1. **State the problem:** We need to find the equation of a line in slope-intercept form, which is $$y = mx + b$$ where $m$ is the slope and $b$ is the y-intercept. 2. **Identify given points:** The line crosses the y-axis at $(0, -2)$, so $b = -2$. 3. **Find the slope $m$:** Use two points on the line, for example $(0, -2)$ and $(-5, 0)$. 4. **Calculate slope formula:** $$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - (-2)}{-5 - 0} = \frac{2}{-5} = -\frac{2}{5}$$ 5. **Write the equation:** Substitute $m = -\frac{2}{5}$ and $b = -2$ into slope-intercept form: $$y = -\frac{2}{5}x - 2$$ 6. **Check with other points:** The problem states the line passes through $(-6, 0)$ and $(3, -3)$. Check $(-6, 0)$: $$y = -\frac{2}{5}(-6) - 2 = \frac{12}{5} - 2 = \frac{12}{5} - \frac{10}{5} = \frac{2}{5} \neq 0$$ This means the slope $-\frac{2}{5}$ does not fit the point $(-6, 0)$. 7. **Recalculate slope using points $(0, -2)$ and $(-6, 0)$:** $$m = \frac{0 - (-2)}{-6 - 0} = \frac{2}{-6} = -\frac{1}{3}$$ 8. **Write the corrected equation:** $$y = -\frac{1}{3}x - 2$$ 9. **Verify with point $(3, -3)$:** $$y = -\frac{1}{3}(3) - 2 = -1 - 2 = -3$$ which matches the point. **Final answer:** $$\boxed{y = -\frac{1}{3}x - 2}$$