1. **State the problem:** We need to find how many adult tickets ($d$) and student tickets ($s$) were sold to fill 500 seats and raise exactly 2350. 2. **Set up equations:** - Total tickets sold: $$d + s = 500$$ - Total money raised: $$6.5d + 3.5s = 2350$$ 3. **Solve the system:** From the first equation, express $s$ as $$s = 500 - d$$ Substitute into the second equation: $$6.5d + 3.5(500 - d) = 2350$$ 4. **Simplify and solve for $d$:** $$6.5d + 1750 - 3.5d = 2350$$ $$\cancel{6.5d} + 1750 - \cancel{3.5d} = 2350$$ $$3d + 1750 = 2350$$ $$3d = 2350 - 1750$$ $$3d = 600$$ $$d = \frac{600}{3} = 200$$ 5. **Find $s$:** $$s = 500 - 200 = 300$$ 6. **Answer for first part:** The members sold 200 adult tickets and 300 student tickets. 7. **Second part problem:** At one performance, there were two times as many student tickets as adult tickets, and 300 tickets sold total. Let adult tickets be $a$, student tickets be $b$. 8. **Set up equations:** $$b = 2a$$ $$a + b = 300$$ 9. **Solve for $a$ and $b$:** Substitute $b$: $$a + 2a = 300$$ $$3a = 300$$ $$a = 100$$ $$b = 2 \times 100 = 200$$ 10. **Calculate money raised:** $$6.5 \times 100 + 3.5 \times 200 = 650 + 700 = 1350$$ 11. **Calculate how much below the goal:** $$2350 - 1350 = 1000$$ 12. **Final answers:** - Adult tickets sold: 200 - Student tickets sold: 300 - At the performance with 300 tickets, sales fell 1000 below the goal of 2350.