Question

write an equation of the line that passes through (0,6) and is ______ to the line $y = 3x$

1. parallel 2. perpendicular

write an equation of the line that passes through (-3, 2) and is ______ to the line $y = \frac{1}{3}x + 1$

1. parallel 2. perpendicular

write two equations that are parallel to each other. write two equations that are perpendicular to each other.

determine whether the following pairs of lines are parallel, perpendicular, or neither.

1. $3x - 2y = 9$ $y = -\frac{5}{3}x - 12$
2. $y = 6x + 16$ $y - 2 = 6(x - 1)$
3. $x + y = 0$ $y = x + 10$

ohio geometry state test question:
31 what is the equation of the line perpendicular to the line $3x + 2y = 8$ that passes through the point (-4, 1)?
a $y - 1 = \frac{2}{3}(x + 4)$
b $y + 1 = \frac{2}{3}(x - 4)$
c $y - 1 = -\frac{3}{2}(x + 4)$
d $y + 1 = -\frac{3}{2}(x - 4)$

Explanation:

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Part 1: Line through (0,6) relative to $y=3x$

1. Parallel

Step1: Identify slope of parallel line

Parallel lines have equal slopes. The slope of $y=3x$ is $m=3$.

Step2: Use slope-intercept form

The line passes through $(0,6)$, so the y-intercept $b=6$.
$y = mx + b$
$y = 3x + 6$

2. Perpendicular

Step1: Find perpendicular slope

Perpendicular slopes multiply to -1: $m_{\perp} = -\frac{1}{3}$.

Step2: Use slope-intercept form

Line passes through $(0,6)$, so $b=6$.
$y = -\frac{1}{3}x + 6$

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Part 2: Line through (-3,2) relative to $y=\frac{1}{3}x + 1$

1. Parallel

Step1: Identify parallel slope

Parallel slope equals $\frac{1}{3}$.

Step2: Use point-slope form

Point-slope formula: $y - y_1 = m(x - x_1)$
$y - 2 = \frac{1}{3}(x + 3)$
Simplify to slope-intercept: $y = \frac{1}{3}x + 3$

2. Perpendicular

Step1: Find perpendicular slope

$m_{\perp} = -3$ (since $\frac{1}{3} \times -3 = -1$).

Step2: Use point-slope form

$y - 2 = -3(x + 3)$
Simplify to slope-intercept: $y = -3x -7$

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Part 3: Write parallel/perpendicular equations

Parallel equations

Step1: Choose equal slopes

Use slope $m=2$ for both lines, different y-intercepts.
$y = 2x + 1$
$y = 2x - 4$

Perpendicular equations

Step1: Choose slopes multiplying to -1

Use $m=4$ and $m_{\perp}=-\frac{1}{4}$.
$y = 4x + 2$
$y = -\frac{1}{4}x + 5$

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Part 4: Classify line pairs

1. $3x - 2y = 9$ and $y = -\frac{5}{3}x -12$

Step1: Convert first line to slope-intercept

$3x - 2y = 9 \implies y = \frac{3}{2}x - \frac{9}{2}$

Step2: Compare slopes

Slopes: $\frac{3}{2}$ and $-\frac{5}{3}$. They are not equal, and $\frac{3}{2} \times -\frac{5}{3} = -\frac{5}{2}
eq -1$.
Conclusion: Neither

2. $y = 6x + 16$ and $y - 2 = 6(x - 1)$

Step1: Simplify second line

$y - 2 = 6(x - 1) \implies y = 6x -4$

Step2: Compare slopes

Slopes are both 6.
Conclusion: Parallel

3. $x + y = 0$ and $y = x + 10$

Step1: Convert first line to slope-intercept

$x + y = 0 \implies y = -x$

Step2: Check slopes

Slopes: $-1$ and $1$. $-1 \times 1 = -1$.
Conclusion: Perpendicular

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Part 5: Ohio Geometry Test Question

Step1: Find slope of given line

Rewrite $3x + 2y = 8$: $y = -\frac{3}{2}x + 4$. Slope $m = -\frac{3}{2}$.

Step2: Find perpendicular slope

$m_{\perp} = \frac{2}{3}$ (since $-\frac{3}{2} \times \frac{2}{3} = -1$).

Step3: Use point-slope form

Line passes through $(-4,1)$:
$y - 1 = \frac{2}{3}(x + 4)$

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Answer:

1. Line through (0,6):

• Parallel: $y = 3x + 6$
• Perpendicular: $y = -\frac{1}{3}x + 6$

1. Line through (-3,2):

• Parallel: $y = \frac{1}{3}x + 3$
• Perpendicular: $y = -3x -7$

1. Example equations:

• Parallel: $y = 2x + 1$ and $y = 2x - 4$
• Perpendicular: $y = 4x + 2$ and $y = -\frac{1}{4}x + 5$

1. Line pair classification:
2. Neither
3. Parallel
4. Perpendicular

1. Ohio Geometry Test Answer: A. $y - 1 = \frac{2}{3}(x + 4)$