1. **State the problem:** Solve the linear equation $$3x + 9\left(-\frac{5}{9}x + 5\right) = 45$$. 2. **Apply the distributive property:** Multiply 9 by each term inside the parentheses: $$3x + 9 \times \left(-\frac{5}{9}x\right) + 9 \times 5 = 45$$ 3. **Simplify each term:** $$3x - 9 \times \frac{5}{9}x + 45 = 45$$ 4. **Cancel common factors:** $$3x - \cancel{9} \times \frac{5}{\cancel{9}}x + 45 = 45$$ which simplifies to $$3x - 5x + 45 = 45$$ 5. **Combine like terms:** $$3x - 5x = -2x$$ So the equation becomes $$-2x + 45 = 45$$ 6. **Isolate the variable term:** Subtract 45 from both sides: $$-2x + 45 - 45 = 45 - 45$$ $$-2x = 0$$ 7. **Solve for $x$:** Divide both sides by $-2$: $$\frac{-2x}{-2} = \frac{0}{-2}$$ $$x = 0$$ **Final answer:** $$x = 0$$