1. **State the problem:** We are given the system of equations: $$x - 2y = 4$$ $$-3x + 2y = 8$$ We need to find the value of $x$. 2. **Add the two equations to eliminate $y$:** $$ (x - 2y) + (-3x + 2y) = 4 + 8 $$ This simplifies to: $$ x - 2y - 3x + 2y = 12 $$ 3. **Simplify the left side:** $$ (x - 3x) + (-2y + 2y) = 12 $$ $$ -2x + 0 = 12 $$ $$ -2x = 12 $$ 4. **Solve for $x$ by dividing both sides by $-2$:** $$ \cancel{-2}x = \frac{12}{\cancel{-2}} $$ $$ x = -6 $$ 5. **Final answer:** $$ x = -6 $$