Solve the following system of equations in the set of real numbers:

$$
|y-x|-\frac{|x|}{x}+1=0 ;|2 x-y|+|x+y-1|+|x-y|+y-1=0
$$
The first equation only makes sense if $x \neq 0$.

1. Case: $x > 0$. In this case, from the first equation, $|y - x| = 0$, which means $x = y$. Substituting this into the second equation and using the equality $|x| = x$: $|2x - 1| = -(2x - 1)$. This is only true if $2x - 1 \leq 0$, so in this case, the solution is: $0 < x = y \leq 0.5$.
2. Case: $x < 0$. In this case, according to the first equation, $|y - x| = -2$, which is impossible. Therefore, the solution to the problem is given by the $(x, y)$ pairs obtained in the first case.

Based on the work of Zsuzsanna Vaik (Budapest, Berzsenyi D. Gymnasium, II. grade)