Respuesta :
Answer:
The sum of the x-coordinates of all solutions is -2
Step-by-step explanation:
We are given system of equations as
[tex]7x^2+3y^2=187[/tex]
[tex]3y^2-7x=47[/tex]
Firstly, we will isolate x
[tex]7x=3y^2-47[/tex]
[tex]x=\frac{3y^2-47}{7}[/tex]
now, we can plug back in first equation
[tex]7(\frac{3y^2-47}{7})^2+3y^2=187[/tex]
now, we can solve for y
[tex]\frac{9y^4}{7}-\frac{261y^2}{7}+\frac{2209}{7}=187[/tex]
[tex]\frac{9y^4}{7}-\frac{261y^2}{7}+\frac{2209}{7}-187=0[/tex]
[tex](y^2-25)(y^2-4)=0[/tex]
[tex]y=-5,y=5,y=-2,y=2[/tex]
now, we can find x-values
At y=-5:
[tex]x=\frac{3(-5)^2-47}{7}[/tex]
[tex]x=4[/tex]
At y=5:
[tex]x=\frac{3(5)^2-47}{7}[/tex]
[tex]x=4[/tex]
At y=-2:
[tex]x=\frac{3(-2)^2-47}{7}[/tex]
[tex]x=-5[/tex]
At y=2:
[tex]x=\frac{3(2)^2-47}{7}[/tex]
[tex]x=-5[/tex]
now, we can add all x-coordinate solution values
[tex]=4+4-5-5[/tex]
[tex]=-2[/tex]
So,
the sum of the x-coordinates of all solutions is -2
