Why Algebraic Expressions with Parentheses?

Date: 03/18/2003 at 10:27:34
From: Jason
Subject: Algebraic Expressions Without Parentheses

How would you write these algebraic expressions without parentheses?

 -(2x-3y-6)

and

 -(5x-13y-1)

I was told that you can, but I don't think you can without solving 
the problem. The problem is, there is not enough information there to 
solve the problem. So, if it is possible to write the expressions 
without the parentheses, then how do you do it? Also if you can do 
it, then why are the parentheses there in the first place?

Date: 03/18/2003 at 12:48:26
From: Doctor Ian
Subject: Re: Algebraic Expressions Without Parentheses

Hi Jason, 

I think you mean that these are expressions, and not equations, so
there's no way to determine unique values for x and y.  If so, you're
absolutely right. 

One way to write the expressions without the parentheses is to 
translate

  -whatever

into 

  -1 * whatever

since those are equivalent. Let's see what happens when we do that:

    -(2x-3y-6)

  = -1 * (2x - 3y - 6)

Now we can apply the distributive property:

  = ((-1)2x - (-1)3y - (-1)6)

  = (-2x + 3y + 6)

After you do this enough times, you'll notice that you can just flip
the signs, e.g.:

    -(5x - 13y - 1)

  = (-5x + 13y + 1) 

>Also if you can do 
>it, then why are the parentheses there in the first place?

Sometimes the parentheses are there because the expression came from
somewhere else, and had to be substituted as a whole. For example, you 
might start with something like

  (number of zibbles) = (number of brizzles) - (number of wilmons)

                      = (3x + 4y - 4) - (5x - 13y + 1)

And now your life will be easier if you move the minus sign inside:

                      = (3x + 4y - 4) + (-5x + 13y - 1)

because now you can just drop all the parentheses:

                      = 3x + 4y - 4 + -5x + 13y - 1

But just because you had to substitute the expression using 
parentheses, that doesn't mean you want to keep the parentheses around 
any longer than you have to. 

Does that make sense? 

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