1. **Problem Statement:** Find the equation of the straight line joining the points $(-5, 2)$ and $(3, -4)$. 2. **Formula Used:** The slope $m$ of a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ The equation of the line in point-slope form is $$y - y_1 = m(x - x_1)$$ 3. **Calculate the slope:** $$m = \frac{-4 - 2}{3 - (-5)} = \frac{-6}{3 + 5} = \frac{-6}{8} = -\frac{3}{4}$$ 4. **Write the equation using point-slope form with point $(-5, 2)$:** $$y - 2 = -\frac{3}{4}(x - (-5))$$ $$y - 2 = -\frac{3}{4}(x + 5)$$ 5. **Expand and simplify:** $$y - 2 = -\frac{3}{4}x - \frac{3}{4} \times 5$$ $$y - 2 = -\frac{3}{4}x - \frac{15}{4}$$ 6. **Add 2 to both sides:** $$y = -\frac{3}{4}x - \frac{15}{4} + 2$$ Express 2 as $\frac{8}{4}$ to combine: $$y = -\frac{3}{4}x - \frac{15}{4} + \frac{8}{4}$$ $$y = -\frac{3}{4}x - \frac{7}{4}$$ **Final answer:** $$\boxed{y = -\frac{3}{4}x - \frac{7}{4}}$$