Question

How do you determine whether the lines for each pair of equations 3x+2y=-5 y=-2/3x+6 are parallel, perpendicular, or neither?

Answer 1

The lines are not parallell, nor are they perpendicular.

Explanation:

First, we get the two linear equations into `y=mx+b` form:

`L_1: y=-2/3x+6 -> m=-2/3`

`L_2: 3x+2y=-5`

`L_2: 2y=-3x-5`

`L_2: y=-3/2x-5 -> m=-3/2`

If the lines were parallell, they would have the same `m`-value, which they don't, so they cannot be parallell.

If the two lines are perpendicular, their `m`-values would be negative reciprocals of each other. In the case of `L_1`, the negative reciprocal would be:
`-1/(-2/3)=-(-3/2)=3/2`

This is almost the negative reciprocal, but we're off by a minus sign, so the lines are not perpendicular.

Answer 2

Neither parallel nor perpendicular

Explanation:

Rearranging the `1` st equation as `y=mx+c`,we get,

`y=-3/2x -(5/2)` hence, slope =`-3/2`

the other equation is, `y=-2/3x+6` ,slope is `-2/3`

Now,slope of both the equations are not equal,so they are not parallel lines.

Again,product of their slope is `-3/2 * (-2/3)=1`

But,for two lines to be perpendicular, product of their slope has to be `-1`

So,they are not perpendicular as well.