1. **State the problem:** Solve the system of equations: $$\begin{cases} x - y = 1 \\ x + y = 3 \end{cases}$$ 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:** $$ (x - y) + (x + y) = 1 + 3 $$ Simplify: $$ x - y + x + y = 4 $$ $$ 2x + \cancel{-y} + \cancel{y} = 4 $$ $$ 2x = 4 $$ 4. **Solve for $x$:** $$ x = \frac{4}{2} $$ $$ x = 2 $$ 5. **Substitute $x=2$ into one of the original equations, for example $x + y = 3$:** $$ 2 + y = 3 $$ 6. **Solve for $y$:** $$ y = 3 - 2 $$ $$ y = 1 $$ 7. **Final answer:** The solution to the system is $$ (x,y) = (2,1) $$ This matches the given solution.