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
