System of Absolute Value Equations

Let's analyze the cases for $x$ and $y$:

1. If $x \ge 0$, then $|x| = x$, so $2x + y = 15 \implies y = 15 - 2x$.
2. For the second equation $x + |y| - y = 20$:
   • If $y \ge 0$, then $|y| = y$, giving $x = 20$. Then $y = 15 - 2(20) = -25$, which contradicts $y \ge 0$.
   • If $y < 0$, then $|y| = -y$, giving $x - 2y = 20$.

Substituting $y = 15 - 2x$ into $x - 2y = 20$: $x - 2(15 - 2x) = 20$ $5x - 30 = 20 \implies 5x = 50 \implies x = 10$ Then $y = 15 - 2(10) = -5$ (which satisfies $y < 0$).

Thus, $x - y = 10 - (-5) = 15$.