Let $a,b,c$  be positive real numbers. The following system of equations

$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$ , $\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$  and $-\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$  has

Let $\frac{x^2}{a^2}=X$ , $\frac{y^2}{b^2}=Y$ , $\frac{z^2}{c^2}=Z$  so that the equations become

$X+Y-Z=1$

$X-Y+Z=1$

$-X+Y+Z=1$

This system can be written as $AX=B$ , where

$A=\left[\begin{array}{ccc}1& 1& -1\\ 1& -1& 1\\ -1& 1& 1\end{array}\right]$

Now $\left|A\right|=\left|\begin{array}{ccc}1& 1& -1\\ 1& -1& 1\\ -1& 1& 1\end{array}\right|=\left|\begin{array}{ccc}2& 0& 0\\ 1& -1& 1\\ -1& 1& 1\end{array}\right|$

Operating $R_1\to R_1+R_2$

                $=2\left(-1-1\right)=-4\ne 0$ .

$\therefore$  Given system has a unique solution.