

smaller ?

Answer 10

Exactly. So 3 and 4 are out, since they are multiplying the bigger variables instead of the smaller ones. Now we have 1 and 2 left.

Answer 11

Would it be 1 since 2 is bigger ?

Answer 12

No, although I can see why you might think that. 1 would be eliminating the z variable. if you multiply z by 9, then you would be able to subtract 9z-9z via the substitution method. 1, on the other hand, will give you -5y, which is the negative of 5y. If you add these two together, you'll get rid of the y variable, making 2 the right answer. Do you understand how I got that. or would you like me to explain it more?

Answer 13

I understand

Answer 14

Ok, then that's good.

Answer 15

so to eliminate y, you have to multiply (mostly) by a non zero number, such that, it will have the effect that your new y if added to the y in the othe equation, the result will be zero. so in this case, only -5y + 5y will eliminate y. so equation p times -5
B