1. **State the problem:** Solve the system of linear equations: $$3x + 2y = 12$$ $$-x + 11y = 31$$ 2. **Choose a method:** We will use the substitution or elimination method. Here, elimination is convenient. 3. **Eliminate one variable:** Multiply the second equation by 3 to align coefficients of $x$: $$3(-x + 11y) = 3(31)$$ $$-3x + 33y = 93$$ 4. **Add the first and modified second equations:** $$3x + 2y = 12$$ $$-3x + 33y = 93$$ Adding gives: $$\cancel{3x} + 2y + \cancel{-3x} + 33y = 12 + 93$$ $$35y = 105$$ 5. **Solve for $y$:** $$y = \frac{105}{35} = 3$$ 6. **Substitute $y=3$ into the first equation:** $$3x + 2(3) = 12$$ $$3x + 6 = 12$$ 7. **Solve for $x$:** $$3x = 12 - 6 = 6$$ $$x = \frac{6}{3} = 2$$ **Final answer:** $$x = 2, \quad y = 3$$