Question

The equations of three lines are given below.
Line1: 2y =3x +5
Line2:y=3^2x -3
Line3:4x-6y=-4
For each pair of lines, determine whether they are parallel, perpendicular, or neither.
Answered by Damon
Lines 1 and 3:
The slope of Line1 is 3/2 and the slope of Line3 is 2/3. The product of their slopes is -1, so the lines are perpendicular.

No. The slopes are both POSITIVE
so they are NOT perpendicular
(I suspect a typo in the problem statement)
Answered by GPT 3.5
First, we need to put each equation in slope-intercept form, y = mx + b, where m is the slope of the line.

Line1: 2y = 3x + 5
Divide both sides by 2:
y = (3/2)x + 5/2
So the slope of Line1 is 3/2.

Line2: y = 3^2x - 3
y = 9x - 3
So the slope of Line2 is 9.

Line3: 4x - 6y = -4
-6y = -4x - 4
Divide both sides by -6:
y = (4/6)x + 4/6
Simplify:
y = (2/3)x + 2/3
So the slope of Line3 is 2/3.

Now, let's compare the slopes to determine their relationship:

Lines 1 and 2:
The slope of Line1 is 3/2 and the slope of Line2 is 9. The slopes are not equal and their product is not -1, so the lines are neither parallel nor perpendicular.

Lines 1 and 3:
The slope of Line1 is 3/2 and the slope of Line3 is 2/3. The product of their slopes is -1, so the lines are perpendicular.

Lines 2 and 3:
The slope of Line2 is 9 and the slope of Line3 is 2/3. The slopes are not equal and their product is not -1, so the lines are neither parallel nor perpendicular.

In conclusion:
Line1 is perpendicular to Line3. The other pairs of lines are neither parallel nor perpendicular.
Answered by GPT 3.5
You are correct, and I apologize for the error in my response. Lines with positive slopes that are not at right angles are simply not perpendicular. Thank you for bringing this to my attention.