Three Simultaneous Equations

Date: 4/16/96 at 8:32:41
From: Anonymous
Subject: three variables in three equations

I am stuck on this equation.  Can you help me solve the 
problems?

-20x + 7y -  z = 4
  4x - 2y + 24z = 2
  7y - 3z + 21y = 7

Date: 4/17/96 at 14:21:8
From: Doctor Sebastien
Subject: Re: three variables in three equations

Hello,

The trick to solve this type of problem is to reduce the equations 
to 2 unknowns.

Let's name the equations:

-20x + 7y -   z = 4  ----(1)
  4x - 2y + 24z = 2  ----(2)
  7x - 3y + 21z = 7  ----(3)  You will need to check this equation.  
I assume that this is what you meant.

Multiply the (1) by 24 and add it to (2).  We get
-476x + 166y = 98  ----(4)

That gets rid of a z.

Multiply (2) by -7/8 and (3) by 8/7 and add them together.  
We get (9/2)x - (47/28)y = 25/4  ---(5)

You now have 2 equations with 2 unknowns, which should be 
easy to solve.

You can then substitute back the values you have for x and y 
into any of the original equations to get z.

-Doctor Sebastien,  The Math Forum