System of quadratic equations that is symmetric

Question

Solve for $z$: $z^2-3z+1=x, x^2-3x+1=z$

I see that it is symmetric, but not anything else. Hints would be great, but please do not spoil the answer. Thanks!

Answer 1

$$z^2-3z+1=x, x^2-3x+1=z \implies$$

$$(z-1)^2=x+z,(x-1)^2=z+x \implies$$

$$(z-1)^2=(x-1)^2$$