1. **State the problem:** Solve the system of linear equations: $$x - y = 1$$ $$x + y = 3$$ 2. **Formula and rules:** To solve a system of two 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:** $$ (x - y) + (x + y) = 1 + 3 $$ $$ 2x = 4 $$ 4. **Solve for $x$:** $$ x = \frac{4}{2} = 2 $$ 5. **Substitute $x=2$ into one of the original equations:** Using $x + y = 3$: $$ 2 + y = 3 $$ 6. **Solve for $y$:** $$ y = 3 - 2 = 1 $$ 7. **Final answer:** $$ x = 2, \quad y = 1 $$ This means the solution to the system is the point $(2,1)$ where the two lines intersect.