Equations with Variables

Date: 9/23/96 at 19:34:48
From: daugherty
Subject: Variables on Both Sides

Dr. Math,

How do you solve problems with variables on both sides?  Here's one 
I'm working on:

         6 - 2y = 7y + 13

I just can't figure it out!  Would you please walk me through it and 
explain WHY you're doing things?

Thanks a million!
Rachel Daugherty

Date: 10/18/96 at 17:49:6
From: Doctor Lynn
Subject: Re: Variables on Both Sides

Hello, Rachel.

This kind of problem isn't too difficult once you get the knack of it.
What you need to do is to get rid of one of the variables from one 
side. In your equation, 

   6 - 2y = 7y + 13

you can either get rid of the -2y or the 7y, but it's generally easier
if you get rid of the negatives, so I'll show how to get rid of the
2y.

The most important rule for equations is that you always do the same 
thing to both sides of the equation.

Now, to get rid of the -2y we need to find what operation we can do
to the lefthand side of the equation so that the -2y becomes zero.

Well, the opposite of a negative is a positive, so we add 2y to the 
lefthand side of the equation. We must also do the same to the 
righthand side, like this:

  6 - 2y + 2y = 7y + 13 +2y

The equality sign "=" means that both sides of the equation are 
exactly the same. We have done the same thing to both sides, so the 
two sides are still the same.

But now on the lefthand side, we have -2y+2y, which is clearly zero.
This means that we can rewrite the equation without these terms:

 6 = 9y + 13

We now cancel the +13 in the same way, so that the numbers and 
variables are on separate sides of the "=":

        6 = 9y + 13
   6 - 13 = 9y + 13 - 13
       -7 = 9y

Finally, we will cancel the implied multiplication by 9 by dividing
both sides by 9:

 -7/9 = y

So in summary, the points to remember are:

   Every operation has an opposite.  To "get rid of a term", you
      simply apply the opposite operation to both sides.

   You are trying to get the numbers on one side of the "=" and the
      variables on the other.

I hope that helps.  Write back if you have any more problems.

-Doctor Lynn,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/