1. **Problem statement:** Use linear interpolation to find the value of $y$ when $x=26.5$ given points $(30, 51)$ and $(20, 36)$. 2. **Formula for linear interpolation:** $$y = y_1 + \frac{(x - x_1)(y_2 - y_1)}{x_2 - x_1}$$ where $(x_1, y_1)$ and $(x_2, y_2)$ are known points. 3. **Assign values:** $$x_1 = 20, y_1 = 36, x_2 = 30, y_2 = 51, x = 26.5$$ 4. **Calculate numerator and denominator:** $$x - x_1 = 26.5 - 20 = 6.5$$ $$y_2 - y_1 = 51 - 36 = 15$$ $$x_2 - x_1 = 30 - 20 = 10$$ 5. **Apply formula:** $$y = 36 + \frac{6.5 \times 15}{10}$$ 6. **Simplify fraction:** $$y = 36 + \frac{\cancel{6.5} \times 15}{\cancel{10}} = 36 + 9.75$$ 7. **Final answer:** $$y = 45.75$$ This means when $x=26.5$, $y$ is approximately $45.75$ by linear interpolation.