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. Parallel Perpendicular Neither −2x+y=123x+2y=−2y=23x−1−2x+3y=11

Answer 1

We are given first equation: −2x+3y=12.

Converting it in slope-intercept form

3y=2x + 12

y= 2/3 x + 12/3

y= 2/3 x + 4.

Slope of the given equation is 2/3.

Note: When slopes are same, lines would be parallel.

When slope are negative reciprocals, lines would be perpendicular.

When neither same nor negative reciprocal, the lines neither parallel nor perpendicular.

1) First option : -2x+y=12.

In slope-intercept form y = 2x +12.

Slope is 2 there.

Neither parallel nor perpendicular to the line −2x+3y=12.

2) 3x+2y=-2

In slope-intercept form y = -3/2 x - 1.

Slope = -3/2.

Slope are negative reciprocal to slope of −2x+3y=12 equation.

Therefore, lines are perpendicular.

3) y=2/3x-1

Slope = 2/3.

Slopes are same therefore lines are parallel.

4) -2x+3y=11

y = 2/3 x + 11/3.

Slopes are same therefore lines are parallel.

Answer 2

Solution

To solve this problem we will use the form of line y=mx+c with slope of the line= m.

Two lines are parallel if their slopes are equal that is m1=m2

And two lines are perpendicular if the product of their slopes = -1 that is m1.m2= -1

given line -2x+3y=12 ⇒ 3y=12+2x ⇒y=4+2x/3 ⇒ slope(m1) = 2/3

(1) L1: -2x+y=12 ⇒ y=2x+12 ⇒ slope (m2)= 2

By observation m1≠m2 and m1×m2≠ -1

∴ this line is neither parallel nor perpendicular.

(2) L2: 3x+2y= -2 ⇒ y=-3x/2-1 ⇒ slope (m2)= -3/2

By observation m1×m2= -1

∴ this line is perpendicular to given line.

(3) L3: y=23x-1 ⇒ slope (m2)= 23

By observation m1≠m2 and m1×m2≠ -1

∴ this line is neither parallel nor perpendicular

(4) L4: -2x+3y=11 ⇒ y=2x/3+11/3 ⇒ slope (m2)= 2/3

By observation m1=m2

∴ this line is parallel to given line.