solve non linear system with two variables

Question

I want to solve a non linear system of two equations $x$ and $z$. I have two equations : $$ x^2-z^2-2 x z +1 =0,$$ $$ z^2-x^2 -2 x z +1 =0.$$ I know that there are two solutions $(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$ and $(\frac{-\sqrt{2}}{2},\frac{-\sqrt{2}}{2})$, but I don't know how to find them.

I tried to substract the first equation with the second one and I get : $$ x^2-z^2+x^2-z^2=0$$ but I think that I do not have to do this operation on nonlinear system.

Presumably you are in the reals. Subtracting gives $x^2=z^2$. There are only two possibilities, Once can be easily eliminated. — copper.hat, Oct 21 '21 at 20:47
$x^2=z^2$ give me all solution on $R$ it's not right ? why $ \frac{\sqrt{2}{2}$ ? — ZchGarinch, Oct 21 '21 at 20:49
If you substitute $x=y$ (one of the possibilities you will get the equation $2x^2 = 1$. Solve it. — copper.hat, Oct 21 '21 at 20:51

score 2 · Accepted Answer · answered Oct 21 '21 at 20:53

$$ x^2-z^2-2 x z +1 =0$$ $$ z^2-x^2 -2 x z +1 =0$$

Subtracting equations

$$ x^2-z^2+x^2-z^2=0$$

$$x^2=z^2$$

$$x=\mp z$$

Let $x=z$

Substituting in the first equation $$2x^2=1$$

$$x=z=-\frac{1}{\sqrt2}$$ or $$x=z=\frac{1}{\sqrt2}$$

Let $x=-z$

Substituting in the first equation

$$2x^2=-1$$ no solution

solve non linear system with two variables

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