Question

homework 11: parallel vs.directions: review the following notes box below and complete the problems. be sure to show all the work.example 2 identifying parallel and perpendicular linesdetermine which of the lines are parallel and which of the lines are perpendicular.solutionfind the slope of each line.line a: $m = \frac{3 - 2}{0 - (-3)} = \frac{1}{3}$line b: $m = \frac{0 - (-1)}{2 - 0} = \frac{1}{2}$line c: $m = \frac{-4 - (-5)}{1 - (-1)} = \frac{1}{2}$line d: $m = \frac{2 - 0}{-3 - (-2)} = -2$because lines b and c have the same slope, lines b and c are parallel. because $\frac{1}{2}(-2) = -1$, lines b and d are perpendicular and lines c and d are perpendicular.directions: write and graph the equation that goes through given point a that is parallel to the original line.1)2)3)4)

Explanation:

Problem 1

Step1: Find original line slope

Original line passes through $(0,6)$ and $(3,0)$.
$m = \frac{0-6}{3-0} = -2$

Step2: Use point-slope form

Point A is $(-3,4)$. Parallel lines have equal slope.
$y - 4 = -2(x - (-3))$

Step3: Simplify to slope-intercept

$y - 4 = -2x - 6$
$y = -2x - 2$

Problem 2

Step1: Find original line slope

Original line is horizontal: $y=-2$, slope $m=0$.

Step2: Use point-slope form

Point A is $(2,3)$. Parallel lines have equal slope.
$y - 3 = 0(x - 2)$

Step3: Simplify to slope-intercept

$y = 3$

Problem 3

Step1: Find original line slope

Original line is vertical: $x=1$, slope is undefined.

Step2: Write parallel line equation

Vertical parallel lines have form $x=k$. Point A is $(-4,-3)$, so $x=-4$.

Problem 4

Step1: Find original line slope

Original line passes through $(0,-4)$ and $(6,2)$.
$m = \frac{2-(-4)}{6-0} = \frac{6}{6} = 1$

Step2: Use point-slope form

Point A is $(-2,1)$. Parallel lines have equal slope.
$y - 1 = 1(x - (-2))$

Step3: Simplify to slope-intercept

$y - 1 = x + 2$
$y = x + 3$

Answer:

1. Equation: $y = -2x - 2$ (Graph: line through $(-3,4)$ with slope $-2$)
2. Equation: $y = 3$ (Graph: horizontal line through $(2,3)$)
3. Equation: $x = -4$ (Graph: vertical line through $(-4,-3)$)
4. Equation: $y = x + 3$ (Graph: line through $(-2,1)$ with slope $1$)