Multi-Variable Equations

Date: 02/07/97 at 09:26:27
From: lisa
Subject: Multi-Variable Equations

Dear Dr. Math,

I'm tyring to solve for s in the following equation and every time I 
try to do it, I end up with a different answer. The equation is as 
follows:

	       log(f/e) + s * s * 0.5 * t
	d =   ---------------------------
		         s * s * sqrt(t)

Any help you could give me would be greatly appreciated.  Thanks.

Regards, Lisa

Date: 02/07/97 at 15:31:23
From: Doctor Mike
Subject: Re: Multi-Variable Equations

Hi Lisa,  
  
I suggest you start out by simplifying. The variables or numbers other 
than "s" may be obscuring the basic idea of this equation for you.  
I'll use upper case letters for the new stuff to emphasize what I am 
doing:

    A = log(f/e)
    B = t/2
    C = square root of t
    D = d

Then the equation becomes:
 
             A + s*s*B  
       D = -------------
               s*s*C 
  
It's the same equation but with fewer messy details do deal with.  
  
My next suggestion is to dig way back for a few key ideas you probably 
have seen before. They are among the very most important ideas there 
are in high school math courses. 
 
1. If you have a true equation, and you do exactly the same thing to 
   both sides of that equation, then the new equation you get is also 
   true.  An example is subtracting seven from both sides of 
   2*X+7 = 33  to get  2X=26. 
  
2. The distributive law, which has many forms including:
 
      R*(S+T) = R*S + R*T
      R*(S-T) = R*S - R*T
      (R+S+T-U-V)*Z = R*Z + S*Z + T*Z - U*Z - V*Z

  Does this kind of thing look pretty familiar?  Good.
  
Now, using (1) you can multiply both sides of the equation by  s*s*C  
to get:  
      
       s*s*C*D = A + s*s*B
  
Using (1) again you can subtract s*s*B from both sides of the equation 
to get:
  
       s*s*C*D - s*s*B = A
  
Then you can use the distributive law on that equation to get: 
  
       s*s*(C*D - B) = A 
   
Then divide both sides of the equation by the part in parentheses:
  
                 A      
       s*s = ---------  
              C*D - B   
  
Now you know what "s squared" is, and s is just the square root of 
that. Of course, to get the "final final" answer in terms of the 
original variables and/or numbers (f, e and t), you replace the upper 
case letters by what they equal. 
  
I hope this helps.  
  
-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 02/07/97 at 15:49:40
From: Lisa
Subject: Re: Multi-Variable Equations

Dear Dr. Math,

Thank you so much for your help. I forgot how much fun math can be!

- A grateful person who now vows to throw away her calculator!