Solving Linear Equations and the Scale Metaphor

Date: 08/26/2004 at 21:53:01
From: Lisa 
Subject: The logic? behind the procedure used to isolate variables

As a 30-something adult I am trying to conquer a life-long math phobia 
and I am hoping that by understanding exactly what is happening when 
we go through certain steps (isolating x for example) in algebra to 
solve a problem it will help.  There is nothing intuitive about the 
process to me and I've heard math-types say that it is.    

For example, when isolating X in linear equations, I know that you 
have to perform the same operation on both sides of the equal sign, 
but I don't understand why.   

Hope this isn't a stupid question and thanks.

Date: 08/26/2004 at 23:15:58
From: Doctor Peterson
Subject: Re: The logic? behind the procedure used to isolate variables

Hi, Lisa.

It's not a stupid question, but it can be hard to answer!  The trouble
is that there are many ways to explain it, and without knowing what
explanations you have seen and been unimpressed by, I can't be sure
what explanation will be most likely to help you.  But trying to find
it can be a good exercise for both of us--for you to find a way to 
understand, and for me to find a way to explain something that to me
is indeed "natural".

(By the way, nothing is "intuitive" until your intuition is trained to
see it that way!  To an adult, it is intuitive that someone who leaves
the room still exists, but that has to be learned by babies.  So we
think of it as intuitive only because we have gotten used to it, and
connected it to our experience.  I want to learn how to do that for
people who are resistant to having their intuitions trained; I suspect
that it may be because you are more skeptical and less gullible, 
rather than more "stupid".)

Let's start with one of the more basic explanations, using a balance
scale as a model.  We'll use the simple equation

  x + 7 = 12

Imagine that the two sides represent the total weight on each side of
a scale:

  x+7     12
  ---     ---
   |       |
   +---+---+
       |
      -+-

The 7 and the 12 can stand for the weights of two known objects, which
weigh 7 and 12 grams respectively.  The x is some object whose weight
we don't know; I like to think of it as a black box we can't look
inside of, but whose contents we are trying to figure out.

What can we do to find the weight?  Well, we wish we just had the x
balancing something known, because then we'd know its weight!  And we
can accomplish that.

Think of the 7 and the 12 now as 7 1-gram weights and 12 1-gram 
weights respectively.  Then if we take off 7 of them from each side,
the scale should still balance.  Do you see why that is true?

Once we do that, we have only the x left on the left, and 12-7, or 5,
1-gram weights on the right:

   x       5
  ---     ---
   |       |
   +---+---+
       |
      -+-

So the weight of object x is 5 grams.

What we do with the equation is identical.  We subtract 7 from both
sides, and the equation will remain true.  Consider what is really
happening now that we know that x is 5.  On the left we have 5+7 to
start with, which is 12; on the right we have 12.  Subtracting 7 from
12, no matter how it is represented, always gives the same answer, 5.
So subtracting 7 from x+7 (by just dropping the 7) on the left, and
subtracting 7 from 12 (by actually getting 5) on the right, will both
give the same answer.

Does that help?  If not, let me know where you have trouble--which
sentence doesn't convince you, and why (if you can tell)--and I'll see 
what I can do to make the explanation better, or try something 
altogether different.

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

Date: 08/27/2004 at 07:50:43
From: Lisa 
Subject: Thank you (The logic? behind the procedure used to isolate
variables)

Excellent!  The scale metaphor was perfect--if you subtract 7 grams
from both sides, the scale's reading will remain exactly the same so
you haven't altered the outcome of the problem.  Thanks again!