1. **Problem Statement:** Find the slope of the lines passing through each pair of points and determine if the lines are parallel or perpendicular. 2. **Formula for slope:** The slope $m$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by: $$m=\frac{y_2 - y_1}{x_2 - x_1}$$ 3. **Important rules:** - Lines are parallel if their slopes are equal. - Lines are perpendicular if the product of their slopes is $-1$. --- **(a)** - Points 1: $(2,7)$ and $(-4,3)$ $$m_1=\frac{3-7}{-4-2}=\frac{-4}{-6}=\frac{2}{3}$$ - Points 2: $(1,1)$ and $(4,-\frac{7}{2})$ $$m_2=\frac{-\frac{7}{2}-1}{4-1}=\frac{-\frac{7}{2}-\frac{2}{2}}{3}=\frac{-\frac{9}{2}}{3}=-\frac{3}{2}$$ - Check parallel/perpendicular: $$m_1 \neq m_2$$ $$m_1 \times m_2 = \frac{2}{3} \times -\frac{3}{2} = -1$$ So, lines are **perpendicular**. **(b)** - Points 1: $(3,8)$ and $(4,2)$ $$m_1=\frac{2-8}{4-3}=\frac{-6}{1}=-6$$ - Points 2: $(2,7)$ and $(-4,8)$ $$m_2=\frac{8-7}{-4-2}=\frac{1}{-6}=-\frac{1}{6}$$ - Check: $$m_1 \neq m_2$$ $$m_1 \times m_2 = -6 \times -\frac{1}{6} = 1$$ Not $-1$, so lines are **neither parallel nor perpendicular**. **(c)** - Points 1: $(12,9)$ and $(3,7)$ $$m_1=\frac{7-9}{3-12}=\frac{-2}{-9}=\frac{2}{9}$$ - Points 2: $(1,0)$ and $(4,\frac{2}{3})$ $$m_2=\frac{\frac{2}{3}-0}{4-1}=\frac{\frac{2}{3}}{3}=\frac{2}{9}$$ - Check: $$m_1 = m_2 = \frac{2}{9}$$ Lines are **parallel**. **(d)** - Points 1: $(6,-8)$ and $(-4,2)$ $$m_1=\frac{2-(-8)}{-4-6}=\frac{10}{-10}=-1$$ - Points 2: $(-2,6)$ and $(-1,5)$ $$m_2=\frac{5-6}{-1-(-2)}=\frac{-1}{1}=-1$$ - Check: $$m_1 = m_2 = -1$$ Lines are **parallel**. --- 4. **Problem Statement:** Find the length of the ramp given slope $\frac{1}{15}$ and vertical height 4.5 ft. 5. **Formula:** Slope $m = \frac{\text{vertical rise}}{\text{horizontal run}}$ 6. Given: $$m=\frac{1}{15}, \quad \text{vertical rise} = 4.5$$ 7. Find horizontal run $x$: $$\frac{1}{15} = \frac{4.5}{x} \implies x = 4.5 \times 15 = 67.5$$ 8. **Answer:** The ramp must extend 67.5 ft from the home. --- 9. **Problem Statement:** Compare two phone plans. 10. **Company A:** Monthly fee 30, cost per minute 0.15. $$C_A = 30 + 0.15m$$ 11. **Company B:** Monthly fee 5, cost per minute 0.20. $$C_B = 5 + 0.20m$$ 12. **Find minutes where costs are equal:** $$30 + 0.15m = 5 + 0.20m$$ $$30 - 5 = 0.20m - 0.15m$$ $$25 = 0.05m$$ $$m = \frac{25}{0.05} = 500$$ 13. **Compare costs at 300 minutes:** - Company A: $$C_A = 30 + 0.15 \times 300 = 30 + 45 = 75$$ - Company B: $$C_B = 5 + 0.20 \times 300 = 5 + 60 = 65$$ 14. **Conclusion:** For 300 minutes, Company B is cheaper. --- **Final answers:** - (a) Slopes $\frac{2}{3}$ and $-\frac{3}{2}$, lines are perpendicular. - (b) Slopes $-6$ and $-\frac{1}{6}$, neither parallel nor perpendicular. - (c) Slopes $\frac{2}{9}$ and $\frac{2}{9}$, lines are parallel. - (d) Slopes $-1$ and $-1$, lines are parallel. - Ramp length: 67.5 ft. - Company A cost model: $C_A=30+0.15m$. - Company B cost model: $C_B=5+0.20m$. - Plans equal at 500 minutes. - For 300 minutes, choose Company B.