Changing from Slope-Intercept to Standard Form

Date: 06/02/2005 at 00:35:36
From: Tiffani
Subject: standard form Ax+By+C=0

Hi Doctor Math.  I know a little about standard form, but I am having
a hard time going from slope-intercept form to standard form.  I'm
trying to do a line with a slope of -2/3 and a y-intercept of 8/3, so
I know the slope-intercept form is y = (-2/3)x + 8/3.  But how do I
get that into Ax + By + C = 0 form?

If you can help me, I would greatly appreciate it.

Date: 06/02/2005 at 23:05:57
From: Doctor Peterson
Subject: Re: standard form Ax+By+C=0

Hi, Tiffani.

The process is something like how you solve an equation, but in 
reverse.  In solving an equation, you want to transform it into an
equivalent equation of the form "y = ?", which you do by taking a
series of steps that gradually get y by itself.  What you want to do
here is to transform it into an equation of the form "?x + ?y + ? =
0", which means that x and y and the constant are all on the same
side, rather than one variable being alone on the right.

Let's take an example.  Say we're given

  y = (1/2)x - 2/5

We want x, y, and the constant on the same side, so let's just 
subtract y from both sides and we get

  0 = (1/2)x - y - 2/5

Everything looks right except that the zero is on the left; I've
already arranged for the terms on the right to be in the right order.
So we can just write

  (1/2)x - y - 2/5 = 0

Now, there's one thing we can do to make this look nicer; we'd like A,
B, and C to be whole numbers rather than fractions, since fractions
complicate things.  To eliminate fractions, we can multiply the whole
equation by the LCD, which is 10.  Do that, and we get

  (10/2)x - 10y - 20/5 = 0

  5x - 10y - 4 = 0

And we're done!

It was really pretty easy: subtract y from both sides and then
multiply by the LCD.  But more important than the specific steps is 
the basic principle, which applies to many kinds of problems: 
visualize the form you want to end up with, then do one step at a time 
to get it that way.  We first saw that the y was not with the other 
terms, so we fixed that; then we fixed up the order; and then we got 
rid of fractions, which were not part of our image of the ideal 
"standard" form.

If you have any further questions, feel free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 06/02/2005 at 23:43:48
From: Tiffani
Subject: Thank you (standard form Ax+By+C=0)

Thank you Dr. Math, I think I finally understand it.  If I have any
more problems, I'll be sure to ask but this was very helpful.  Thanks
again.