Solve for x, y, and z

Date: 7/27/96 at 3:9:31
From: Anonymous
Subject: Solving System of Equations

My friend sent me these equations. I'm completely stuck! Help!

   Solve the following system

          1/2y + 1/3z = 26
          1/3x + 1/4z = 23
          1/2x + 1/4y = 28
- T.Y.

Date: 7/29/96 at 13:28:24
From: Doctor Jerry
Subject: Re: Solving System of Equations

The solution is x = 42, y = 28, and z = 36.

Your friend has given you a system of three 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.  A less formal method that works for this 
system is to multiply the first equation by 3 and the second by -4 and 
add the results.  This gives

 3/2y + z = 3*26
-4/3x - z = -4*23 

Adding these equations gives

 -4/3x +3/2y = -14

Now, look at this equation together with the third of the equations 
you gave.

-4/3x + 3/2y = -14
 1/2x + 1/4y = 28

To get rid of the fractions, multiply the first equation by 6 and the 
second by 4.  This gives

-8x + 9y = -84
 2x +  y = 112

Now multiply the second equation by 4 and add the two equations. This 
gives

13y = 364.

So, y = 28.  From the equation 2x+y = 112, it follows that x = 42.  
From the equation 1/2y + 1/3z = 26, it follows that z = 36.

Have you followed this simple, but multi-step reasoning?  Please ask 
again if not.

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