Question 1032649
<pre><b>
That's no explanation at all!  It's just a magic trick
method that works.
9x - 3y = -18
{His magic trick: Swap the letters on the
left 

9y - 3x 

change one of the signs, 

9y + 3x

substitute in
the point (1,5) 

9(5) + 3(1) = 45 + 3 = 48

set 9y + 3x equal to 48


9y + 3x = 48

Divided through by 3

3y + x = 16

and abracadabra, you have the answer!

But that's math-a-magic!, not mathematics.

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We are given a point (1,5) and the equation of a line that we 
can easily graph by finding its intercepts (-2,0) and (0,6).

like this:

{{{drawing(3600/17,400,-5,4,-5,12,
locate(1,5.6,"(1,5)"),
graph(3600/17,400,-5,4,-5,12), line(8,30,-6,-12),

circle(1,5,0.15),circle(1,5,0.13),circle(1,5,0.11),circle(1,5,0.09),circle(1,5,0.07),circle(1,5,0.05),circle(1,5,0.03),circle(1,5,0.01) )}}}  

and we want a line that goes through the given point
that is perpendicular to the given line. In other words,
we want this red line:

{{{drawing(3600/17,400,-5,4,-5,12,
locate(1,5.6,"(1,5)"),
graph(3600/17,400,-5,4,-5,12), line(8,30,-6,-12),
red(line(-11,9,10,2)),
circle(1,5,0.15),circle(1,5,0.13),circle(1,5,0.11),circle(1,5,0.09),circle(1,5,0.07),circle(1,5,0.05),circle(1,5,0.03),circle(1,5,0.01) )}}} 

The given line has equation

{{{9x - 3y = -18}}}

We get it in slope-intercept form by solving
for y:

{{{-3y=-9x-18}}}

Divide through by -3

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

Compare that to

{{{y=mx+b}}}

So the given line's slope is m=3.

m=3/1 has a rise of 3 and a run of 1.  The slope is
positive so it slopes uphill to the right.

A line perpendicular to it will have the rise and
run swapped but it will slope downhill to the right.
So its slope will have the run and rise swapped as
1/3 and will be negative because it slants downhill
to the right.  So its slope is -1/3.

Then we use the point-slope formula:

Point-slope formula:

y - y<sub>1</sub> = m(x - x<sub>1</sub>)

where m=-1/3 and (x<sub>1</sub>,y<sub>1</sub>) = (1,5)

{{{y-(5)}}}{{{""=""}}}{{{expr(-1/3)(x-1)}}}

{{{y-5}}}{{{""=""}}}{{{expr(-1/3)*x+1/3)}}}

Multiply through by LCD = 3

{{{3y-15}}}{{{""=""}}}{{{3*expr(-1/3)*x+3*expr(1/3))}}}

{{{3y-15}}}{{{""=""}}}{{{cross(3)*expr(-1/cross(3))*x+cross(3)*expr(1/cross(3)))}}}

{{{3y-15}}}{{{""=""}}}{{{-x+1))}}}

Add x to both sides and then 15 to both sides:

{{{3y + x = 16}}}

Same as with the magic trick.

Edwin</pre></b>