Equation of a Line

Date: 02/14/2002 at 00:32:12
From: Victor Squillacioti
Subject: Proofs

Prove the point-slope equation property, which is y-y1 = m(x-x1). 

It gave me a hint, but it wasn't very useful to me. It was:

Show that any point (x2,y2) on the line described must satisfy the 
equation, and that any point satisfying the equation must be on the 
line described.

Please give me any useful information to help me. Thank you in 
advance.

Victor Squillacioti

Date: 02/14/2002 at 16:52:15
From: Doctor Peterson
Subject: Re: Proofs

Hi, Victor.

To show that this is the equation of the line through (x1,y1) with 
slope m, you have to show two things:

  (1) every point on the line satisfies the equation;
  (2) every point that satisfies the equation is on the line.

That is what the hint said.

Now, what does it mean for a point to be on the line? Since we know 
the slope of the line, which means that the slope between any two 
points on the line has the value m, we can say that the slope between 
points (x1,y1), which is given as on the line, and the point (x2,y2) 
which we are assuming is on the line, will be m. Write an equation 
that says this, and use that to show that the equation will be true 
when x = x2 and y = y2. You will have shown that (1) is true.

Now to go the other way, suppose that you have a point (x2,y2) that 
satisfies the equation. Write out what the equation tells you about x2 
and y2. You want to show that the slope between (x1,y1) and (x2,y2) is 
m, which will prove that (x2,y2) is on the line. Write out the 
definition of this slope, and see if you can show it is equal to m.

Here is some information on the equations of a line that touches on 
this idea:

   Formulas for the Equation of a Line
   http://mathforum.org/dr.math/problems/michelle9.28.98.html   


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