Subtracting as Adding a Negative

Date: 01/14/2002 at 13:42:06
From: Laura koornneef
Subject: Application of integers and microcomputers

This is the question:

     5(x-y) -2(x+y) = 3x - 7y 

My teacher says to think of the question as a negative number, not a 
minus number, but I don't know how he's doing that.
   
Thanks so much,
Laura the Canadian

Date: 01/14/2002 at 14:10:33
From: Doctor Peterson
Subject: Re: Application of integers and microcomputers

Hi, Laura.

Let's work on a similar problem so you can see how to do it:

    Simplify the expression 4(a-b) - 3(a+b)

We can start by expanding each product using the distributive 
property:

    4a - 4b - 3a - 3b

(Where did that come from? I'll get back to this in a minute, but when 
you're used to these things, you'll be able to just write that out at 
once, and I want you to see it that way.)

Now we can collect and combine like terms, namely the "a" terms and 
the "b" terms:

    4a - 3a - 4b - 3b

    (4 - 3)a - (4 + 3)b

    1a - 7b

    a - 7b

There's a lot going on here, and probably the parts that are hard for 
you are the signs. I'll go through it all again, but explain things 
slowly. We start with

    4(a-b) - 3(a+b)

Now, you should know the distributive property in something like this 
form:

    x(y+z) = xy + xz

That is, if you multiply a sum, it's the same as multiplying each term 
in the sum and adding the products.

But how do the subtractions fit in? That's probably where your teacher 
wants you to think "negative" rather than "minus." When I see 
something like a-b, I see it as a + -b; that is, subtracting is the 
same as adding the negative of a number. If you always see subtraction 
this way, you don't have to worry about separate rules for 
subtraction. So let's do the whole simplification project, written 
this way:

    4(a + -b) + -3(a+b)

Now we can distribute:

    4a + 4(-b) + (-3)a + (-3)b

See what's going on? The -b and -3 are just numbers; I've put 
parentheses around them just to hold them together and keep the 
negative sign from looking like a subtraction.

Now I can gather the a's and the b's together:

    4a + (-3)a  +  4(-b) + (-3)b

Ah, but one of those terms has -b instead of b; let's move the sign 
over to the 4, since the product will be negative either way:

    4a + (-3)a  +  (-4)b + (-3)b

Now we can use the distributive property in reverse to combine terms:

    (4 + -3)a + (-4 + -3)b

Now add:

   (1)a + (-7)b

Now stop looking at it as negatives, and write it as a subtraction 
again:

    a - 7b

This is what went on in my mind as I did it quickly before.

Now try doing the same thing on your problem. If you have trouble, 
please write out what you did as well as you can, and show me, so I 
can diagnose your difficulty and suggest ways to do better.

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