1. **Problem statement:** Solve the system of equations using the method of elimination: $$\begin{cases} 3x + y = 13 \\ x - y = 3 \end{cases}$$ 2. **Method of elimination:** Add or subtract equations to eliminate one variable. 3. Add the two equations: $$ (3x + y) + (x - y) = 13 + 3 $$ $$ 3x + y + x - y = 16 $$ $$ 4x = 16 $$ 4. Solve for $x$: $$ x = \frac{16}{4} $$ $$ x = 4 $$ 5. Substitute $x=4$ into the second original equation: $$ 4 - y = 3 $$ 6. Solve for $y$: $$ -y = 3 - 4 $$ $$ -y = -1 $$ $$ y = 1 $$ 7. **Final answer:** $$ (x, y) = (4, 1) $$ This completes the solution for the first system.