1. **State the problem:** Solve the system of equations using the elimination method: $$6x + 3y + 4 = 0$$ $$5y = -9x - 6$$ 2. **Rewrite the equations in standard form:** From the first equation: $$6x + 3y = -4$$ From the second equation: $$5y = -9x - 6 \implies 9x + 5y = -6$$ 3. **Set up the system:** $$\begin{cases} 6x + 3y = -4 \\ 9x + 5y = -6 \end{cases}$$ 4. **Eliminate one variable:** Multiply the first equation by 3 and the second by 2 to align coefficients of $x$: $$3(6x + 3y) = 3(-4) \implies 18x + 9y = -12$$ $$2(9x + 5y) = 2(-6) \implies 18x + 10y = -12$$ 5. **Subtract the first new equation from the second:** $$(18x + 10y) - (18x + 9y) = -12 - (-12)$$ $$18x - 18x + 10y - 9y = 0$$ $$y = 0$$ 6. **Substitute $y=0$ into one original equation:** Using $6x + 3y = -4$: $$6x + 3(0) = -4 \implies 6x = -4 \implies x = -\frac{2}{3}$$ 7. **Final solution:** $$\boxed{\left(-\frac{2}{3}, 0\right)}$$ This is the solution to the system using the elimination method.