Question

How do you determine if the lines `-12x+9y=15` and `-4x+3y=5` are paralle , perpendicular or neither?

Answer 1

I would look at the slope of the lines represented by these equations (it representes the inclination of the lines):

first write them isolating `y`;
1) `y=12/9x+15/9=4/3x+5/3`
and
2) `y=4/3x+5/3`

The slope is the coeficient of `x`;

You can have:
1) same slope=parallel lines;
2)slope1=-1/slope2 =perpendicular lines;
3) slope1`!=`slope2 crossing lines.

In this case they are parallel (and coincident).