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Math  /  Algebra

QuestionThe equations of three lines are given below. Line 1: y=−2x−8y=-2 x-8 Line 2: 3x−6y=−63 x-6 y=-6 Line 3: y=−2x+1y=-2 x+1
For each pair of lines, determine whether they are parallel, perpendicular, or neith
Line 1 and Line 2 : Parallel Perpendicular Neither Line 1 and Line 3: Parallel Perpendicular Neither Line 2 and Line 3 : Parallel Perpendicular Neither

Studdy Solution

STEP 1

1. Two lines are parallel if they have the same slope.
2. Two lines are perpendicular if the product of their slopes is −1-1.
3. Lines in the form y=mx+b y = mx + b have a slope m m .
4. Lines in the form Ax+By=C Ax + By = C can be rewritten in slope-intercept form to identify their slope.

STEP 2

1. Identify the slope of Line 1.
2. Identify the slope of Line 2.
3. Identify the slope of Line 3.
4. Compare slopes of Line 1 and Line 2.
5. Compare slopes of Line 1 and Line 3.
6. Compare slopes of Line 2 and Line 3.

STEP 3

Line 1 is given in slope-intercept form y=−2x−8 y = -2x - 8 . The slope m1 m_1 is:
m1=−2 m_1 = -2

STEP 4

Line 2 is given in standard form 3x−6y=−6 3x - 6y = -6 . To find the slope, convert it to slope-intercept form:
3x−6y=−6 3x - 6y = -6
Subtract 3x 3x from both sides:
−6y=−3x−6 -6y = -3x - 6
Divide every term by −6-6:
y=12x+1 y = \frac{1}{2}x + 1
The slope m2 m_2 is:
m2=12 m_2 = \frac{1}{2}

STEP 5

Line 3 is given in slope-intercept form y=−2x+1 y = -2x + 1 . The slope m3 m_3 is:
m3=−2 m_3 = -2

STEP 6

Compare the slopes of Line 1 and Line 2:
- Slope of Line 1: m1=−2 m_1 = -2 - Slope of Line 2: m2=12 m_2 = \frac{1}{2}
Since m1≠m2 m_1 \neq m_2 and m1⋅m2=−2⋅12=−1 m_1 \cdot m_2 = -2 \cdot \frac{1}{2} = -1 , the lines are perpendicular.

STEP 7

Compare the slopes of Line 1 and Line 3:
- Slope of Line 1: m1=−2 m_1 = -2 - Slope of Line 3: m3=−2 m_3 = -2
Since m1=m3 m_1 = m_3 , the lines are parallel.

STEP 8

Compare the slopes of Line 2 and Line 3:
- Slope of Line 2: m2=12 m_2 = \frac{1}{2} - Slope of Line 3: m3=−2 m_3 = -2
Since m2≠m3 m_2 \neq m_3 and m2⋅m3=12⋅−2=−1 m_2 \cdot m_3 = \frac{1}{2} \cdot -2 = -1 , the lines are perpendicular.
The results are: - Line 1 and Line 2: Perpendicular - Line 1 and Line 3: Parallel - Line 2 and Line 3: Perpendicular

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