Solving Systems of Three Equations

Date: 05/07/98 at 20:26:26
From: Claire
Subject: Solving linear systems

    3x + 4y + 8z = 5
    9x - 5y + 6z = 2
   15x + 2y - 4z = 5

I tried solving for x, then y and the z but I still ended up with 
extra variables. Help!

Date: 05/09/98 at 02:41:33
From: Doctor Alain
Subject: Re: Solving linear systems

This is a system of three linear equations in the three unknowns 
x, y, and z. Geometrically, each equation describes a plane in space.  
The solution is the one point of space common to the three planes.

There are systematic ways of solving such systems. One is called the 
Gauss/Jordan algorithm. I will give a solution based on the Gauss/
Jordan algorithm but avoiding fractions. I will leave the symbols 
x, y and z in the calculations; however, these are normally left out 
in the Gauss/Jordan algorithm.

First we want to have zeros in the first column under the first row. 
We do this by subtracting 3 times row 1 from row 2 and 5 times row 1 
from row 3:

   3x +  4y +  8z =   5
   0x - 17y - 18z = -13
   0x - 18y - 44z = -20

Then we want to put a zero in column 2 under row 2. We do this in two 
steps. First we multiply row 2 by -18 and row 3 by 17. We get:

   3x +   4y +   8z =    5
   0x + 306y + 324z =  234
   0x - 306y - 748z = -340

Now we add row 2 to row 3:

   3x +   4y +   8z =    5
   0x + 306y + 324z =  234
   0x +   0y - 424z = -106

Now you can find z with row 3. Once you find z, you can find y with 
row 2, and when that is done, you can find x using row 1 and the 
values of y and z. 

-Doctor Alain,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/