Question

Are the pairs of lines parallel, perpendicular, or intersecting?

y=−23x+7
−2x−3y=12

y−6=14(x+2)
y=−4x−10

2x−y=7
3x−4y=−8

Answer 1

To determine whether the pairs of lines are parallel, perpendicular, or intersecting but neither parallel nor perpendicular, we need to compare the slopes of each line. If two lines have the same slope, they are parallel. If the product of their slopes is -1, they are perpendicular. Otherwise, they are intersecting but not parallel or perpendicular.

First, let's rewrite each equation in slope-intercept form (y = mx + b), where m is the slope:

1. y=−23x+7
This first equation is already in slope-intercept form with a slope of (-23).

2. −2x−3y=12
We must isolate y:
−3y = 2x + 12
y = (-2/3)x - 4
The slope here is (-2/3).

3. y−6=14(x+2)
We need to distribute the 14 and isolate y:
y - 6 = 14x + 28
y = 14x + 34
The slope here is 14.

4. y=−4x−10
This equation is already in slope-intercept form, and the slope is -4.

5. 2x−y=7
Isolating y gives us:
-y = -2x + 7
y = 2x - 7
The slope here is 2.

6. 3x−4y=−8
Isolating y gives us:
-4y = -3x - 8
y = (3/4)x + 2
The slope here is (3/4).

Now, let's compare the slopes:

First pair of lines: y=−23x+7 and −2x−3y=12 (slopes are -23 and -2/3)
These slopes are not the same and their product is not -1, so the lines are neither parallel nor perpendicular. Therefore, they are intersecting.

Second pair of lines: y−6=14(x+2) and y=−4x−10 (slopes are 14 and -4)
Again, these slopes are not the same and their product is 14 * (-4) = -56, which is not -1. So, they are intersecting.

Third pair of lines: 2x−y=7 and 3x−4y=−8 (slopes are 2 and 3/4)
Here, the slopes are not the same (2 and 3/4) and their product is not -1 (2 * 3/4 = 3/2), hence these lines are also intersecting.

In summary, none of the pairs of lines are parallel or perpendicular; all pairs are intersecting but neither parallel nor perpendicular.