How to solve this system of equations ??

Question

I am having trouble solving this system ... Always running in circles ...

$$x^2 + 2yz = 1$$

$$y^2 + 2xz = 2$$

$$z^2 + 2xy = 2$$

Answer 1

Hint:

If you add all the equations you obtain

$(x+y+z)^2=5$

hence

$x+y+z=\pm\sqrt{5}$

Let's go for $x+y+z=\sqrt{5}$

If you subtract the second equation from the first you obtain

$(x-z)(x+z-2y)=(x-z)(x+y+z-3y)=(x-z)(\sqrt{5}-3y)=-1$

Similarlu subtracting the third from the first:

$(x-y)(\sqrt{5}-3z)=-1$

And subtracting the last one from the second

$(y-z)(\sqrt{5}-3x)=0$

From this last equation you find two solutions:

1) $z=y$ and then...

2) $x=\frac{\sqrt{5}}{3}$ and then...

I think you can keep on from here...