1. **Problem:** Determine if the graphs of the given equations are parallel, perpendicular, or neither for problems 7-17, then write equations of lines perpendicular to given lines through given points for problems 19-24. 2. **Recall:** - Two lines are **parallel** if their slopes are equal. - Two lines are **perpendicular** if the product of their slopes is $-1$. - Slope-intercept form: $y = mx + b$, where $m$ is the slope. - Standard form: $Ax + By = C$, slope $m = -\frac{A}{B}$. --- **7.** Point $(1,3)$; line $y = -3x + 2$ - Check if point lies on line: $y = -3(1) + 2 = -3 + 2 = -1 \neq 3$ so point not on line. **8.** Point $(2,-2)$; line $y = -x - 2$ - Check point: $y = -2 - 2 = -4 \neq -2$ no. **9.** Point $(1,-3)$; line $y + 2 = 4(x - 1)$ - Rewrite: $y = 4x - 4 - 2 = 4x - 6$ - Check point: $-3 = 4(1) - 6 = -2$ no. **10.** Point $(2,-1)$; line $y = -\frac{3}{2}x + 6$ - Check: $y = -3 + 6 = 3 \neq -1$ no. **11.** Point $(0,0)$; line $y = \frac{2}{3}x + 1$ - Check: $y = 0 + 1 = 1 \neq 0$ no. **12.** Point $(4,2)$; line $x = -3$ - Vertical line at $x = -3$, point $x=4$ no. --- **13.** Lines: $y = x + 11$ slope $m_1 = 1$ $y = -x + 2$ slope $m_2 = -1$ - Product $1 \times (-1) = -1$ so **perpendicular**. **14.** Lines: $y = \frac{3}{4}x - 1$ slope $m_1 = \frac{3}{4}$ $y = -\frac{3}{4}x + 29$ slope $m_2 = -\frac{3}{4}$ - Slopes not equal, product $= -\frac{9}{16} \neq -1$ so **neither**. **15.** Lines: $y = -2x + 3$ slope $m_1 = -2$ $2x + y = 7 \Rightarrow y = -2x + 7$ slope $m_2 = -2$ - Slopes equal, so **parallel**. **16.** Lines: $y - 4 = 3(x + 2) \Rightarrow y = 3x + 6 + 4 = 3x + 10$ slope $m_1 = 3$ $2x + 6y = 10 \Rightarrow 6y = -2x + 10 \Rightarrow y = -\frac{1}{3}x + \frac{5}{3}$ slope $m_2 = -\frac{1}{3}$ - Product $3 \times (-\frac{1}{3}) = -1$ so **perpendicular**. **17.** Lines: $y = -7$ slope $m_1 = 0$ (horizontal) $x = 2$ vertical line slope undefined - Horizontal and vertical lines are **perpendicular**. --- **19.** Point $(0,0)$; line $y = -3x + 2$ - Slope $m = -3$ - Perpendicular slope $m_p = \frac{1}{3}$ - Equation through $(0,0)$: $y = \frac{1}{3}x$ **20.** Point $(-2,3)$; line $y = \frac{1}{2}x - 1$ - Slope $m = \frac{1}{2}$ - Perpendicular slope $m_p = -2$ - Use point-slope: $y - 3 = -2(x + 2)$ - Simplify: $y - 3 = -2x - 4 \Rightarrow y = -2x - 1$ **21.** Point $(1,-2)$; line $y = 5x + 4$ - Slope $m = 5$ - Perpendicular slope $m_p = -\frac{1}{5}$ - Equation: $y + 2 = -\frac{1}{5}(x - 1)$ - Simplify: $y = -\frac{1}{5}x + \frac{1}{5} - 2 = -\frac{1}{5}x - \frac{9}{5}$ **22.** Point $(-3,2)$; line $x - 2y = 7$ - Rewrite: $-2y = -x + 7 \Rightarrow y = \frac{1}{2}x - \frac{7}{2}$ slope $m = \frac{1}{2}$ - Perpendicular slope $m_p = -2$ - Equation: $y - 2 = -2(x + 3)$ - Simplify: $y - 2 = -2x - 6 \Rightarrow y = -2x - 4$ **23.** Point $(5,0)$; line $y + 1 = 2(x - 3)$ - Slope $m = 2$ - Perpendicular slope $m_p = -\frac{1}{2}$ - Equation: $y - 0 = -\frac{1}{2}(x - 5)$ - Simplify: $y = -\frac{1}{2}x + \frac{5}{2}$ **24.** Point $(1,-6)$; line $x - 2y = 4$ - Rewrite: $-2y = -x + 4 \Rightarrow y = \frac{1}{2}x - 2$ slope $m = \frac{1}{2}$ - Perpendicular slope $m_p = -2$ - Equation: $y + 6 = -2(x - 1)$ - Simplify: $y + 6 = -2x + 2 \Rightarrow y = -2x - 4$ --- **Final answers:** 7-12: Point-line membership checked. 13: Perpendicular 14: Neither 15: Parallel 16: Perpendicular 17: Perpendicular 19: $y = \frac{1}{3}x$ 20: $y = -2x - 1$ 21: $y = -\frac{1}{5}x - \frac{9}{5}$ 22: $y = -2x - 4$ 23: $y = -\frac{1}{2}x + \frac{5}{2}$ 24: $y = -2x - 4$