Equations That Make No Sense

Date: 01/15/2003 at 00:16:37
From: Nicholas
Subject: Rational Equations 

   x-8/5 = x+2/6

I would assume you find the lowest common denominator of 5 and 6. Then 
multiply it by x. Not sure if it's correct though.

Date: 01/16/2003 at 11:30:46
From: Doctor Ian
Subject: Re: Rational Equations 

Hi Nicholas,

You could certainly proceed by finding a common denominator, but it
would be kind of like driving to New York on your way from Chicago to
Los Angeles.

Let's write the equation in a slightly fuzzier way:

  x - this = x + that

Think about the number line for a moment.  Suppose we choose a number
on it, somewhere:

   <-------------|------------->
                 x

If we subtract something from it, we end up to the left of where we
started:

   <---|----------|------------->
     x-this       x

And if we add something to it, we end up to the right of where we 
started:

   <---|----------|---------|---->
     x-this       x        x+that

Is there _anywhere_ on the number line that we could start so that
these two new points are in the same place?  This is what we're asking
when we write the equation

  x - this = x + that

There is no such place. So there is _no_ value of x that will make
this equation true. How can we show that 'by algebra'? 

We can start by noting that we can subtract the same thing from both
sides of an equation. Why not subtract x from both sides? Then we get

      x - 8/5 - x = x + 2/6 - x

             -8/5 = 2/6

Now, this clearly isn't true. But we derived it from the original
equation using valid rules. So what does that mean? It means that
the original equation can't be true, either.  

As I said, we could also proceed by finding a common denominator for
the fractions:

              x - 8/5 = x + 2/6 

            x - 48/30 = x + 10/30

    x - 48/30 + 48/30 = x + 10/30 + 48/30

                    x = x + 58/30

And again, we have something that obviously can't be true. There is
no number such that you can add a non-zero number to it, and end up
with the same thing.  

So what should you learn from this? One important lesson, I think, is
that it's possible to write equations that make no sense, just as it's 
possible to write sentences that make no sense: 

   Context, Language, and False Equations
   http://mathforum.org/library/drmath/view/52359.html 
  
A second important lesson is that before you jump in and start
executing the first technique that occurs to you, it's a good idea to
step back and think about whether there might be an easier way to get
the answer you're looking for:

   Factoring vs. an Equation
   http://mathforum.org/library/drmath/view/61333.html 

I hope this helps.  Write back if you'd like to talk more about this,
or anything else. 

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