Simplifying Variable Expressions

Date: 09/07/2006 at 11:34:14
From: Lori
Subject: Order of operation using variables 

I am having problems following the proper order of operations.  I 
just need a good example of how to work it out.  Please help.

  -16[42 - 3(c + 2)] - 16

I seem to be confusing myself the more I try to work out the problem, 
coming up with different answers each time.

Date: 09/07/2006 at 13:01:51
From: Doctor Peterson
Subject: Re: Order of operation using variables

Hi, Lori.

Properly speaking, the order of operations is used to _evaluate_ an 
expression, which in this case would require a value for the 
variable.  Were you given one?  Or were you told to _simplify_ the 
expression?  In that case, the order of operations plays a role, but 
the main idea is to apply properties like the distributive property 
to eliminate parentheses and combine like terms. 

I'll do a similar problem, so you can try doing the same with yours.

Let's simplify the expression

  3 - [4 - 2(3x - 5)]

I've put in plenty of negatives to make this as hard as your own!

The order of operations tells us that if we were evaluating the 
expression, we would start on the inside, with 3x - 5; we do the same 
in simplifying.  Since 3x - 5 is already as simple as it can get, we 
move outward one step and look at what is being done to it: it is 
multiplied by -2.

You might say that it's actually being multiplied by 2, and then 
subtracted; but experienced people find it easiest to see subtraction 
as if it were an addition of a negative.  That is, I see our 
expression this way:

  3 + -[4 + -2(3x + -5)]

So I want to simplify -2(3x + -5).  I eliminate the parentheses by 
distributing: each term, 3x and -5, is multiplied by -2 to give -6x 
+ +10. So now our expression is

  3 + -[4 + -6x + 10]

Now we move up another notch and look at the entire expression inside 
the [ ], namely 4 + -6x + 10.  We can simplify this by combining like 
terms.  There is only one term with an x, but there are two constant 
terms to combine.  We get -6x + 14.  So we now have

  3 + -[-6x + 14]

Once again, since we've simplified what's inside the [ ], we consider 
what is done to it on the outside: it is multiplied by -1 (which is 
what taking a negative does).  So we distribute the -1:

  3 + 6x + -14

Here multiplying each term by -1 just changed its sign.  Now we 
combine like terms one more time, and we're done:

  6x + -11 or just 6x - 11

Notice that we kept alternating between combining like terms and 
distributing a multiplication, as we worked our way outward.  I'll 
rewrite the whole process so you can see it all together:

  3 + -[4 + -2(3x + -5)]
            \_________/
  3 + -[4 +  -6x + 10  ]
        \_____________/
  3 + -[    -6x + 14   ]
      \________________/
  3 +        6x - 14
  \________________/
       6x - 11

When I write something like this, I don't usually write the "+ -" 
everywhere, but leave it as subtraction; that's because when I look 
at the subtraction I SEE the sign attached to the following number. 
You may find it helpful to write it out this way until you get used 
to it.

If you have any further questions, feel free to write back.  I'd like 
to see your work, written out something like mine, so I can tell 
where you are making mistakes.


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

Date: 09/08/2006 at 21:35:44
From: Lori
Subject: Thank you (Order of operation using variables )

Thank you very much.  It still took me awhile, about an hour to figure
out the positive and negative sign placement but I finally got it.

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