Formulas for the Equation of a Line

Date: 09/28/98 at 23:25:57
From: Michelle
Subject: Algebra II: point slope formula

My question isn't about how to use point slope formula, but more of 
what it is used for. Given a problem, I can accurately use:

   y - y1 = m(x - x1) 

to get the answer, but I don't understand what its specific meaning 
is. Is it just another way of getting to a y = mx + b style formula 
suitable for graphing, or does it serve some other purpose?

Thank you,
Michelle

Date: 09/29/98 at 12:56:20
From: Doctor Peterson
Subject: Re: Algebra II: point slope formula

Hi, Michelle. Good question! Too often people learn formulas but don't 
stop to ask what they are for.

There are several different ways to write the equation of a line, and 
each is designed to be used when you have certain pieces of information 
to start with. You could do everything with one form, such as slope-
intercept, but often it's easier to use a form specially designed for 
one case. For example, there are:

   Slope-intercept:    y = mx + b

   Slope-x-intercept:  y = m(x - a)

   Point-slope:        y - y1 = m(x - x1)

   Two-point:          y - y1   y2 - y1
                       ------ = -------
                       x - x1   x2 - x1

   Two-intercept:       x     y
                       --- + --- = 1
                        a     b

These are all equivalent, and which one you use just depends on what 
you are given. But you don't have to memorize all of them. If you 
understand how graphs work, you can figure everything out when you need 
to. All these forms (except for the two-intercept form) are just ways 
of saying that the slope is constant. Either you are given the slope 
(m), or you figure out the slope (y2 - y1) / (x2 - x1), and then you 
compare the general point (x, y) with either an intecept (0, b) or 
(a, 0), or a general point (x1, y1).

So the "meaning" of the point-slope form is simply that you can get the 
rise (y - y1) by multiplying the run (x - x1) by the given slope. You 
can graph it as it stands (by identifying the point and the slope), or 
you can change it into any other form you want. It's yours to use any 
way you like.

I hope that helps!

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