1. Find the equation of the line passing through the points (2,7) and (5,−13). Step 1: State the problem. Find the equation of the line through points (2,7) and (5,−13). Step 2: Use the two-point formula for slope: $$m=\frac{y_2 - y_1}{x_2 - x_1} = \frac{-13 - 7}{5 - 2} = \frac{-20}{3}$$ Step 3: Use point-slope formula: $$y - y_1 = m(x - x_1)$$ Using point (2,7): $$y - 7 = -\frac{20}{3}(x - 2)$$ Step 4: Simplify: $$y - 7 = -\frac{20}{3}x + \frac{40}{3}$$ $$y = -\frac{20}{3}x + \frac{40}{3} + 7$$ $$y = -\frac{20}{3}x + \frac{40}{3} + \frac{21}{3} = -\frac{20}{3}x + \frac{61}{3}$$ Step 5: Using two-point formula for line equation: $$y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$$ $$y - 7 = -\frac{20}{3}(x - 2)$$ Same as above. --- 2. Find the equation of the line passing through the points (−3,4) and (1,−2). Step 1: Calculate slope: $$m=\frac{-2 - 4}{1 - (-3)} = \frac{-6}{4} = -\frac{3}{2}$$ Step 2: Use point-slope formula with point (-3,4): $$y - 4 = -\frac{3}{2}(x + 3)$$ Step 3: Simplify: $$y - 4 = -\frac{3}{2}x - \frac{9}{2}$$ $$y = -\frac{3}{2}x - \frac{9}{2} + 4 = -\frac{3}{2}x - \frac{9}{2} + \frac{8}{2} = -\frac{3}{2}x - \frac{1}{2}$$ --- 3. Find the equation of the line passing through the points (0,5) and (4,−3). Step 1: Calculate slope: $$m=\frac{-3 - 5}{4 - 0} = \frac{-8}{4} = -2$$ Step 2: Use point-slope formula with point (0,5): $$y - 5 = -2(x - 0)$$ Step 3: Simplify: $$y - 5 = -2x$$ $$y = -2x + 5$$ --- 4. Find the equation of the line passing through the points (−2,−1) and (3,9). Step 1: Calculate slope: $$m=\frac{9 - (-1)}{3 - (-2)} = \frac{10}{5} = 2$$ Step 2: Use point-slope formula with point (-2,-1): $$y + 1 = 2(x + 2)$$ Step 3: Simplify: $$y + 1 = 2x + 4$$ $$y = 2x + 3$$ --- 5. Find the equation of the line passing through the points (5,0) and (−1,6). Step 1: Calculate slope: $$m=\frac{6 - 0}{-1 - 5} = \frac{6}{-6} = -1$$ Step 2: Use point-slope formula with point (5,0): $$y - 0 = -1(x - 5)$$ Step 3: Simplify: $$y = -x + 5$$ --- 6. Find the equation of the line passing through the points (2,−3) and (2,7). What special type of line is this? Step 1: Calculate slope: $$m=\frac{7 - (-3)}{2 - 2} = \frac{10}{0}$$ Slope is undefined. Step 2: Equation is vertical line: $$x = 2$$ Special type: Vertical line. --- 7. Find the equation of the line passing through the points (−4,3) and (6,3). What special type of line is this? Step 1: Calculate slope: $$m=\frac{3 - 3}{6 - (-4)} = \frac{0}{10} = 0$$ Step 2: Equation is horizontal line: $$y = 3$$ Special type: Horizontal line. --- 8. Find the equation of the line passing through the points (1,−2) with slope $m=\frac{3}{4}$. Step 1: Use point-slope formula: $$y - (-2) = \frac{3}{4}(x - 1)$$ Step 2: Simplify: $$y + 2 = \frac{3}{4}x - \frac{3}{4}$$ $$y = \frac{3}{4}x - \frac{3}{4} - 2 = \frac{3}{4}x - \frac{3}{4} - \frac{8}{4} = \frac{3}{4}x - \frac{11}{4}$$ --- 9. Find the equation of the line passing through the point (−3,5) with slope $m = -2$. Step 1: Use point-slope formula: $$y - 5 = -2(x + 3)$$ Step 2: Simplify: $$y - 5 = -2x - 6$$ $$y = -2x - 6 + 5 = -2x - 1$$ --- 10. Find the equation of the line passing through the point (0,−4) with slope $m = \frac{1}{2}$. Step 1: Use point-slope formula: $$y - (-4) = \frac{1}{2}(x - 0)$$ Step 2: Simplify: $$y + 4 = \frac{1}{2}x$$ $$y = \frac{1}{2}x - 4$$