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### Don't Multiply by Zero

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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

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/
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Associated Topics:
High School Basic Algebra

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