1. **State the problem:** Solve the system of equations using the method of elimination: $$\begin{cases} 3x + y = 13 \\ x - y = 3 \end{cases}$$ 2. **Write down the equations:** Equation 1: $3x + y = 13$ Equation 2: $x - y = 3$ 3. **Add the two equations to eliminate $y$:** $$ (3x + y) + (x - y) = 13 + 3 $$ Simplify: $$ 3x + y + x - y = 16 $$ $$ (3x + x) + (y - y) = 16 $$ $$ 4x + \cancel{y - y} = 16 $$ $$ 4x = 16 $$ 4. **Solve for $x$:** $$ x = \frac{16}{4} $$ $$ x = 4 $$ 5. **Substitute $x=4$ into one of the original equations to find $y$:** Using Equation 2: $$ 4 - y = 3 $$ $$ -y = 3 - 4 $$ $$ -y = -1 $$ $$ y = 1 $$ 6. **Final solution:** $$ x = 4, \quad y = 1 $$ This means the solution to the system is the point $(4,1)$ where both equations intersect.