1. **State the problem:** Find the slope of the line passing through the points $(0,-1)$ and $(4,2)$ on the coordinate plane. 2. **Formula for slope:** The slope $m$ of a line passing 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, $(x_1,y_1) = (0,-1)$ and $(x_2,y_2) = (4,2)$, so: $$m=\frac{2 - (-1)}{4 - 0}$$ 4. **Simplify the numerator and denominator:** $$m=\frac{2 + 1}{4} = \frac{3}{4}$$ 5. **Final answer:** The slope of the line is $\boxed{\frac{3}{4}}$. This means for every 4 units moved horizontally, the line rises 3 units vertically.