Here,
we are looking for a line perpendicular to 2x - y = 6 and passes through the point (0 , 0).
2x - y = 6
-y = -2x + 6
y = -(-2x + 6)
y = 2x - 6
Hence comparing to y = mx + c, the slope is m and in y = 2x - 6, the slope = 2
But for line to be perpendicular to one whose slope is 2, we would have to take the reciprocal and negate it.
Reciprocal of 2 = 1/2
Negate it = -1/2
Hence slope of the perpendicular line = -1/2
and since it passes through (0, 0)
y = mx + c
y = (-1/2)x + c, substituting x = 0 and y = 0
0 = (-1/2)*0 + c
0 = 0 + c
0 = c
c = 0
y = (-1/2)x + c = (-1/2)x + 0 = -(1/2)x
So the equation of the perpendicular line to 2x - y = 6 and passes through the point (0 ,0) is y = (-1/2)x
Hope this explains it.