1. **State the problem:** Solve the system of linear equations using the elimination method: $$3x + 2y = 8$$ $$4x - y = 22$$ 2. **Elimination method formula:** We aim to eliminate one variable by making the coefficients of that variable opposites in both equations, then adding or subtracting the equations. 3. **Step 1: Make coefficients of $y$ opposites.** Multiply the second equation by 2: $$2 \times (4x - y) = 2 \times 22$$ $$8x - 2y = 44$$ 4. **Step 2: Add the first equation and the new equation:** $$3x + 2y = 8$$ $$8x - 2y = 44$$ \hline $$(3x + 8x) + (2y - 2y) = 8 + 44$$ $$11x + \cancel{2y - 2y} = 52$$ $$11x = 52$$ 5. **Step 3: Solve for $x$:** $$x = \frac{52}{11}$$ 6. **Step 4: Substitute $x$ back into one of the original equations to find $y$.** Use the first equation: $$3x + 2y = 8$$ $$3 \times \frac{52}{11} + 2y = 8$$ $$\frac{156}{11} + 2y = 8$$ 7. **Step 5: Isolate $y$:** $$2y = 8 - \frac{156}{11}$$ $$2y = \frac{88}{11} - \frac{156}{11}$$ $$2y = -\frac{68}{11}$$ 8. **Step 6: Solve for $y$:** $$y = \frac{-\frac{68}{11}}{2} = -\frac{68}{11} \times \frac{1}{2} = -\frac{68}{22} = -\frac{34}{11}$$ **Final answer:** $$x = \frac{52}{11}, \quad y = -\frac{34}{11}$$