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Distributive Law

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Date: 8/2/96 at 17:50:40
From: Anonymous
Subject: Distribution of Subtraction

I am an English teacher trying to teach Algebra to high school
students and I need a lot of help.

Regarding the following equation:

[(2y)(squared) + 8] - (y (squared) - y) = 3y sq

The 1st step of the solution in my text is:
(4y sq + 8) - y sq + y = 3y sq

My question: Why does "-(y sq - y)" become "-y sq + y?"  Why does the

Thank you.
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Date: 8/5/96 at 0:6:44
From: Doctor Mike
Subject: Re: Distribution of Subtraction

Hello,

You are NOT the first person in history to get hung up on this point.
This is a form of (or really a consequence of) the Distributive Law.

The most common version of the Distributive Law is:

A*(B + C) = A*B + A*C

It says that to multiply a sum of 2 numbers by A, you multiply each
one of them individually by A, and then add those 2 results together.

Another common version of the Distributive Law is:

A*(B - C) = A*B - A*C

This looks different from the first one above, but it really isn't a
new and different property of numbers.  Here's why it follows :

Since "B - C" (B minus C) is the same as B + -C (B plus the negative
of C), it is true that "A*(B - C)" is the same as "A*(B + -C)".
So using the first version of the Distributive Law from above,

A*(B - C) = A*(B + -C)
= A*B + A*(-C)
= A*B - A*C

Now, finally, what you have been waiting for.  An expression such as
-(B - C) is the same as (-1)*(B - C), so

-(B - C) = (-1)*(B - C)
= (-1)*B - (-1)*C
= -B -(-C)
= -B + C

By the way, the reason for that last step was that the "negative of
the negative of C" is just C.  See?

In your particular example "-(y sq - y)", B is y squared and C is y.

I hope this gets you and your class started in the right direction.

-Doctor Mike,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
Middle School Negative Numbers

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