Parallel and Perpendicular Lines

Date: 01/14/99 at 20:51:19
From: Anonymous
Subject: Algebra: linear equations

How do you tell if the graphs of these equations are parallel, 
perpendicular, or neither without graphing?

   4y - 5 = 3x + 1 and 12 = -6x + 8y - 3

Date: 01/15/99 at 15:47:54
From: Doctor Teeple
Subject: Re: Algebra: linear equations

Hello,

Thanks for writing to Dr. Math.

The best way to get information on how two lines relate to each other 
(i.e.: if they are parallel, perpendicular, or neither) is to look at 
their slopes. And the easiest way to find the slopes is to get your 
equations into slope intercept form, which means they look like this:

   y = m x + b

Then we can read the slope right off the equation: it is the number 
represented by m. 

I'll help you get one into this form:

Start with 4y - 5 = 3x + 1.
Then add 5 to both sides: 4y = 3x + 6.
Then divide both sides by 4: y = 3/4 x + 6/4.

This means that the slope is 3/4. 

Try the other one on your own and reduce the slope to the smallest 
possible fraction. 

Now, what does this mean? If the slopes are the same, it means the two 
lines are parallel. If one slope is the negative reciprocal of the 
other, then the two lines are perpendicular. To get the negative 
reciprocal of a number, put one over the number and then make it 
negative. So:

   if the number is      its negative reciprocal is
         2                        -1/2
        3/10                     -10/3
       -3/2                        2/3
        
So if the two lines had these respective slopes, they would be 
perpendicular. 

If the slopes are neither the same nor negative reciprocals, the lines 
are neither parallel nor perpendicular. 

If you need more about putting equations in slope intercept form or on
parallel and perpendicular lines, try searching the archives. For 
example, I found:

   Using the Slope-Intercept Formula
  http://mathforum.org/dr.math/problems/plotkin27.html   

   Perpendicular/Parallel Lines
  http://mathforum.org/dr.math/problems/hartge1.17.97.html   

If you need more help with these ideas, please write back.

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