Question

The lines with the given equations below are given here; are they parallel, perpendicular, or neither? (1) `2x-5y=8, 5x-2y=2` (2) `y=5/6x+8, y= -6/5x-4` (3) `x-2y=12, 3x-6y=10`

Answer 1

1. Neither
2. Perpendicular
3. Parallel

Explanation:

Parallel lines have the same slope (y=mx+b, where m is the slope) and different y-intercepts, perpendicular lines have slopes that are opposite (negative/positive) reciprocals (multiplicative inverses) of each other.

Examples:

Parallel slopes: 3 and 3

Perpendicular slopes: 2 and `-1/2`

Neither: 4 and -8

To find out if the first two lines are parallel or perpendicular or neither, first put both equations into slope-intercept (y=mx+b) form.

Problem 1:
`2x-5y=8`

`2x=8+5y`

`5y=2x-8`

`y=2/5x-8/5 rarr` The first line has a slope of `2/5`

`5x-2y=2`

`5x=2+2y`

`2y=5x-2`

`y=5/2x-1 rarr` The second line has a slope of `5/2`, this is not the opposite reciprocal of the first slope or the same as the first slope; these two lines are neither perpendicular or parallel

Problem 2:

`y=5/6x+8 rarr` The slope is `5/6`

`y=-6/5x-4 rarr` The lines are perpendicular; `-6/5` is the opposite reciprocal of `5/6`

Problem 3:

`x-2y=12`

`x=2y+12`

`2y=x-12`

`y=1/2x-6 rarr` The first line has a slope of `1/2` and a y-intercept of -6

`3x-6y=10`

`3x=6y+10`

`6y=3x-10`

`y=3/6x-10/6`

`y=1/2x-5/3 rarr` This line has a slope of `1/2` (same as the first one), but a different y-intercept. These two lines are parallel.