1. **State the problem:** Find the equation of the line passing through the point $(5, -3)$ with slope $\frac{2}{5}$. 2. **Formula used:** The point-slope form of a line is given by: $$y - y_1 = m(x - x_1)$$ where $m$ is the slope and $(x_1, y_1)$ is a point on the line. 3. **Substitute the known values:** $$y - (-3) = \frac{2}{5}(x - 5)$$ which simplifies to $$y + 3 = \frac{2}{5}(x - 5)$$ 4. **Distribute the slope:** $$y + 3 = \frac{2}{5}x - 2$$ 5. **Isolate $y$ to get slope-intercept form:** $$y = \frac{2}{5}x - 2 - 3$$ $$y = \frac{2}{5}x - 5$$ **Final answer:** The equation of the line is $$y = \frac{2}{5}x - 5$$