1. **Problem Statement:** Find the equation of the straight line passing through each pair of points given. 2. **Formula:** The equation of a line through points $(x_1, y_1)$ and $(x_2, y_2)$ is found using the slope-intercept form: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ $$y - y_1 = m(x - x_1)$$ where $m$ is the slope. 3. **Important Rules:** - If $x_1 = x_2$, the line is vertical and equation is $x = x_1$. - If $y_1 = y_2$, the line is horizontal and equation is $y = y_1$. 4. **Solutions:** a. Points $(1,4)$ and $(3,10)$: $$m = \frac{10 - 4}{3 - 1} = \frac{6}{2} = 3$$ Equation: $$y - 4 = 3(x - 1) \Rightarrow y = 3x + 1$$ b. Points $(1,5)$ and $(2,7)$: $$m = \frac{7 - 5}{2 - 1} = 2$$ Equation: $$y - 5 = 2(x - 1) \Rightarrow y = 2x + 3$$ c. Points $(1,6)$ and $(2,11)$: $$m = \frac{11 - 6}{2 - 1} = 5$$ Equation: $$y - 6 = 5(x - 1) \Rightarrow y = 5x + 1$$ d. Points $(0,4)$ and $(1,3)$: $$m = \frac{3 - 4}{1 - 0} = -1$$ Equation: $$y - 4 = -1(x - 0) \Rightarrow y = -x + 4$$ e. Points $(3,-4)$ and $(10,-18)$: $$m = \frac{-18 + 4}{10 - 3} = \frac{-14}{7} = -2$$ Equation: $$y + 4 = -2(x - 3) \Rightarrow y = -2x + 2$$ f. Points $(0,-1)$ and $(1,-4)$: $$m = \frac{-4 + 1}{1 - 0} = -3$$ Equation: $$y + 1 = -3(x - 0) \Rightarrow y = -3x - 1$$ g. Points $(-5,3)$ and $(2,4)$: $$m = \frac{4 - 3}{2 + 5} = \frac{1}{7}$$ Equation: $$y - 3 = \frac{1}{7}(x + 5) \Rightarrow y = \frac{1}{7}x + \frac{5}{7} + 3 = \frac{1}{7}x + \frac{26}{7}$$ h. Points $(-3,-2)$ and $(4,4)$: $$m = \frac{4 + 2}{4 + 3} = \frac{6}{7}$$ Equation: $$y + 2 = \frac{6}{7}(x + 3) \Rightarrow y = \frac{6}{7}x + \frac{18}{7} - 2 = \frac{6}{7}x + \frac{4}{7}$$ i. Points $(-7,-3)$ and $(-1,6)$: $$m = \frac{6 + 3}{-1 + 7} = \frac{9}{6} = \frac{3}{2}$$ Equation: $$y + 3 = \frac{3}{2}(x + 7) \Rightarrow y = \frac{3}{2}x + \frac{21}{2} - 3 = \frac{3}{2}x + \frac{15}{2}$$ j. Points $(2,5)$ and $(1,-4)$: $$m = \frac{-4 - 5}{1 - 2} = \frac{-9}{-1} = 9$$ Equation: $$y - 5 = 9(x - 2) \Rightarrow y = 9x - 18 + 5 = 9x - 13$$ k. Points $(-3,4)$ and $(5,0)$: $$m = \frac{0 - 4}{5 + 3} = \frac{-4}{8} = -\frac{1}{2}$$ Equation: $$y - 4 = -\frac{1}{2}(x + 3) \Rightarrow y = -\frac{1}{2}x - \frac{3}{2} + 4 = -\frac{1}{2}x + \frac{5}{2}$$ l. Points $(6,4)$ and $(-7,7)$: $$m = \frac{7 - 4}{-7 - 6} = \frac{3}{-13} = -\frac{3}{13}$$ Equation: $$y - 4 = -\frac{3}{13}(x - 6) \Rightarrow y = -\frac{3}{13}x + \frac{18}{13} + 4 = -\frac{3}{13}x + \frac{70}{13}$$ m. Points $(-5,2)$ and $(6,2)$: Since $y$ values are equal, line is horizontal: $$y = 2$$ n. Points $(1,-3)$ and $(-2,6)$: $$m = \frac{6 + 3}{-2 - 1} = \frac{9}{-3} = -3$$ Equation: $$y + 3 = -3(x - 1) \Rightarrow y = -3x + 3 - 3 = -3x$$ o. Points $(6,-4)$ and $(6,6)$: Since $x$ values are equal, line is vertical: $$x = 6$$ **Final answers:** a. $y = 3x + 1$ b. $y = 2x + 3$ c. $y = 5x + 1$ d. $y = -x + 4$ e. $y = -2x + 2$ f. $y = -3x - 1$ g. $y = \frac{1}{7}x + \frac{26}{7}$ h. $y = \frac{6}{7}x + \frac{4}{7}$ i. $y = \frac{3}{2}x + \frac{15}{2}$ j. $y = 9x - 13$ k. $y = -\frac{1}{2}x + \frac{5}{2}$ l. $y = -\frac{3}{13}x + \frac{70}{13}$ m. $y = 2$ n. $y = -3x$ o. $x = 6$