1. **State the problem:** Find the slope of the line passing through the points $(-2, 9)$ and $(8, 34)$. 2. **Formula for slope:** The slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ This formula calculates the "rise" over the "run," or the change in $y$ divided by the change in $x$. 3. **Substitute the points:** Using $(-2, 9)$ as $(x_1, y_1)$ and $(8, 34)$ as $(x_2, y_2)$, we get $$m = \frac{34 - 9}{8 - (-2)}$$ 4. **Simplify the numerator and denominator:** $$m = \frac{25}{8 + 2} = \frac{25}{10}$$ 5. **Simplify the fraction:** $$m = \frac{25}{10} = \frac{5}{2}$$ 6. **Interpretation:** The slope of the line is $\frac{5}{2}$, which means for every 2 units moved horizontally to the right, the line rises 5 units vertically. **Final answer:** The slope of the line passing through the points $(-2, 9)$ and $(8, 34)$ is $\boxed{\frac{5}{2}}$.