1. **State the problem:** Solve the system of equations: $$x + y = 5$$ $$x - y = 1$$ 2. **Formula and rules:** To solve a system of linear equations, we can use the addition (elimination) method or substitution method. Here, we use addition to eliminate one variable. 3. **Add the two equations:** $$\begin{aligned} (x + y) + (x - y) &= 5 + 1 \\ x + y + x - y &= 6 \\ 2x &= 6 \end{aligned}$$ 4. **Simplify and solve for $x$:** $$x = \frac{6}{2}$$ $$x = 3$$ 5. **Substitute $x=3$ into the first equation to find $y$:** $$3 + y = 5$$ $$y = 5 - 3$$ $$y = 2$$ 6. **Final answer:** $$x = 3, \quad y = 2$$ This means the solution to the system is the point $(3, 2)$ where both equations intersect.