1. **State the problem:** Find the equation of the line passing through points $A(-5, 3)$ and $B(0, -7)$.\n\n2. **Formula used:** The slope $m$ of the line through points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $$m = \frac{y_2 - y_1}{x_2 - x_1}.$$\n\n3. **Calculate the slope:** Substitute $A(-5, 3)$ and $B(0, -7)$ into the slope formula: $$m = \frac{-7 - 3}{0 - (-5)} = \frac{-10}{5} = -2.$$\n\n4. **Use point-slope form:** The equation of the line is $$y - y_1 = m(x - x_1).$$ Using point $A(-5, 3)$: $$y - 3 = -2(x - (-5)) = -2(x + 5).$$\n\n5. **Simplify the equation:** $$y - 3 = -2x - 10.$$\n\n6. **Isolate $y$ to get slope-intercept form:** $$y = -2x - 10 + 3 = -2x - 7.$$\n\n**Final answer:** The equation of the line is $$y = -2x - 7.$$