1. **Stating the problem:** Solve the system of equations: $$\begin{cases} x + y = 7 \\ x - y = 1 \end{cases}$$ 2. **Formula and rules:** To solve a system of linear equations, we can use the addition (elimination) method or substitution method. Here, elimination is straightforward because adding the two equations will eliminate $y$. 3. **Step 1: Add the two equations to eliminate $y$:** $$ (x + y) + (x - y) = 7 + 1 $$ Simplify: $$ x + y + x - y = 8 $$ $$ 2x = 8 $$ 4. **Step 2: Solve for $x$:** $$ x = \frac{8}{2} $$ Intermediate step showing cancellation: $$ x = \frac{\cancel{8}}{\cancel{2}} = 4 $$ 5. **Step 3: Substitute $x=4$ into one of the original equations to find $y$:** Using $x + y = 7$: $$ 4 + y = 7 $$ 6. **Step 4: Solve for $y$:** $$ y = 7 - 4 $$ $$ y = 3 $$ **Final answer:** $$ \boxed{x=4, y=3} $$