1. **Problem statement:** Given points $A(2, -3)$ and $B(-4, 1)$, (a) calculate the length $AB$, and (b) find the equation of the line passing through $A$ and $B$. 2. **Length of segment $AB$:** Use the distance formula between two points $A(x_1, y_1)$ and $B(x_2, y_2)$: $$AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 3. Substitute the coordinates: $$AB = \sqrt{(-4 - 2)^2 + (1 - (-3))^2} = \sqrt{(-6)^2 + (4)^2} = \sqrt{36 + 16} = \sqrt{52}$$ 4. Simplify $\sqrt{52}$: $$\sqrt{52} = \sqrt{4 \times 13} = 2\sqrt{13}$$ 5. **Equation of line $AB$:** The line equation is $y = mx + c$, where $m$ is the slope and $c$ the y-intercept. 6. Calculate slope $m$: $$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - (-3)}{-4 - 2} = \frac{4}{-6} = \frac{\cancel{4}}{\cancel{6}} \times \frac{1}{-1} = -\frac{2}{3}$$ 7. Use point-slope form with point $A(2, -3)$: $$y - y_1 = m(x - x_1)$$ $$y - (-3) = -\frac{2}{3}(x - 2)$$ $$y + 3 = -\frac{2}{3}x + \frac{4}{3}$$ 8. Solve for $y$: $$y = -\frac{2}{3}x + \frac{4}{3} - 3$$ $$y = -\frac{2}{3}x + \frac{4}{3} - \frac{9}{3}$$ $$y = -\frac{2}{3}x - \frac{5}{3}$$ **Final answers:** - Length $AB = 2\sqrt{13}$ units - Equation of line $AB$: $y = -\frac{2}{3}x - \frac{5}{3}$