Question

In the system below, what is the sum of the x-coordinates of all solutions?

7x^2+3y^2=187
3y^2-7x=47

Answer 1

The sum of the x-coordinates of all solutions is -2

Explanation:

We are given system of equations as

`7x^2+3y^2=187`

`3y^2-7x=47`

Firstly, we will isolate x

`7x=3y^2-47`

`x=(3y^2-47)/(7)`

now, we can plug back in first equation

`7((3y^2-47)/(7))^2+3y^2=187`

now, we can solve for y

`(9y^4)/(7)-(261y^2)/(7)+(2209)/(7)=187`

`(9y^4)/(7)-(261y^2)/(7)+(2209)/(7)-187=0`

`(y^2-25)(y^2-4)=0`

`y=-5,y=5,y=-2,y=2`

now, we can find x-values

At y=-5:

`x=(3(-5)^2-47)/(7)`

`x=4`

At y=5:

`x=(3(5)^2-47)/(7)`

`x=4`

At y=-2:

`x=(3(-2)^2-47)/(7)`

`x=-5`

At y=2:

`x=(3(2)^2-47)/(7)`

`x=-5`

now, we can add all x-coordinate solution values

`=4+4-5-5`

`=-2`

So,

the sum of the x-coordinates of all solutions is -2