Question

parallel and perpendicular lines
identify the slope of each function.

1. $y = 2x - 1$ 2. $y = \frac{1}{3}x + 5$ 3. $y - 4 = \frac{2}{3}(x - 6)$
2. $y + 1 = 4(x + 9)$ 5. $5x + 2y = 10$ 6. $- 3x - 6y = 12$

given the slope, determine the parallel and perpendicular slope.

slope	parallel slope	perpendicular slope
-7
$\frac{3}{5}$
$-\frac{9}{4}$
5

draw a line parallel to the given line and going through the point.
draw a line perpendicular to the given line and going through the point.

Explanation:

Part 1: Identify the slope of each function

Step1: Use slope-intercept form $y=mx+b$

For equations already in $y=mx+b$, $m$ is slope.

1. $y=2x-1$: Slope $m=2$
2. $y=\frac{1}{3}x+5$: Slope $m=\frac{1}{3}$

Step2: Use point-slope form $y-y_1=m(x-x_1)$

For equations in this form, $m$ is slope.

1. $y-4=\frac{2}{3}(x-6)$: Slope $m=\frac{2}{3}$
2. $y+1=4(x+9)$: Slope $m=4$

Step3: Rewrite to slope-intercept form

Isolate $y$ to get $y=mx+b$.

1. $5x+2y=10$

$2y=-5x+10$
$y=-\frac{5}{2}x+5$: Slope $m=-\frac{5}{2}$

1. $-3x-6y=12$

$-6y=3x+12$
$y=-\frac{1}{2}x-2$: Slope $m=-\frac{1}{2}$

Step1: Parallel slope rule

Parallel lines have equal slopes: $m_{\text{parallel}}=m$

Step2: Perpendicular slope rule

Perpendicular slopes are negative reciprocals: $m_{\text{perpendicular}}=-\frac{1}{m}$

1. For $m=2$:

Parallel slope $=2$, Perpendicular slope $=-\frac{1}{2}$

1. For $m=-7$:

Parallel slope $=-7$, Perpendicular slope $=\frac{1}{7}$

1. For $m=\frac{3}{5}$:

Parallel slope $=\frac{3}{5}$, Perpendicular slope $=-\frac{5}{3}$

1. For $m=-\frac{9}{4}$:

Parallel slope $=-\frac{9}{4}$, Perpendicular slope $=\frac{4}{9}$

1. For $m=5$:

Parallel slope $=5$, Perpendicular slope $=-\frac{1}{5}$

Parallel Line:

1. First, calculate the slope of the given line: using two points on the line (e.g., $(0,4)$ and $(8,0)$), slope $m=\frac{0-4}{8-0}=-\frac{1}{2}$
2. Use the same slope $-\frac{1}{2}$ and the marked point $(4,1)$. Plot a second point by moving 2 right and 1 down from $(4,1)$ to $(6,0)$, then draw a straight line through $(4,1)$ and $(6,0)$.

Perpendicular Line:

1. The given line has slope $-\frac{1}{2}$, so the perpendicular slope is the negative reciprocal: $2$
2. Use slope $2$ and the marked point $(4,1)$. Plot a second point by moving 1 right and 2 up from $(4,1)$ to $(5,3)$, then draw a straight line through $(4,1)$ and $(5,3)$.

Answer:

1. $2$
2. $\frac{1}{3}$
3. $\frac{2}{3}$
4. $4$
5. $-\frac{5}{2}$
6. $-\frac{1}{2}$

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Part 2: Determine parallel and perpendicular slopes