Is the graph of x-3y=6 and x-3y=9 parallel, perpendicular, or neither? is the graph of y=4x+1 and y=-4x-2 parallel, perpendicular, or neither?

x - 3y = 6 and x - 3y = 9 ⇒ Parallel

y = 4x + 1 and y = -4x - 2 ⇒ Neither

Step-by-step explanation:

Parallel lines have the same slope.  Given the definition of parallel lines, it is easier to tell that the graph of x - 3y = 6 and x - 3y = 9 will have parallel lines, since both equations have the same slope of -1/3.

Perpendicular lines have negative reciprocal slopes.  This means that if you multiply the slopes of both equations, it will result with a product of -1. Another important aspect of perpendicular lines is that they have a single point of intersection, and that they intersect at a 90° angle.

The graph of equations y = 4x + 1 and y = -4x - 2  intersect, but are not perpendicular from each other because they do not have negative reciprocal slopes. The negative reciprocal slope of 4 is -¼; likewise, the negative reciprocal slope of -4 is ¼.

Therefore, the graph of y = 4x + 1 and y = -4x - 2 are neither parallel or perpendicular.