1. **Problem:** Find the equation of the line joining the points (4,1) and (7,3). 2. **Formula:** The slope $m$ of the line through points $(x_1,y_1)$ and $(x_2,y_2)$ is given by: $$m=\frac{y_2 - y_1}{x_2 - x_1}$$ The equation of the line in point-slope form is: $$y - y_1 = m(x - x_1)$$ 3. **Calculate slope for (4,1) and (7,3):** $$m=\frac{3 - 1}{7 - 4} = \frac{2}{3}$$ 4. **Write equation using point (4,1):** $$y - 1 = \frac{2}{3}(x - 4)$$ 5. **Simplify:** $$y - 1 = \frac{2}{3}x - \frac{8}{3}$$ 6. **Add 1 to both sides:** $$y = \frac{2}{3}x - \frac{8}{3} + 1$$ 7. **Convert 1 to fraction:** $$1 = \frac{3}{3}$$ 8. **Combine constants:** $$y = \frac{2}{3}x - \frac{8}{3} + \frac{3}{3} = \frac{2}{3}x - \frac{5}{3}$$ --- 9. **Problem:** Find the equation of the line joining the points (-5,1) and (-3,-2). 10. **Calculate slope:** $$m=\frac{-2 - 1}{-3 - (-5)} = \frac{-3}{2} = -\frac{3}{2}$$ 11. **Write equation using point (-5,1):** $$y - 1 = -\frac{3}{2}(x + 5)$$ 12. **Simplify:** $$y - 1 = -\frac{3}{2}x - \frac{15}{2}$$ 13. **Add 1 to both sides:** $$y = -\frac{3}{2}x - \frac{15}{2} + 1$$ 14. **Convert 1 to fraction:** $$1 = \frac{2}{2}$$ 15. **Combine constants:** $$y = -\frac{3}{2}x - \frac{15}{2} + \frac{2}{2} = -\frac{3}{2}x - \frac{13}{2}$$ **Final answers:** - For points (4,1) and (7,3): $$y = \frac{2}{3}x - \frac{5}{3}$$ - For points (-5,1) and (-3,-2): $$y = -\frac{3}{2}x - \frac{13}{2}$$