Question

In the system shown below, what is the sum of all of the x-coordinates of all solutions?
x^2+4y^2=100
4y-x^2=-20
please please explain I am very stuck

Answer 1

Try to use elimination method:

\[ ~~~~x^2+4y^2=100 \\
\underline{+~-x^2+4y=-20}\\~~~~~~~~~~~~~~?\]

Answer 2

add equations together, you should eliminate \(x^2\), leaves you \(4y^2+4y=80\)

Answer 3

They want the sum of all the x-coordinates of the solution.
Eliminate y. Find y from the second equation, substitute it in y and you will have an equation in x.

Answer 4

Explain how to eliminate y please @aum

Answer 5

4y-x^2=-20
add x^2
4y = x^2-20
divide by 4
y = (x^2-20)/4

Put this in the first equation and simplify.

Answer 6

it's more of a substitution than elimination, but that's good move though

Answer 7

I put (x^2-20)/4 back into the equation and my answer was bizarre

Answer 8

I know, right lol...

So you have \(x^2+4\left(\dfrac{x^2-20}{4}\right)^2=100\)

You can simplify \(\left(\dfrac{x^2-20}{4}\right)^2\) to \(\dfrac{(x^2-20)^2}{4^2}\)

What is \((x^2-20)^2\) ?

Answer 9

x^4-40x^2-400

Answer 10

Almost. it's \(\Large x^4-40x^2+400\)

Answer 11

sorry typo haha

Answer 12

so I multiplied it out and got 4x^4-160x^2+1600

Answer 13

so it would be 4x^4-158x^2+1600=100?

Answer 14

lol okay, so you now have \(x^2+4\cdot\dfrac{x^4-40x^2+400}{4^2}=100 \\ x^2+\cancel{4}\cdot\dfrac{x^4-40x^2+400}{4^{\cancel2}}=100\\x^2+\dfrac{x^4-40x^2+400}{4}=100\)

Hey, you hate denominator, right? me too, so let multiply both sides by 4 to get \(4x^2+x^4-40x^2+400=400\)

Answer 15

wow im way off, so then we solve for x okay

Answer 16

do you understand what I just did, right?

Answer 17

Yes, I see it now

Answer 18

ok great, so we now have \(4x^2+x^4-40x^2+400=400 \\\rightarrow x^4-36x^2+400=400\\\rightarrow x^4-36x^2=0\)
Do you know what to do next?

Answer 19

take out a greatest common factor (x^2)

Answer 20

so the answers are x=6 x=-6?

Answer 21

and x=0 too.

Question wants sum of all of the x-coordinates of all solutions

So you have -6+0+6=?

Answer 22

So zero

Answer 23

right. your answer is 0

Answer 24

thank you so much!

Answer 25

love how lengthy and harsh problem ended up get us 0 haha

Answer 26

welcome!

Answer 27

I know right