System Of Equations With Products Of Variables

Date: 10/29/2003 at 18:17:24
From: Katie Siemens
Subject: solving systems of equations

I'm trying to solve the following system of equations for (x,y,z):

     xy + zx = -1
     xy + yz = -9
     yz + zx = -4

I know that I need to solve for a variable and substitute it in and 
solve but I can never get rid of the varriables to get actual answers.
My answers just keep getting more and more complex!  For example, I
can figure out that:

     x = -1/(y + z)
     y = -9/(x + z)
     z = -4/(y + x)

but it doesn't seem to get me anywhere.  Can you help?

Date: 10/29/2003 at 23:39:39
From: Doctor Ian
Subject: Re: solving systems of equations

Hi Katie,

In this case, elimination may work better than substitution.  Suppose
we subtract the second equation from the first:

   xy + zx      = -1
  
- (xy + yz      = -9)
   -----------------------
        zx - yz = -1 - -9

        zx - yz = 8

Now, what if we add that new equation to the third original equation?

    zx - yz = 8

 +  zx + yz = -4
    ----------------
   2zx      = 8 + -4

        2zx = 4

         zx = 2

Okay, that's a start!  Now you can use substitution to go back and
find out what xy and yz are.  Then you'll have a system that looks
like this:

   xy = ?

   yz = ?

   zx = 2

Can you take it from there? 

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