Negative Coefficients, Subtraction -- and Combining Like Terms?

Date: 09/21/2009 at 23:48:28
From: Anuj
Subject: How do you combine like terms

The directions say to combine like terms and simplify: 

  -9x - y + 1 + 5y + 5x - 10.

My answer was 4x - 4y - 9.

But I'm not sure if I'm supposed to add or subtract when the
numeral/variable is negative/positive.  Am I adding and subtracting them
right?

Date: 09/22/2009 at 11:27:13
From: Doctor Ian
Subject: Re: How do you combine like terms

Hi Anuj,

The basic idea behind combining terms is that you're applying the
distributive property.  With constants, this might look like

  3*5 + 8*5 = (3 + 8)*5 = 11*5

Does that look familiar?  Well, you can do the same thing with variable
terms, e.g.,

  3x + 5x = 3*x + 5*x = (3 + 5)*x = 8*x = 8x

The only thing that changes really is that some of the multiplication
symbols are left implied, rather than written out.

Suppose I have something like

  3x + 4y + 5x + 6y

Addition is commutative, so I can change the order to get "like" terms
together,

  3x + 5x + 4y + 6y

and then I can combine them,

  (3 + 5)x + (4 + 6)y

      = 8x + 10y

So far, so good?

Okay, what if I have subtractions -- like you had in your expression?

  3x - 4y - 5x + 6y

Now, subtraction is NOT commutative; but every subtraction can be
written as an addition:

  3x + -4y + -5x + 6y

And this puts me back in the same boat as before:

    3x + -4y + -5x + 6y

 =  3x + -5x + -4y + 6y

 = (3 + -5)x + (-4 + 6)y

       = -2x + 2y

One last wrinkle: sometimes you don't see a coefficient, e.g., 

  x + 4x + 7x

If it helps, take that implied coefficient of 1, and make it visible:

    x + 4x + 7x

 = 1x + 4x + 7x

 = (1 + 4 + 7)x 

 = 12x

(Are you seeing a pattern here?  An awful lot of math is just looking at
what's in front of you, and asking yourself if there's some way to
rewrite it so that it will be easier to work with!)

Does that make sense?  Try your expression again, and let me know what
you come up with, okay? 

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

Date: 09/22/2009 at 22:39:31
From: Anuj
Subject: Thank you (How do you combine like terms)

Thank you very much.  I got a A+++ on my test because of your help. 
Thank you so much!