1. **State the problem:** Solve the system of equations using the elimination method: $$\begin{cases} x + y = 7 \\ x - y = 1 \end{cases}$$ 2. **Explain the elimination method:** The goal is to eliminate one variable by adding or subtracting the equations. 3. **Add the two equations:** $$ (x + y) + (x - y) = 7 + 1 $$ Simplify: $$ x + y + x - y = 8 $$ $$ 2x = 8 $$ 4. **Solve for $x$:** $$ x = \frac{8}{2} = 4 $$ 5. **Substitute $x=4$ into one of the original equations, for example $x + y = 7$:** $$ 4 + y = 7 $$ 6. **Solve for $y$:** $$ y = 7 - 4 = 3 $$ 7. **Final answer:** $$ (x, y) = (4, 3) $$ This means the solution to the system is $x=4$ and $y=3$.