1. **State the problem:** Solve the system of linear equations: $$x - 2y = 2$$ $$2x - 3y = -1$$ 2. **Formula and method:** We can use the substitution or elimination method. Here, we'll use elimination. 3. **Elimination method:** Multiply the first equation by 2 to align coefficients of $x$: $$2(x - 2y) = 2 \times 2 \Rightarrow 2x - 4y = 4$$ 4. **Subtract the second equation from this new equation:** $$(2x - 4y) - (2x - 3y) = 4 - (-1)$$ $$2x - 4y - 2x + 3y = 4 + 1$$ $$-y = 5$$ 5. **Solve for $y$:** $$y = -5$$ 6. **Substitute $y = -5$ into the first original equation:** $$x - 2(-5) = 2$$ $$x + 10 = 2$$ $$x = 2 - 10 = -8$$ 7. **Final answer:** $$x = -8, \quad y = -5$$