1. **State the problem:** Solve the system of linear equations using the elimination method: $$3x + 2y = 8$$ $$4x - y = 2$$ 2. **Goal:** Eliminate one variable by making the coefficients of either $x$ or $y$ the same (or opposites) in both equations. 3. **Eliminate $y$:** Multiply the second equation by 2 to match the coefficient of $y$ in the first equation: $$2 \times (4x - y) = 2 \times 2$$ $$8x - 2y = 4$$ 4. **Add the two equations:** $$3x + 2y = 8$$ $$8x - 2y = 4$$ Adding gives: $$3x + 2y + 8x - 2y = 8 + 4$$ $$ (3x + 8x) + (2y - 2y) = 12$$ $$11x + \cancel{2y - 2y} = 12$$ $$11x = 12$$ 5. **Solve for $x$:** $$x = \frac{12}{11}$$ 6. **Substitute $x$ back into one original equation to find $y$:** Use the second equation: $$4x - y = 2$$ $$4 \times \frac{12}{11} - y = 2$$ $$\frac{48}{11} - y = 2$$ 7. **Isolate $y$:** $$-y = 2 - \frac{48}{11}$$ $$-y = \frac{22}{11} - \frac{48}{11}$$ $$-y = -\frac{26}{11}$$ 8. **Multiply both sides by $-1$ to solve for $y$:** $$y = \cancel{-1} \times \cancel{-1} \frac{26}{11}$$ $$y = \frac{26}{11}$$ **Final answer:** $$x = \frac{12}{11}, \quad y = \frac{26}{11}$$