Write an equation of the line through (7,-7) and perpendicular to 6y=x-12

To write an equation through a point, it would be easiest to use point-slope form (y - y1) = m(x - x1). Since we already know x1 and y1 (7, -7), we just need to find slope.
Slopes of perpendicular lines are negative inverses, so to find the slope of the line, first find the slope of the line that we are given:
6y = x - 12
y = (1/6)x - 2
So our given slope is (1/6). The negative inverse of this slope is -6. So now we have everything needed to write an equation for the line through (7, -7).
(y - (-7)) = -6(x - 7)
y - 7 = -6x +42
y = -6x + 49