Solving an Equation

Date: 11/30/2002 at 11:11:59
From: Amanda Parrish
Subject: Solving an Equation

Dear Dr. Math,
           
In math class we have been assigned to answer the following question:

How do you solve an equation?

I know how to solve an equation if it is given to me, but I do not 
know how to describe how to solve any equation. I hope you can help.

Thanks,
Amanda Parrish

Date: 11/30/2002 at 12:00:33
From: Doctor Ian
Subject: Re: Solving an Equation

Hi Amanda,

To solve an equation, you apply inverses to each side of the equation
until the variable you want to solve for is alone by itself on one
side of the equation, and appears nowhere on the other side. 

What is an inverse?  An inverse is an operation that 'undoes' another
operation. Suppose you have something like 

  x + 5 = 10

The additive inverse of 5 is -5, so you can 'invert' the addition by
adding -5:

  x + 5 + -5 = 10 + -5

           x = 5

Suppose you have something like 

  2x = 10

The multiplicative inverse of 2 is 1/2, so you can 'invert' the
multiplication by multiplying by 1/2:

  2x * (1/2) = 10 * (1/2)

           x = 5

Suppose you have something like

  log x = 3

The inverse of a logarithm is an exponent, so you can 'invert' the log
by making both sides exponents of the base of the log:

    log x     3
  10      = 10

              3
        x = 10

And sometimes you have to combine these into multiple steps:

    log(2x + 5) = 3


    log(2x + 5)     3
  10            = 10              Invert the log.


         2x + 5 = 1000


    2x + 5 + -5 = 1000 + -5       Invert the addition


             2x = 995      
 
        
     2x * (1/2) = 995 * (1/2)     Invert the multiplication


              x = 995/2


A general question like "How do you solve an equation" deserves a
general answer, so I'd probably just restate my initial sentence:

  To solve an equation, you apply inverses to each side of the 
  equation until the variable you want to solve for is alone by 
  itself on one side of the equation, and appears nowhere on the 
  other side.  Obviously which inverses you apply, and the order 
  in which you apply them, depends on the operations that need
  to be inverted.  

Does this help? 

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