Math

QuestionDetermine if the following line pairs are parallel, perpendicular, or neither:
Line 1: y=23x8y=-\frac{2}{3} x-8, Line 2: 6x4y=66 x-4 y=-6, Line 3: 3y=2x+23 y=-2 x+2.

Studdy Solution

STEP 1

Assumptions1. The equations of the lines are given in the form y=mx+cy = mx + c, where mm is the slope of the line and cc is the y-intercept. . Two lines are parallel if their slopes are equal.
3. Two lines are perpendicular if the product of their slopes is -1.
4. If the lines are neither parallel nor perpendicular, they are classified as neither.

STEP 2

First, we need to rewrite the equations of the lines in the slope-intercept form, y=mx+cy = mx + c.
Line1 is already in this form y=2x8y=-\frac{2}{} x-8.
Line2 6x4y=66 x-4 y=-6 can be rewritten as y=2x+2y = \frac{}{2}x + \frac{}{2}.
Line y=2x+2 y=-2 x+2 can be rewritten as y=2x+2y = -\frac{2}{}x + \frac{2}{}.

STEP 3

Now, we can compare the slopes of each pair of lines to determine if they are parallel, perpendicular, or neither.

STEP 4

Compare the slopes of Line1 and Line2lope of Line1 m1=23m1 = -\frac{2}{3}lope of Line2 m2=32m2 = \frac{3}{2}Since m1m2m1 \neq m2 and m1m21m1 \cdot m2 \neq -1, Line1 and Line2 are neither parallel nor perpendicular.

STEP 5

Compare the slopes of Line1 and Line3lope of Line1 m1=23m1 = -\frac{2}{3}lope of Line3 m3=23m3 = -\frac{2}{3}Since m1=m3m1 = m3, Line1 and Line3 are parallel.

STEP 6

Compare the slopes of Line2 and Line3lope of Line2 m2=32m2 = \frac{3}{2}lope of Line3 m3=23m3 = -\frac{2}{3}Since m2m3=1m2 \cdot m3 = -1, Line2 and Line3 are perpendicular.
So, the pairs of lines areLine1 and Line2 NeitherLine1 and Line3 ParallelLine2 and Line3 Perpendicular

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