1. **State the problem:** We need to find the equation that models the distance $y$ (in miles) an athlete runs during week $x$ based on the given data: Week: 1, 2, 3, 4, 5, 6 Distance: 2, 3.5, 5, 6.5, 8, 9.5 2. **Identify the type of relationship:** The distances increase by a constant amount each week, suggesting a linear relationship of the form $$y = mx + b$$ where $m$ is the slope (rate of change) and $b$ is the y-intercept (distance at week 0). 3. **Calculate the slope $m$:** The slope is the change in distance divided by the change in week number. $$m = \frac{3.5 - 2}{2 - 1} = \frac{1.5}{1} = 1.5$$ 4. **Find the y-intercept $b$:** Use the point $(1, 2)$ and the slope $1.5$: $$2 = 1.5 \times 1 + b$$ $$b = 2 - 1.5 = 0.5$$ 5. **Write the equation:** $$y = 1.5x + 0.5$$ 6. **Check the equation with another point:** For $x=3$: $$y = 1.5 \times 3 + 0.5 = 4.5 + 0.5 = 5$$ This matches the given data. **Final answer:** The equation that models the distance is: $$\boxed{y = 1.5x + 0.5}$$