Question

Is each line parallel, perpendicular, or neither parallel nor perpendicular to the line −2x+3y=12? Drag each choice into the boxes to correctly complete the table. Is it Parallel -Perpendicular -Neither

-2x+y=12 3x+2y=-2 y=2/3x-1 -2x+3y=11

Answer 1

When the original equation of the line -2x+3y=12 is simplified, the equation comes out as y=2x/3 + 4, meaning the slope is 2/3 and the y-int is 4

-When -2x+y=12 is simplified, it becomes y=2x+12, so it is neither

-When 3x+2y=-2 is simplified, it becomes y= -3x/2 - 1, the slope is the opposite reciprocal of the original equation's slope, meaning this line is perpendicular to the original line

-For y=2/3x - 1, the slope is equal to the slope of the original line but the y-intercept is different, meaning this line is parallel

-For the line -2x + 3y=11, when it's simplified it becomes y=2/3x + 11/3, the slope is equal to the slope of the original line but it has a different y-int, so the line is parallel.

Answer 2

-2x+y=12: Neither, 3x+2y=-2: Perpendicular, y=2/3x-1: Parallel, -2x+3y=11: Neither

To determine the relationship between each line and the given line −2x+3y=12, we consider their slopes. The original line has a slope of 2/3 when expressed in slope-intercept form (y=mx+b). Lines that are parallel have the same slope, while lines that are perpendicular have slopes that are negative reciprocals of each other.

Analyzing each line:

−2x+y=12: When rearranged into slope-intercept form, this line has slope of 2. Since it does not have the same slope or a negative reciprocal slope, it is neither parallel nor perpendicular to the original line.

3x+2y=−2: This line, when expressed in slope-intercept form, has a slope of −3/2. The negative reciprocal of 2/3 is −3/2, indicating a perpendicular relationship with the original line.

y= 2/3x−1: This line has the same slope (2/3) as the original line, demonstrating a parallel relationship.

−2x+3y=11: When transformed into slope-intercept form, this line has a slope of 2/3, similar to the original line. Therefore, it is neither parallel nor perpendicular to the given line.