1. **State the problem:** Solve the system of equations: $$\begin{cases} 4x + 9y = 12 \\ -2x + y = -6 \end{cases}$$ 2. **Use substitution or elimination method.** Here, we use substitution from the second equation: $$y = -6 + 2x$$ 3. **Substitute $y$ into the first equation:** $$4x + 9(-6 + 2x) = 12$$ 4. **Simplify:** $$4x - 54 + 18x = 12$$ $$22x - 54 = 12$$ 5. **Add 54 to both sides:** $$22x - \cancel{54} + \cancel{54} = 12 + 54$$ $$22x = 66$$ 6. **Divide both sides by 22:** $$\frac{22x}{\cancel{22}} = \frac{66}{\cancel{22}}$$ $$x = 3$$ 7. **Substitute $x=3$ back into $y = -6 + 2x$:** $$y = -6 + 2(3) = -6 + 6 = 0$$ 8. **Solution:** The system intersects at the point **$(3, 0)$**. This matches the graph's intersection point. **Final answer:** $(3, 0)$