Definition of 'Solve'

Date: 11/29/2005 at 19:09:57
From: Skyler
Subject: Mathematical definition

I am a 9th grade student.  In algebra class today I was told by my
teacher that a problem such as 2 + 2  cannot be solved.  In order to
solve a problem, there has to be a variable involved.  If this is
correct, what would the solution to any math question that did not 
have a variable be called?  What is the definition of "solve" in
mathematics?

I think that "solve" is what you would call a resolution to any math
problem, not limited to questions involving variables.

Date: 11/29/2005 at 23:35:24
From: Doctor Peterson
Subject: Re: Mathematical definition

Hi, Skyler.

In everyday English, we talk about "solving" any kind of problem,
meaning "finding a solution".  In math, we want to be as clear as we
can, so we restrict certain words to certain specific kinds of problem
solving.  This is mostly a way to save words.

For example, when you write 2 + 2, that is not a "problem", really--
it is just an "expression", that is, a statement that 2 is being added
to 2.  When you do the addition, we call it "evaluating" the 
expression, which means finding its value.  The reason we want to use
specific words for this is that there are other things you can do with
an expression, such as "simplifying" it.

So in this broad sense, this could be called a problem, and you could
solve it:

  What is the value of 2 + 2?

This could be another problem to solve:

  Simplify the expression 2x + 2x.

But when I just give you the expression 2 + 2 without saying what to
do with it, you can't "solve" it because I haven't said what to do. 
What you can do is to "evaluate" it or "simplify" it.  Those words
have more specific meaning than the general term "solve".

In the specific sense, we use the term "solve" to mean "find a value 
of the variable that makes an equation true".  So we use the word 
solve (in this sense) only with reference to an equation.  And a
"solution" means a value of the variable that makes the equation true.

So it's not quite that you can't solve anything without a variable;
it's that you can "solve" two things, in different senses: a "problem"
(in which we specifically state what needs to be done), or an 
"equation" (in which case we have defined solving as doing a specific
thing with it).

I hope that helps a bit.  If you have any further questions, feel free
to write back.


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