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.

So, if I had:
                    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
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