First we find the slope of the line 9x+6y=174
by solving it for y to get it in slope-y-intercept form,
which is :








We compare that to the slope-y-intercept form 



we see that the slope is 

Then we remember that perpendicular lines have slopes 
which are opposite signed reciprocals of each other.

Therefore the slope of a line perpendicular to a line
with slope  has slope .

Now we use the point-slope formula

 with (x1, y1) = (18,14),
and 



Multiply both sides by 3 to clear the fraction:







We want it to begin with a positive term in x, so that
it will be in the same form as the given equation, so we
we multiply through by -1 





The Green line is the graph of the Given line 9x+6y=174.
The Red line is the graph of the Required line 2x-3y=-6.

Edwin