Substituting in the Point-Slope Form of a Linear Equation

Date: 02/05/2005 at 08:34:04
From: Julie
Subject: Point-slope equation

In the point-slope equation y - y1 = m(x - x1) why is the point you 
are using, such as (3,-5), always substituted for the y1 and x1?  Why
can't it be substituted for the 'plain' x and the 'plain' y?

I teach algebra and the students asked me the above question and I 
didn't really have a good answer, except that is 'math tradition' 
(smile).  Thanks.

Date: 02/08/2005 at 13:20:16
From: Doctor Justin
Subject: Re: Point-slope equation

Hi Julie,

The short answer is that you can substitute the known point in for 
(x,y) or for (x1,y1) and you will get the same final result for your
equation.  The only way you can go wrong is if you substitute for
(x,y1) or (x1,y), which is a common mistake for early algebra students.

Here are some ideas on why it doesn't matter which of the two correct
substitutions you use.  If you accept that the slope of a line is the
same for every pair of points (a, b) and (c, d) on the line, then you
can make an equation using our definition of slope.  

The equation I'm talking about is the one you mentioned above,
y_0 - y_1 = m(x_0 - x_1).  Once you have the value for m, since m is
the same everywhere, we can substitute the variable point (x, y) and
the known point (a, b) into the equation.  

It does not matter if you substitute the variable point into the 
equation for (x_0, y_0) or (x_1, y_1).  Let me show you why:

Consider the line with points (a, b) and (c, d).  The slope of this 
line is,

            b - d    -(d - b)   d - b
       m = ------- = -------- = ----- 
            a - c    -(c - a)   c - a

That's some proof that the order of the substituted points doesn't 
matter, but let's take it a little further by substituting the 
variable point (x, y) and the known point (a, b) into the equation.

       y - b = m(x - a)

Factor out a -1,

    -(b - y) = -m(a - x)

Multiply both sides by -1,

       b - y = m(a - x)

So, as you can see, whether you put (a, b) or (x, y) in for (x_0, y_0) 
doesn't really matter.  The same equation will still follow.  

I hope this helps.

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