Cramer's Rule in Action

Date: 05/08/98 at 15:19:52
From: Moriah
Subject: Cramer's Rule

Could you explain Cramer's Rule to me? I thought I got it in class, 
but when I went home to do the homework, I had no clue how to use it. 

Thanks!

Date: 05/09/98 at 07:53:30
From: Doctor Anthony
Subject: Re: Cramer's Rule

As an example, let's solve the following system using Cramer's rule:

    2x + y - z = 0
    x  - y + z = 6
    x + 2y + z = 3 

Write the equations with all terms on the left, and the zero on the 
right:

   2x  + y - z + 0 = 0
    x  - y + z - 6 = 0
    x + 2y + z - 3 = 0

Lay out the determinants as follows. ALWAYS express the numerators as:

   x, -y, z, and -1

Now, for the determinant under x, write down the columns representing 
the coefficients of y, z, and the constant term as they appear in the 
set of equations. For the determinant under -y, you leave out the y 
column of coefficients. For the determinant under z, you leave out the 
coefficients in the z column; and under -1, you leave out the column 
of constant terms:

         x             - y              z              -1
   ------------  =  -----------  =  -----------  =  -----------
   | 1  -1   0|     |2  -1   0|     |2   1   0|     |2   1  -1|
   |-1   1  -6|     |1   1  -6|     |1  -1  -6|     |1  -1   1|
   | 2   1  -3|     |1   1  -3|     |1   2  -3|     |1   2   1|

Taking the determinants:

               x                         -y             
    ------------------------  = ----------------------- 
    -3 + 12 + 0 - 0 + 6  + 3    -6 + 6 + 0 - 0 + 12 - 3

                z                        -1
     = ---------------------- = ----------------------
       6 - 6 + 0 - 0 + 24 + 3   -2 + 1 - 2 - 1 - 4 - 1

Thus:

     x        -y         z      -1
    ----  =  -----  =  ----  = ----- 
     18        9        27      -9

And from these equations:

   x = 2

   y = -1

   z = 3

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