Finding Equation of a Line Given the Slope and a Point on It

Date: 08/24/2006 at 19:26:45
From: Sara
Subject: getting the equation of a line from y=mx+b to ax+by+c=0

I'm trying to figure out how to convert a line from slope-intercept
form to general form, ax + by + c = 0.  I'm given one set of points
and the slope: (2,-1) and m = 1/4.  I don't know where to start. 
Thanks!

Date: 08/24/2006 at 21:24:26
From: Doctor Rick
Subject: Re: getting the equation of a line from y=mx+b to ax+by+c=0

Hi, Sara.

Let's work through a different problem so that you can do yours on 
your own.  I'll do

  (-4,2) m = 3

First of all, many of us would start with the point-slope form, not 
the slope-intercept form, because we know a point and the slope, not 
the slope and the y-intercept.  The point-slope form is

  y = m(x-a) + b

where the slope is m and the point is (a,b).  Plugging in the numbers, 
I get

  y = 3(x - -4) + 2

I simplify this equation; simplification pretty much leads me to the 
slope-intercept form, because that's the simplest form.

  y = 3(x + 4) + 2
  y = 3x + 3*4 + 2
  y = 3x + 14

If you don't want to remember one more formula, you can certainly use 
the slope-intercept form, but it takes more than just plugging in 
numbers.  I start by plugging in the slope, which I know:

  y = mx + b
  y = 3x + b

To find b, I use the fact that (-4,2) is a solution to the equation. 
I replace x by -4 and y by 2:

  2 = 3*(-4) + b

Then I solve this equation for b:

  2 = -12 + b
  2 + 12 = b
  14 = b

Going back to the incomplete slope-intercept form, I can now replace 
b by 14:

  y = 3x + b
  y = 3x + 14

That's the same answer I got the first way.  It's your choice: whether 
you want to have to remember more (the point-slope form) or do more 
work (solving for b).

Now we've got the equation in simplest (slope-intercept) form.  We 
want to put the equation

  y = 3x + 14 into the form

  ax + by + c = 0

This is a bit odd because it means making the equation *less* simple.  
Let's start by making one side zero, since we know we want that.  I'll 
make the left side zero (because it has less on it to start with), 
then switch sides:

  0 = 3x + 14 - y
  3x + 14 - y = 0

Now let's rearrange the terms to the correct order (term containing 
x, term containing y, constant term):

  3x - y + 14 = 0

To me, that's all we need to do.  If it doesn't look like the exact 
form to you, remember that subtracting a number is the same as 
adding the opposite of it:

  3x + -y + 14 = 0

and that the opposite of a number can be obtained by multiplying it 
by -1:

  3x + -1y + 14 = 0

Now we can see explicitly that a = 3, b = -1, and c = 14.

For the sake of neatness and uniformity (and to help the teacher by 
having everyone write the answer in exactly the same form), we like 
to make all the coefficients (a, b, and c) integers if possible-- 
it isn't always possible--and to make "a" positive.  If I had gotten 
to

  -x + (1/3)y - 14/3 = 0

I would want to multiply through by the denominator, 3:

  3(-x + (1/3)y - 14/3) = 3*0
  3(-x) + 3(1/3)y - 3(14/3) = 0
  -3x + y - 14 = 0

Then I'd multiply by -1 to make the coefficient of x positive:

  -1(-3x + y - 14) = -1*0
  3x - y + 14 = 0

I'd like to see you do your problem following the same method (or 
rather, one of these methods) to see if you've got the idea.  If 
you're still confused about something, feel free to ask me about it.

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