1. **State the problem:** Bill's remaining distance to Philadelphia is a linear function of his driving time. Given two points: after 35 minutes, distance is 45 miles; after 53 minutes, distance is 32.4 miles. We need to find the remaining distance after 57 minutes. 2. **Formula and rules:** The linear function can be written as $$d = mt + b$$ where $d$ is distance, $t$ is time, $m$ is slope, and $b$ is y-intercept. 3. **Find the slope $m$:** $$m = \frac{d_2 - d_1}{t_2 - t_1} = \frac{32.4 - 45}{53 - 35} = \frac{-12.6}{18} = -0.7$$ 4. **Find the y-intercept $b$ using one point, say $(35, 45)$:** $$45 = -0.7 \times 35 + b$$ $$45 = -24.5 + b$$ $$b = 45 + 24.5 = 69.5$$ 5. **Write the linear equation:** $$d = -0.7t + 69.5$$ 6. **Calculate distance after 57 minutes:** $$d = -0.7 \times 57 + 69.5 = -39.9 + 69.5 = 29.6$$ **Final answer:** After 57 minutes, Bill will have **29.6 miles** remaining to his destination.