What is x?

Date: 01/07/2002 at 06:35:18
From: MARTIN TAYLOR
Subject: How do I work out 5x + 3 = 38

Please can you tell me how to work out  5x + 3 = 38 ?

Thanks in advance.

Date: 01/07/2002 at 08:57:14
From: Doctor Ian
Subject: Re: How do I work out 5x + 3 = 38

Hi Martin,

Suppose that we're talking about bags of rocks, and x represents the 
contents of one bag, which we'll show as () below. We put these on a 
balance beam, and it looks like 

                                 r r r r r r r r
                                 r r r r r r r r r r
                                 r r r r r r r r r r 
     () () () () () r r r        r r r r r r r r r r
   \_____________________/     \_____________________/

That is, the left side contains 5 bags and 3 loose rocks, and the 
right side contains 38 loose rocks.  And since the problem tells us 
that these are equal, the beam is balanced.  

We can remove three rocks from each side of the balance beam without 
changing the balance, right? 

                                 r r r r r 
                                 r r r r r r r r r r
                                 r r r r r r r r r r 
     () () () () ()              r r r r r r r r r r
   \_____________________/     \_____________________/

If the bags themselves weigh nothing, then each bag must be holding 
1/5 of the number of rocks on the right side. So if we count the rocks 
on the right, and divide by 5, we get the number of rocks in each bag. 


                 () ---------->  r r r r r r r
                 () ---------->  r r r r r r r
                 () ---------->  r r r r r r r
                 () ---------->  r r r r r r r
                 () ---------->  r r r r r r r
 \_____________________/     \_____________________/


Using 'algebra', it looks like this:

      5 * (? rocks per bag) + 3 = 38

  5 * (? rocks per bag) + 3 - 3 = 38 - 3         

          5 * (? rocks per bag) = 35

          5 * (? rocks per bag)   35
          --------------------- = --
                   5               5

                ? rocks per bag = 7

As you can imagine, it gets a little tiring to write (or type) 

  ? rocks per bag

over and over again. Since it doesn't matter too much what we call 
this quantity - so long as we call it the same thing everywhere - we 
can just pick something shorter, like 'rocks', or 'r', or (if we want 
to be conventional) 'x'.  Then it looks like this:

       5x + 3 = 38

   5x + 3 - 3 = 38 - 3

           5x = 35

         5x/5 = 35/5

            x = 7

That looks a little more cryptic and mysterious, but once you get a 
solid grasp on the idea of using letters as names for much longer 
names of particular quantities, it becomes rather normal looking. And 
that's mostly a matter of practice. 

For an introduction to how to think about solving problems like this, 
take a look at

   Basic Tips on Solving for X
   http://mathforum.org/dr.math/problems/megan.11.16.00.html   

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/