1. **State the problem:** Solve the linear equation $5x + 9\left(-\frac{5}{9}x + 5\right) = 45$ for $x$. 2. **Apply the distributive property:** Multiply $9$ by each term inside the parentheses: $$5x + 9 \times \left(-\frac{5}{9}x\right) + 9 \times 5 = 45$$ 3. **Simplify the multiplication:** $$5x - 9 \times \frac{5}{9}x + 45 = 45$$ 4. **Simplify the fraction multiplication:** $$5x - \cancel{9} \times \frac{5}{\cancel{9}}x + 45 = 45$$ which simplifies to $$5x - 5x + 45 = 45$$ 5. **Combine like terms:** $$ (5x - 5x) + 45 = 45$$ $$0 + 45 = 45$$ 6. **Simplify:** $$45 = 45$$ 7. **Interpretation:** The equation is true for all values of $x$, so the solution is all real numbers. **Final answer:** $$\boxed{\text{All real numbers}}$$