1. **Problem:** Find the equation of the line passing through points $A=(1,3)$ and $B=(-2,-3)$. 2. **Formula:** The slope $m$ of a 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:** $$m=\frac{-3 - 3}{-2 - 1} = \frac{-6}{-3} = 2$$ 4. **Write equation using point A:** $$y - 3 = 2(x - 1)$$ Simplify: $$y - 3 = 2x - 2$$ $$y = 2x + 1$$ 5. **Answer:** The equation of the line through points $A$ and $B$ is $$y = 2x + 1$$ --- 6. **Problem:** Find the equation of the line passing through $(3,4)$ and parallel to the line $2x - y = 3$. 7. **Rewrite given line in slope-intercept form:** $$2x - y = 3 \implies y = 2x - 3$$ Slope of given line is $m=2$. 8. **Parallel lines have the same slope, so $m=2$. Use point-slope form:** $$y - 4 = 2(x - 3)$$ Simplify: $$y - 4 = 2x - 6$$ $$y = 2x - 2$$ 9. **Answer:** The equation of the line parallel to $2x - y = 3$ passing through $(3,4)$ is $$y = 2x - 2$$