Question

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

Answer 1

Answer:

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

Step-by-step explanation:

We are given system of equations as

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

$3y^2-7x=47$

Firstly, we will isolate x

$7x=3y^2-47$

$x=\frac{3y^2-47}{7}$

now, we can plug back in first equation

$7(\frac{3y^2-47}{7})^2+3y^2=187$

now, we can solve for y

$\frac{9y^4}{7}-\frac{261y^2}{7}+\frac{2209}{7}=187$

$\frac{9y^4}{7}-\frac{261y^2}{7}+\frac{2209}{7}-187=0$

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

$y=-5,y=5,y=-2,y=2$

now, we can find x-values

At y=-5:

$x=\frac{3(-5)^2-47}{7}$

$x=4$

At y=5:

$x=\frac{3(5)^2-47}{7}$

$x=4$

At y=-2:

$x=\frac{3(-2)^2-47}{7}$

$x=-5$

At y=2:

$x=\frac{3(2)^2-47}{7}$

$x=-5$

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