1. **Problem:** Write an equation in slope-intercept form of the line that passes through the given point and is parallel to the graph of the given equation. 2. **Recall:** Lines that are parallel have the same slope. 3. **Slope-intercept form:** $$y = mx + b$$ where $m$ is the slope and $b$ is the y-intercept. --- **7.** Point $(1,3)$, line $y=3x+2$. - Slope $m=3$. - Use point-slope form: $$y - y_1 = m(x - x_1)$$ - Substitute: $$y - 3 = 3(x - 1)$$ - Simplify: $$y - 3 = 3x - 3$$ - Add 3 both sides: $$y = 3x - 3 + 3$$ - $$y = 3x$$ **8.** Point $(2,-2)$, line $y = -x - 2$. - Slope $m = -1$. - Point-slope: $$y - (-2) = -1(x - 2)$$ - $$y + 2 = -x + 2$$ - $$y = -x + 2 - 2$$ - $$y = -x$$ **9.** Point $(1,-3)$, line $y + 2 = 4(x - 1)$. - Rewrite given line: $$y = 4x - 4 - 2 = 4x - 6$$ - Slope $m=4$. - Point-slope: $$y - (-3) = 4(x - 1)$$ - $$y + 3 = 4x - 4$$ - $$y = 4x - 4 - 3$$ - $$y = 4x - 7$$ **10.** Point $(2,-1)$, line $y = -\frac{3}{2}x + 6$. - Slope $m = -\frac{3}{2}$. - Point-slope: $$y - (-1) = -\frac{3}{2}(x - 2)$$ - $$y + 1 = -\frac{3}{2}x + 3$$ - $$y = -\frac{3}{2}x + 3 - 1$$ - $$y = -\frac{3}{2}x + 2$$ **11.** Point $(0,0)$, line $y = \frac{2}{3}x + 1$. - Slope $m = \frac{2}{3}$. - Point-slope: $$y - 0 = \frac{2}{3}(x - 0)$$ - $$y = \frac{2}{3}x$$ **12.** Point $(4,2)$, line $x = -3$. - This is a vertical line with undefined slope. - Parallel line is vertical through $x=4$. - Equation: $$x = 4$$ --- **13.** Compare $y = x + 11$ and $y = -x + 2$. - Slopes: $1$ and $-1$. - Product: $1 \times (-1) = -1$. - Lines are **perpendicular**. **14.** Compare $y = \frac{3}{4}x - 1$ and $y = \frac{3}{4}x + 29$. - Slopes equal $\frac{3}{4}$. - Lines are **parallel**. **15.** Compare $y = -2x + 3$ and $2x + y = 7$. - Rewrite second: $y = -2x + 7$. - Slopes both $-2$. - Lines are **parallel**. **16.** Compare $y = 4 = 3(x + 2)$ (assumed typo, interpret as $y = 3(x+2)$) and $2x + 6y = 10$. - First: $y = 3x + 6$ slope $3$. - Second: $6y = -2x + 10 \Rightarrow y = -\frac{1}{3}x + \frac{5}{3}$ slope $-\frac{1}{3}$. - Product $3 \times (-\frac{1}{3}) = -1$. - Lines are **perpendicular**. **17.** Compare $y = -7$ and $x = 2$. - Horizontal and vertical lines. - Lines are **perpendicular**. **18.** Compare $y = 4x - 2$ and $-x + 4y = 0$. - Second: $4y = x \Rightarrow y = \frac{1}{4}x$. - Slopes $4$ and $\frac{1}{4}$. - Product $4 \times \frac{1}{4} = 1$. - Lines are **neither parallel nor perpendicular**. --- **19.** Point $(0,0)$, line $y = -3x + 2$. - Slope $m = -3$. - Perpendicular slope $m_p = \frac{1}{3}$. - Equation: $$y - 0 = \frac{1}{3}(x - 0)$$ - $$y = \frac{1}{3}x$$ **20.** Point $(-2,3)$, line $y = \frac{1}{2}x - 1$. - Slope $m = \frac{1}{2}$. - Perpendicular slope $m_p = -2$. - Equation: $$y - 3 = -2(x + 2)$$ - $$y - 3 = -2x - 4$$ - $$y = -2x - 1$$ **21.** Point $(1,-2)$, line $y = 5x + 4$. - Slope $m=5$. - Perpendicular slope $m_p = -\frac{1}{5}$. - Equation: $$y + 2 = -\frac{1}{5}(x - 1)$$ - $$y + 2 = -\frac{1}{5}x + \frac{1}{5}$$ - $$y = -\frac{1}{5}x + \frac{1}{5} - 2$$ - $$y = -\frac{1}{5}x - \frac{9}{5}$$ **22.** Point $(-3,2)$, line $x - 2y = 7$. - Rewrite: $x = 2y + 7$ or $y = \frac{x - 7}{2}$ slope $m = \frac{1}{2}$. - Perpendicular slope $m_p = -2$. - Equation: $$y - 2 = -2(x + 3)$$ - $$y - 2 = -2x - 6$$ - $$y = -2x - 4$$ **23.** Point $(5,0)$, line $y + 1 = 2(x - 3)$. - Rewrite: $y = 2x - 6 - 1 = 2x - 7$ slope $m=2$. - Perpendicular slope $m_p = -\frac{1}{2}$. - Equation: $$y - 0 = -\frac{1}{2}(x - 5)$$ - $$y = -\frac{1}{2}x + \frac{5}{2}$$ **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: $$y + 6 = -2(x - 1)$$ - $$y + 6 = -2x + 2$$ - $$y = -2x + 2 - 6$$ - $$y = -2x - 4$$