Equation of a Line

Date: 01/27/98 at 20:57:30
From: amanda
Subject: Equation of a line

How do I find the equation of a line through origin perpendicular to 
2x + 3y = 5 ?

Date: 01/28/98 at 13:09:19
From: Doctor Jaffee
Subject: Re: Equation of a line

There are several properties of equations of lines that you have to 
know in order to understand how to solve this problem.

First, any time you have an equation in the form y = mx + b, where 
m and b are real numbers, m will be the "slope" of the line and b will 
be the "y-intercept" of the line.

So if we take your equation 2x + 3y = 5 and add -2x to both sides as 
follows:
       -2x + 2x + 3y = -2x + 5         we get
                  3y = -2x + 5         then divide both sides by 3 
                                       and the result is                 
                   y = (-2/3)x + 5/3 

This means that the slope of the line (our value of m) is -2/3 and the 
y-intercept (our value of b) is 5/3. 

Check the Dr. Math archives, select the link to the high school 
section, and then the link to "equations, graphs, and translations" - 
the question from 6/28/96 entitled graphing y = mx + b provides some 
more information about this topic. Also, it would be worth your while 
to try the link entitled "linear equations" in the high school 
section. Look for the problem dated 12/1/94.

Now, back to your problem.  The second important property of equations 
of lines is that if two lines are perpendicular, their slopes are 
negative reciprocals of each other. This means that if one of the 
slopes is negative, the other will be positive, and furthermore, once 
you adjust the sign you must "flip" the number you get. In the 
previous paragraph we calculated that the slope of the given line was 
-2/3, so to find the slope of the perpendicular line we change the 
-2/3 to 2/3 and then flip it so that it ends up as 3/2, which will be 
the slope of the line we are looking for.

Finally, you said that the line goes through the origin, so the 
y-intercept of the line will be 0. If we substitute 3/2 for m and 
0 for b in the equation y = mx + b, the result is y = (3/2)x + 0 or 
y = (3/2)x which is the answer to the problem.

I hope this has helped.
 
-Doctor Jaffee,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/