Question 1091255
<pre>

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 {{{y=m*x+b}}}:

{{{9x+6y=174}}}
{{{6y=-9x+174}}}

{{{expr(6/6)y=expr((-9)/6)x+174/6}}}

{{{y=expr(-3/2)x+29}}}

We compare that to the slope-y-intercept form 

{{{y=m*x+b}}}

we see that the slope is {{{-3/2}}}

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 {{{-3/2}}} has slope {{{"" + 2/3}}}.

Now we use the point-slope formula

{{{y-y[1]=m(x-x[1])}}} with (x<sub>1</sub>, y<sub>1</sub>) = (18,14),
and {{{m=2/3}}}

{{{y-14=expr(2/3)(x-18)}}}

Multiply both sides by 3 to clear the fraction:

{{{3y-42=2(x-18)}}}

{{{3y-42=2x-36}}}

{{{-2x+3y=6}}}

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 

{{{2x-3y=-6}}}

{{{drawing(480,600,-3,25,-3,32,
graph(480,600,-3,25,-3,32),
circle(18,14,.2),locate(18,14,"(18,14)"),
green(line(-8,41,28,-13)), red(line(27,20,-6,-2)) )}}}

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</pre>