1. **State the problem:** We are given the proportion $$\frac{45}{35} = \frac{9}{7a - 7}$$ and need to find the value of $a$. 2. **Recall the property of proportions:** If $$\frac{x}{y} = \frac{m}{n}$$, then $$x \cdot n = y \cdot m$$. 3. **Apply the property:** Multiply both sides crosswise: $$45 \times (7a - 7) = 35 \times 9$$ 4. **Calculate the right side:** $$35 \times 9 = 315$$ 5. **Write the equation:** $$45(7a - 7) = 315$$ 6. **Distribute 45:** $$45 \times 7a - 45 \times 7 = 315$$ $$315a - 315 = 315$$ 7. **Add 315 to both sides:** $$315a - 315 + 315 = 315 + 315$$ $$315a = 630$$ 8. **Divide both sides by 315:** $$\frac{315a}{\cancel{315}} = \frac{630}{\cancel{315}}$$ $$a = 2$$ **Final answer:** $$a = 2$$