Perpendicular line equation for (0,0); 2x-y=6

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.