2x + 4y - 3z = 2
2x - 2y + 3z = 3
3x - 4y + 5z = 4

Rule: For

Ax + By + Cz = E
Fx + Gy + Hz = I
Jx + Ky + Lz = M

Form 4 determinants: 

1. D, the denominator of the value of x, y and z.

2. Dx, the numerator of the value of x.

3. Dy, the numerator of the value of y.

4. Dz, the numerator of the value of z.


    [A B C]
D = [F G H]
    [J K L] 

D is the determinant of the coefficients as they appear left of the
equal sign.  Notice that the column of constants on the right, composed
of E, I, and M is not used in forming D.  However it will be used in 
Dx, Dy, and Dz

To begin forming Dx, the numerator of x, since x is the FIRST variable
in the system of equations, we put the column of constants for the FIRST
column of constants

     [E B C]
Dx = [I G H]
     [X K L]

To begin forming Dy, the numerator of y, since z is the SECOND variable
in the system of equations, we put the column of constants for the SECOND
column of constants


     [  E  ]
Dy = [  I  ]
     [  M  ]

To begin forming Dz, the numerator of z, since z is the THIRD variable
in the system of equations, we put the column of constants for the THIRD
column of constants

     [    E]
Dz = [    I]
     [    M]

Now we finish Dx, Dy, and Dz, by copying in the other two columns from
the determinant D.  

     [E B C]
Dx = [I G H]
     [M K L]  

     [A E C]
Dy = [F I H]
     [J M L]

     [A B E]
Dz = [F G I]
     [J K M]

We evaluate 

    [A B C]   [2  4 -3]
D = [F G H] = [2 -2  3] = 6
    [J K L]   [3 -4  5]

     [E B C]   [2  4 -3]
Dx = [I G H] = [3 -2  3] = 4
     [M K L]   [4 -4  5]

     [A E C]   [2 2 -3]
Dy = [F I H] = [2 3  3] = 7 
     [J M L]   [3 4  5]

     [A B E]   [2  4  2]
Dz = [F G I] = [2 -2  3] = 8
     [J K M]   [3 -4  4]

x = Dx/D = 4/6 = 2/3
y = Dy/D = 7/6 
z = Dz/D = 8/6 = 8/6 = 4/3

Edwin
AnlytcPhil@aol.com