Are the following lines parallel, perpendicular, or neither?

x-6y=12
6x+y=6

To determine if the given lines are parallel, perpendicular, or neither, we need to examine their slopes.

First, let's rewrite the equations in slope-intercept form (y = mx + b), where "m" represents the slope:
1) x - 6y = 12 => -6y = -x + 12 => y = (1/6)x - 2
2) 6x + y = 6 => y = -6x + 6

Now that we have the equations in slope-intercept form, we can compare their slopes. The slopes are the coefficients of "x" in each equation.

For our first equation, y = (1/6)x - 2, the slope is (1/6).
For our second equation, y = -6x + 6, the slope is -6.

Parallel lines have the same slope, so if the slopes of the two lines are equal, then the lines are parallel.
Perpendicular lines have slopes that are negative reciprocals of each other, so if the slopes multiply to -1, then the lines are perpendicular.

Comparing the slopes in this case, we see that (1/6) * -6 = -1.
Since the slopes multiply to -1, we can conclude that the two lines are perpendicular.