1. **State the problem:** Solve the system of linear equations: $$\begin{cases} x + 2y = -3 \\ 5x + y = 12 \end{cases}$$ 2. **Formula and rules:** To solve a system of linear equations, we can use substitution or elimination. Here, we'll use elimination. 3. **Multiply the first equation by 5 to align coefficients of $x$:** $$5(x + 2y) = 5(-3) \Rightarrow 5x + 10y = -15$$ 4. **Write the new system:** $$\begin{cases} 5x + 10y = -15 \\ 5x + y = 12 \end{cases}$$ 5. **Subtract the second equation from the first to eliminate $x$:** $$ (5x + 10y) - (5x + y) = -15 - 12 $$ $$ 5x + 10y - 5x - y = -27 $$ $$ 9y = -27 $$ 6. **Solve for $y$:** $$ y = \frac{-27}{9} = -3 $$ 7. **Substitute $y = -3$ into the first original equation:** $$ x + 2(-3) = -3 $$ $$ x - 6 = -3 $$ $$ x = -3 + 6 = 3 $$ 8. **Final solution:** $$ x = 3, \quad y = -3 $$ 9. **Check the solution in the second equation:** $$ 5(3) + (-3) = 15 - 3 = 12 $$ which is correct. **Answer:** $x=3$, $y=-3$.