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### 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/
```
Associated Topics:
High School Linear Equations
Middle School Equations

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