1. **State the problem:** Find the $x$-value where the two lines $y = 5x + 9$ and $y = -x + 3$ intersect. 2. **Set the equations equal:** Since both expressions equal $y$, set them equal to each other to find $x$: $$5x + 9 = -x + 3$$ 3. **Solve for $x$:** Add $x$ to both sides: $$5x + x + 9 = 3$$ which simplifies to $$6x + 9 = 3$$ 4. Subtract 9 from both sides: $$6x + \cancel{9} - 9 = 3 - 9$$ $$6x = -6$$ 5. Divide both sides by 6: $$\frac{6x}{\cancel{6}} = \frac{-6}{\cancel{6}}$$ $$x = -1$$ **Final answer:** The $x$-value of the solution is $\boxed{-1}$.