System of absolute value equations

Consider cases for signs of $x$ and $y$:

Case 1: $x \ge 0, y < 0$ $|x| = x$ and $|y| = -y$. Equation 1: $3x - 2y + y = 7 \implies 3x - y = 7 \implies y = 3x - 7$ Equation 2: $x + x + 3y = 1 \implies 2x + 3y = 1$

Substitute $y = 3x - 7$ into Equation 2: $2x + 3(3x - 7) = 1 \implies 11x - 21 = 1 \implies 11x = 22 \implies x = 2$ $y = 3(2) - 7 = -1$

Since $x = 2 \ge 0$ and $y = -1 < 0$, this solution is valid.

Now, calculate $x + 2y$: $x + 2y = 2 + 2(-1) = 0$.