Help me
Solve this system of equations to get the sum of all of the x coordinates from the solutions.
x^2+4y^2=100
4y-x^2=-20

Question

Help me
Solve this system of equations to get the sum of all of the x coordinates from the solutions. 
x^2+4y^2=100
4y-x^2=-20

anonymous · Answer

you posted this question earlier

cutiecomittee123 · Answer

Im pretty sure it was involving different equations

anonymous · Answer

The one from before were different equations. But the same idea holds. You want to make some sort of substitution that will let you reduce one of the equations to only x or only y. 

So you're looking for something in common in both equations that will make for an easy substitution. I see that both equations have an x^2 term. So if I were to solve the first equation for x^2, I would get
x^2 = 100 - 4y^2

With me so far?

cutiecomittee123 · Answer

Yes I am with ya

cutiecomittee123 · Answer

wait wouldnt you get 4y^2+4y=80
because you can cancel out the two x^2's since one is negative and if you add the equations it works right?

anonymous · Answer

Well, if we do the substitution for x^2 into the 2nd equation. we get
4y - (100-4y^2) = -20 
4y - 100 + 4y^2 = -20
4y^2 + 4y = 80
SO yes, you're right, good job :) 
Would you know what to do from here?

cutiecomittee123 · Answer

Now move 80 to the other side 
and you get 4y^2+4y-80=0
Then factor

cutiecomittee123 · Answer

thats where I get confused because what adds to get 4 but multiplies to get -80???

anonymous · Answer

Well, conveniently all of those terms are divisible by 4, so let's just divide out the 4 from all terms. Doing this would reduce the quadratic into
y^2 + y - 20 = 0

cutiecomittee123 · Answer

thats a lot cleaner and easier to deal with

cutiecomittee123 · Answer

now what

anonymous · Answer

Well, how does that factor?

cutiecomittee123 · Answer

IDK???

anonymous · Answer

Well, -20 would need to factor into 2 numbers that would also add up to a positive 1. This means:
One of the numbers has to be positive and the other has to be negative (otherwise they would never multiply to a negative 20)
The largest of those 2 numbers must be positive. 

So can you come up with the two numbers? Two factors that multiply to -20 yet add to positive 1

cutiecomittee123 · Answer

no, I bet its really easy but I just cant wrap my head around it.

anonymous · Answer

It is easy. And youll think its easy also once you get used to the process. Okay, so let's look at how 20 could possibly break apart:

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