1. **State the problem:** Solve the system of equations using the elimination method: $$\begin{cases} 3x - 2y = 10 \\ 7x + 2y = 30 \end{cases}$$ 2. **Formula and rules:** The elimination method involves adding or subtracting equations to eliminate one variable, making it easier to solve for the other. 3. **Add the two equations to eliminate $y$:** $$ (3x - 2y) + (7x + 2y) = 10 + 30 $$ $$ 3x + 7x + \cancel{-2y} + \cancel{2y} = 40 $$ $$ 10x = 40 $$ 4. **Solve for $x$:** $$ x = \frac{40}{10} = 4 $$ 5. **Substitute $x=4$ into the first equation to find $y$:** $$ 3(4) - 2y = 10 $$ $$ 12 - 2y = 10 $$ $$ -2y = 10 - 12 $$ $$ -2y = -2 $$ $$ y = \frac{-2}{-2} = 1 $$ 6. **Solution:** The solution to the system is $\boxed{(4, 1)}$. 7. **Check the solution:** Substitute $x=4$ and $y=1$ into the second equation: $$ 7(4) + 2(1) = 28 + 2 = 30 $$ The solution satisfies both equations.