Find Equation for Two Points

Date: 12/12/2002 at 20:39:59
From: Elissa Martin
Subject: Pre-Algebraic equations and lines

I want to know how to find an equation for two given coordinate 
points. I saw the other answer you had for Lauren but it was too hard 
to understand. Will you please explain it to me using the standard 
equation and explain as if I were a fifth-grader?

Thanks a bunch,
Elissa

Date: 12/13/2002 at 18:34:20
From: Doctor Ian
Subject: Re: Pre-Algebraic equations and lines

Hi Elissa,

If you have two points, 

             |           B
             |
             |
             |   A    
             |
             |
             | 
  -----------+----------------
             |

and you want to find the equation of the line that runs through them,
the place to start is by finding the slope.  

To find the slope, you figure out how far you have to move along each
axis to get from one point to the other:


             |           B
             |           .
             |           . change in y
             |   A........    
             |     change
             |     in x
             | 
  -----------+----------------
             |

How do you find those?  By subtracting the coordinates of the points.
 If A is (x_a, y_a) and B is (x_b, y_b) then we have


             |               B
             |               .
             |               .  y_b - y_a
             |               . 
             |   A............    
             |     x_b - x_a
             |    
             | 
  -----------+-------------------
             |


The slope is the ratio of these:

          change in y   (y_b - y_a)
  slope = ----------- = -----------
          change in x   (x_b - x_a)

Once you have the slope, you're halfway done, because the equation of
a line looks like 

  y = slope * x + intercept

Well, think about what it _means_ to have an equation like
  
  y = slope * x + intercept

It means that if you know the value of x, you can plug that in and get
the corresponding value for y. Or, if you know the value of y, you
can plug that in and get the corresponding value for x.  

And this has to be true for _every_ point on the line, including the
two points we started with. So we can substitute those into the
equation, 

                y_a = slope * x_a + intercept

  y_a - slope * x_a = intercept

We know everything on the left side, so we can use it to solve the
right side. And once we know the right side for _any_ point, we know
it for _all_ points.  

So let's try it with an example. Suppose our points are (2,3) and
(4,7).  We can find the slope:

          7 - 3   4
  slope = ----- = - = 2 
          4 - 2   2

So the equation must be 

  y = 2 * x + intercept

Now I can take either point, and substitute for it:

       7 = 2 * 4 + intercept

       7 = 8 + intercept

   7 - 8 = intercept

      -1 = intercept

So the equation must be

  y = 2x - 1

We can check this by trying both points:

  3 = 2 * 2 - 1

    = 4 - 1               (Check!)

and

  7 = 2 * 4 - 1

    = 8 - 1               (Check!)


Does this make sense?  If it's still too complicated, you might want
to look at our Chameleon Graphing lesson on Lines and Slope:

   Graphing Points First - Ursula Whitcher
   http://mathforum.org/cgraph/cslope/pointsfirst.html 
   
I hope this helps.  Write back if you'd like to talk more
about this, or anything else.

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