1. **State the problem:** Solve the system of equations using the addition-subtraction method: $$3x - y = 1$$ $$x + y = 3$$ 2. **Explain the method:** The addition-subtraction method involves adding or subtracting the equations to eliminate one variable. 3. **Add the two equations:** $$(3x - y) + (x + y) = 1 + 3$$ Simplify: $$3x - y + x + y = 4$$ $$4x = 4$$ 4. **Solve for $x$:** $$x = \frac{4}{4} = 1$$ 5. **Substitute $x=1$ into one of the original equations:** Using $$x + y = 3$$: $$1 + y = 3$$ 6. **Solve for $y$:** $$y = 3 - 1 = 2$$ 7. **Final answer:** $$x = 1, \quad y = 2$$