1. **State the problem:** Solve the system of equations: $$2x + 6y = 17$$ $$2x - 10y = 9$$ 2. **Method:** We will use the elimination method to solve for $x$ and $y$. 3. **Step 1: Subtract the second equation from the first to eliminate $x$:** $$\begin{aligned} (2x + 6y) - (2x - 10y) &= 17 - 9 \\ 2x + 6y - 2x + 10y &= 8 \\ (2x - \cancel{2x}) + (6y + 10y) &= 8 \\ 16y &= 8 \end{aligned}$$ 4. **Step 2: Solve for $y$:** $$y = \frac{8}{16} = \frac{1}{2}$$ 5. **Step 3: Substitute $y = \frac{1}{2}$ into the first equation to solve for $x$:** $$2x + 6\left(\frac{1}{2}\right) = 17$$ $$2x + 3 = 17$$ 6. **Step 4: Solve for $x$:** $$2x = 17 - 3 = 14$$ $$x = \frac{14}{2} = 7$$ 7. **Final answer:** $$\boxed{x = 7, y = \frac{1}{2}}$$