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### Solving Equations

```
Date: 01/25/97 at 16:38:49
From: traci
Subject: Equations

n
----  +  6  =  -3
-4

n
----  =  -3
13

x
---- -  8  =  7
5
```

```
Date: 01/25/97 at 18:40:17
From: Doctor Wallace
Subject: Re: Equations

Hi Traci!

Notice that, in all three of these equations, we have a number
underneath our variable.  Let's take a look at how to solve an
equation of this type.

First, we have to understand what the equation is telling us.
If we see something like this:

x
---
3

we understand that to mean "x divided by 3."  If we see something
like this:

1
--- x
3

we understand that to mean "one-third times x."  Why do I mention
this?

Because these two ARE THE SAME THING!  This is very important for you
to understand.  Dividing by 3 and multiplying by 1/3 are the same
thing.  You may see this written either way, and you need to know what
to do with it.

Now, to solve any equation, we want to end up with our variable all
alone on one side, as in x = 10, for example.  So, if we see any other
numbers along with our variable, we need to perform an operation (add,
subtract, multiply, or divide) to get the variable by itself.

In my example of the two ways to write x divided by 3, to get x by
itself we need to do the same thing in both cases (because they are
really 2 ways of writing the same thing, remember).  What we need to
do is multiply by 3.

Why?  Because one third times 3 equals 1.  When a variable is all
alone by itself, it has a 1 in front of it, but we never write that 1
because 1 times anything is itself.  So 1x is the same as x.  But
(1/3)x is 1/3 multiplied by x.  So if we multiply the 1/3 by 3, we get
1x or just x.

The same is true if it is written x/3 or x divided by 3.  We multiply
by 3, which "cancels" the 3 on the bottom.

Do you see now why I said it was important to understand these two
ways of writing it?  (1/3)x or x/3 are the same.  It is easy to get
confused if you think of them as different.

Now, it is also important to remember that what we do to one side of
the equation, we MUST do to the other.  If we don't, then we have put
the equation out of balance - the two sides wouldn't be equal any
more and we couldn't really call it an equation.

x
---  =  4
3

To solve this, I would multiply both sides by 3, which gives me

x  =  12

Now, if I had something else on the left side with the x, like this:

x
--- + 2  =  4
3

Then I would have to subtract 2 from both sides to get the x part by
itself, like this:

x
---  + 2 - 2 = 4 - 2
3

which is:
x
---   =  2
3

Then multiply by 3 to get:

x  =  6

I hope this helps you to solve your three equations.  If you need more
help, the doctor's office is open - please write back.

-Doctor Wallace,  The Math Forum
Check out our web site!
```
Associated Topics:
Middle School Algebra
Middle School Equations

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