1. **Problem 1: Find the sum of all possible y-intercepts for a line passing through (9,0) with negative slope and single-digit prime y-intercept.** 2. The line equation is $y = mx + b$ where $m$ is slope and $b$ is y-intercept. 3. Since the line passes through $(9,0)$, substitute to get $0 = m \cdot 9 + b$ which implies $$m = \frac{-b}{9}$$ 4. The slope $m$ must be negative, so $$\frac{-b}{9} < 0 \implies -b < 0 \implies b > 0$$ 5. The y-intercept $b$ is a single-digit prime number. The single-digit primes are 2, 3, 5, 7. 6. Since $b > 0$ and $b$ is prime, possible $b$ values are 2, 3, 5, 7. 7. Sum all possible y-intercepts: $$2 + 3 + 5 + 7 = 17$$ --- **Final answer:** The sum of all possible y-intercepts is **17**. 2. **Problem 2: Find the number of dimes Brynn has given $5 saved in dimes, quarters, and half-dollars, 30 coins total, and quarters are four times half-dollars.** 3. Let $d$ = number of dimes, $q$ = number of quarters, $h$ = number of half-dollars. 4. Given: $$q = 4h$$ 5. Total coins: $$d + q + h = 30$$ Substitute $q$: $$d + 4h + h = 30 \implies d + 5h = 30$$ 6. Total money: dimes 10 cents, quarters 25 cents, half-dollars 50 cents. 7. Total value in cents: $$10d + 25q + 50h = 500$$ Substitute $q$: $$10d + 25(4h) + 50h = 500$$ 8. Simplify: $$10d + 100h + 50h = 500 \implies 10d + 150h = 500$$ 9. From step 5, express $d$: $$d = 30 - 5h$$ 10. Substitute $d$ into money equation: $$10(30 - 5h) + 150h = 500$$ 11. Simplify: $$300 - 50h + 150h = 500$$ $$300 + 100h = 500$$ 12. Solve for $h$: $$100h = 200 \implies h = 2$$ 13. Find $d$: $$d = 30 - 5(2) = 30 - 10 = 20$$ 14. Find $q$: $$q = 4h = 4 \times 2 = 8$$ --- **Final answer:** Brynn has **20 dimes**.