Just Plain Algebra

Date: 01/07/98 at 23:57:39
From: Ryun Patenaude
Subject: Just plain algebra

I try hard but I just don't get algebra. Do you have any advice or any 
programs you might recommend?

Thank you,
  Ryun Patenaude

Date: 01/12/98 at 12:30:24
From: Doctor Joe
Subject: Re: Just plain algebra

Hi Ryun,

The first thing you must learn in algebra is this golden rule:

    Don't panic and always have a clear mind.

At your age, the type of algebra you encounter (correct me if I'm 
wrong; you might be an expert in higher algebra such as group theory, 
linear spaces, homology and topos theory) should be arithmetic 
operations, the solving of algebraic equations in a unknown, usually 
x, and most difficult of all, word problems that involve the 
formulation of an algebraic equation. 

Follow the following steps.  I hope they are useful but they are by no 
means exhaustive:

I am focusing on the aspect of word-problems that involve the 
formulation of an algebraic equation.

Step 1:

Read the question carefully and find/underline the unknown quantity 
that is involved in the question. Note that this unknown quantity  
will be the one particular quantity that other unknowns depend on.

  Example:

  There are 3 pieces of wire. The length of the first is 20 percent of   
  the length of the second, and the length of the third is 100 percent 
  of the length of the second.

  In this question, clearly the length of the second piece of wire is 
  the desired unknown on which the others depend.

Step 2.

Let x (or any symbol you like) be the unknown quantity.

Step 3.

Define the other quantities in terms of the unknown.

This may prove to be the most difficult step. My advice is to try to
imagine you already know the value of x. Then your job becomes finding 
the other quantities as if you know what x is, and you don't need to 
simplify those expressions in terms of x.

Step 4.

Form the final equation. Usually, this comes in the form of a total or 
a final statement in the question.

Example:

  In the previous example, if it is further given that the 3 pieces of 
  wire total 23 cm in length, then the equation will be

     0.2x + x + 1.1x = 23

Step 5.

Simplify expressions in x (this you can practice by doing more 
simplification of expression exercises).

Step 6.

Add, subtract, multiply or divide by suitable numbers on both sides of 
the equation one step at a time. 

Look at the following examples and you'll know what I mean by Step 6.  
It is more meaningful this way:

Example:  

  Suppose we have the equation:

      2x + 3 = 4 - 5x

  Ask yourself: Isn't it more systematic if we group things of the 
  same type together?  (i.e. the unknowns with the unknowns and the 
  known with the known).

  How do we make this happen? We see a 3 on the left; suppose we 
  subtract 3 from the quantities on both sides? The equality still 
  holds, so we have

      (2x + 3) - 3 = (4 - 5x) - 3
   
  Then, 2x + 3 - 3 = 4 - 5x -3
        2x         = 4 - 3 - 5x
        2x         = 1 - 5x

  Likewise, add 5x on both sides,

       2x + 5x     = 1 - 5x + 5x
       7x          = 1

Now, to eliminate the outstanding quantity 7 and make x stand on its 
own (so to speak), we multiply by 1/7 on both sides:

      (1/7)*7x     = 1/7 * 1
      (1/7 * 7) x  = 1/7
       1 * x       = 1/7
                x  = 1/7

I hope this helps you understand algebra.

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