Expert: Paul Klarreich - 3/13/2007

Question
solve:

7x/x-3 - 21/x +11= 63/x^2-3x

Answer
Questioner:   Mallory
Category:  Advanced Math
 
Subject:  MTH 110
Question:  solve:

7x/x-3 - 21/x +11= 63/x^2-3x
...................................
Hi, Mallory,

It's good that you told me what math course you are taking, but unfortunately, MTH 110 doesn't mean much without a link to your college catalog.  Nevertheless:

I assume your example says:  [USE COURIER FONT TO VIEW THIS.]
7x      21            63
----- -  -- + 11 =  --------
x - 3     x         x^2 - 3x

But it could mean other things.  Next time PARENTHESIZE to avoid misunderstanding.

You are going to CLEAR FRACTIONS using a Lowest Common Denominator.  To find that LCD, you will have to factor:

x^2 - 3x = x(x - 3)  which contains each of the other denominators as a factor, so IT IS THE LCD.

Now multiply every term by your LCD, keeping it in factored form:

x(x-3) 7x       x(x-3)21                   63x(x-3)
     -----   -       -- +  x(x-3)11   =  --------
     x - 3            x                  x(x - 3)

You will see mucho cancellation taking place:

x..... 7x   -   ..(x-3)21 + x(x-3)11  = 63.......

where the dots are in the places where the cancelled factors formerly were.

Your equation is now:

7x^2 - 21x + 63 + 11x^2 - 33x = 63

Simplifying:

18x^2 - 54x = 0

x^2 - 3x = 0

x = 3,  x = 0.

Uh oh.  This is BAD NEWS.  These numbers won't ever check (remember checking?) because they make the denominators of your equation zero, which is a no-no.

These are called EXTRANEOUS ROOTS.  They pop up as a result of the multiplication.  As it happens, this equation has NO SOLUTIONS.  No wonder you had trouble.