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I Solved It. Now What?

Date: 10/16/2003 at 15:56:37
From: Rachel
Subject: algebra

When I see an equation like

  12 = 5x + x + 6

I have no idea what this means and don't have any idea on how to 
figure it out.  Here's what I would try:

  12 = 5x + x + 6

  12 = 6x + 6
  -6       -6        <-- Subtract 6 from 12 and from 6
  --   ------
   6     6           <-- Divide 6 and 6x by 6

   x = 1

Now what?


Date: 10/16/2003 at 16:12:50
From: Doctor Ian
Subject: Re: I guess algebra

Hi Rachel, 

You seem to know more than you think, because your solution was right
on the money.  Nicely done. 

Now let's think about what this final equation is telling us.  Every
time you made a change to an equation, it was a change that preserved
the _meaning_ of the equation, right?  So if you end up with 

  x = 1

that means that

  12 = 5x + x + 6

is really just a fancy way of saying that x is equal to 1!  We can
verify that by substituting 1 for x:

  12 = 5x + x + 6

     = 5*1 + 1 + 6

     = 5 + 1 + 6

     = 12

That is, when x has a value of 1, you get a true statement.  For any
other value you choose, the statement will be false.  (Try some and
see what happens.)

Here's what you're doing, but in terms of language instead of
equations.  Suppose I start with a really complicated sentence like 

    This belongs to the third son of the woman married to
    the man who is married to my mother. 

Now, my mother is married to my father.  So the man who is married to
my mother is my father, and I can shorten it to

    This belongs to the third son of the woman married to
    my father. 

But the woman married to my father is just my mother!

    This belongs to the third son of my mother. 
   
But I'm the third son of my mother:

    This belongs to me.

At each stage, I made the language a little simpler, but I didn't
change the meaning at all.  So when I get to the end, I know that each
sentence means the same thing as the one before it, which means that
the first sentence and the last sentence are just two ways of saying
the same thing.  

And that's exactly what's going on in algebra, when you transform a
more complicated equation into a simpler one.  

This, by the way, is one reason that it's good to get into the habit
of writing a series of transformed equations, rather than trying to
work 'in place', as you were doing.  For example, instead of doing this, 

  12 = 5x + x + 6

  12 = 6x + 6
  -6       -6        

I'd go ahead and write a new equation, which I would then simplify:

  12 - 6 = 6x + 6 - 6
  
       6 = 6x
 
Next, I'd do the division using a new equation, and simplify again:

     6/6 = (6x)/6

       1 = x

This takes a little longer, but it's much easier to go back over your
work and see that each step makes sense, by comparing each equation to
the one before it.  This is especially important if you're showing
your work to someone else. 

So what's the point of doing all this?  What can we do with the
information that x is equal to 1? 

Well, in your math classes, the original equations are just made up by
the authors of your textbooks to give you practice; but in real life,
they arise from situations where we know something _about_ some
quantity, but we don't yet know the quantity itself.  So it's kind of
like detective work:  We start with a description like

  The murderer was a left-handed professional golfer who was 
  born in Sweden between 1958 and 1972, walks with a limp, 
  and drives a green Volkswagen Jetta.  

What we _want_ to do is get from that statement to one that looks like

  The murderer was Lars Edberg.  

What we _do_ with that information depends on why we wanted to know it
in the first place.  In the case of solving a murder, it's usually
pretty clear why we want to know who did it.  

In the case of an equation, perhaps we want to know how many seats we
should put in the airplane we're about to build, or how much money we
should spend on advertising next year, or how much germanium we want
to put in our next batch of semiconductors.  It could be anything. 
But the idea is the same:  We start with a description of a quantity,
and want to determine its identity.
   
Does this make sense? 
 
- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School About Math
High School Linear Equations
Middle School About Math
Middle School Equations

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