A system depending on a parameter

Determinant. $\det=\begin{vmatrix}1&m\\m&1\end{vmatrix}=1-m^2$.

Case $m\neq\pm1$. Unique solution. By Cramer:
$$x=\frac{\begin{vmatrix}1&m\\1&1\end{vmatrix}}{1-m^2}=\frac{1-m}{1-m^2}=\frac{1}{1+m},$$
$$y=\frac{\begin{vmatrix}1&1\\m&1\end{vmatrix}}{1-m^2}=\frac{1-m}{1-m^2}=\frac{1}{1+m}.$$

Case $m=1$. System becomes $x+y=1$ twice, so the solutions are $\{(t,1-t):t\in\mathbb{R}\}$.

Case $m=-1$. System becomes $x-y=1$ and $-x+y=1$, i.e. $x-y=1$ and $x-y=-1$, incompatible. No solution.

Conclusion.
- $m\neq\pm1$: unique solution $(1/(1+m),\,1/(1+m))$.
- $m=1$: infinitely many solutions $(t,1-t)$.
- $m=-1$: no solution.