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### 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

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/
```
Associated Topics:
Middle School Algebra

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