1. **State the problem:** Find the slope of the line passing through the points $(3, 3)$ and $(4, 6)$. 2. **Formula for slope:** The slope $m$ of a line through points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ 3. **Substitute the points:** Here, $x_1 = 3$, $y_1 = 3$, $x_2 = 4$, and $y_2 = 6$. So, $$m = \frac{6 - 3}{4 - 3}$$ 4. **Calculate the differences:** $$m = \frac{3}{1}$$ 5. **Simplify the fraction:** $$m = \cancel{\frac{3}{1}} = 3$$ 6. **Interpretation:** The slope of the line is $3$, which means for every 1 unit increase in $x$, $y$ increases by 3 units. This is why the line rises steeply upwards. **Final answer:** The slope of the line is $3$.