Don't Multiply by Zero

Date: 01/23/2002 at 21:36:48
From: Richard
Subject: A meaningless answer from a fractional equation

Here's the problem:

   3y/y-7 - 3/2 = 21/y-7

solve for y.  

Here's what I did.

I multiplied both sides of the equation by the LCD (Lowest Common 
Denominator), which is 2(y-7). After doing that, I got
 
   6y - 3y + 21 = 42

Then I solved for y:

   3y + 21 = 42
        3y = 21
         y = 7  

But wait a minute, if I plug 7 in for all the y's, I get a total 
answer of 21/0 - 3/2 = 21/0.

What kind of answer is this? Is it a meaningless solution?  

Date: 01/23/2002 at 22:53:06
From: Doctor Peterson
Subject: Re: A meaningless answer from a fractional equation

Hi, Richard.

It's not really a paradox; just a warning to be careful when you 
multiply. Remember that it is only legal to multiply both sides of an 
equation by a NON-ZERO value. So when you finish solving the problem, 
you have to go back and check whether 2(y-7) was zero. Since it was, 
the solution you found is not valid, and there is no solution to the 
equation.

When this happens, I like to try to find another way to work it out 
without doing the illegal step. One approach is to simplify the left 
side by adding the fractions:

    3y/(y-7) - 3/2 = 21/(y-7)

    6y - 3(y-7)     42
    ----------- = ------
       2(y-7)     2(y-7)

Now we can subtract the right side, using the common denominator:

    6y - 3(y-7) - 42
    ---------------- = 0
         2(y-7)

    3y - 21
    ------- = 0
     2(y-7)

    3(y-7)
    ------ = 0
    2(y-7)

Now, if y is not 7, we can further simplify this by canceling the 
common factor:

    3/2 = 0

This is, as I hoped, an equation with no solution, obtained without 
doing anything questionable. It took a little more work this way, but 
gives me more confidence in saying there is no solution.

I hope that gives you a little more confidence, too!

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