Cramer's rule:

I will assume you already know how to get the value of a 3x3
determinant.  If you don't know how, post again asking how
it's done.

There are 4 columns,

1. The column of x-coefficients 

2. The column of y-coefficients 

3. The column of z-coefficients  

4. The column of constants:     

There are four determinants:

1. The determinant  consists of just the three columns
of x, y, and z coefficients. in that order, but does not
contain the column of constants.

. 

It has value . 

2. The determinant  is like the determinant 
except that the column of x-coefficients is replaced by the
column of constants.   does not contain the column 
of x-coefficients.

.

It has value .

3. The determinant  is like the determinant 
except that the column of y-coefficients is replaced by the
column of constants.   does not contain the column 
of y-coefficients.

.

It has value .

4. The determinant  is like the determinant 
except that the column of z-coefficients is replaced by the
column of constants.   does not contain the column 
of z-coefficients.

.

It has value .

Now the formulas for x, y and z are





---------------------------

By matrix inversion.

I will assume you already know how to find the inverse
of a matrix, and how to multiply two matrices. If you don't, 
post again asking how.

First we form three matrices, A, X, and B.

1. Matrix A is the 3x3 coefficient matrix A, which consists 
of just the three columns of x, y, and z coefficients. in 
that order, but does not contain the column of constants.

. 

2. Matrix X is the 3x1 matrix of variables 

3. Matrix B is the 3x1 matrix, whose only column is the
column of constants: 

Next we form the matrix equation:

       

or



To solve the equation

       

we left-multiply both sides by , the inverse of .

 

Then since the associatitive principle holds for matrix multiplication,
(even though the commutative principle DOES NOT!!!), we can move
the parentheses on the left around the first two matrix factors:



Now since , where I is the identity matrix, the
above becomes:



and by the identity property:



Performing these operations with the actual matrices we have
the equation 



Next we form the inverse of A, which is written A-1.



Then we indicate the left multiplication of both sides by
 to get the equation :



Next we use the associative principle to move the parentheses so that
they are around the first two factors to get the equation :



Now we perform the actual multiplications and we get the equation :



Then when we multiply the identity matrix  by the column matrix of
variables, we just get the matrix of variables, or the 
equation 



Edwin