There is another way to solve systems of equations with three variables.
It involves a quantity called the determinant.

Every m×m matrix has a unique determinant. The determinant is
a single number. To find the determinant of a 2×2matrix,
multiply the numbers on the downward diagonal and subtract the product
of the numbers on the upward diagonal:

A =

detA = a_{1}b_{2} - a_{2}b_{1}.
For example,

det = 4(6) - (- 1)(- 2) = 24 - 2 = 22

To find the determinant of a 3×3 matrix, copy the first two
columns of the matrix to the right of the original matrix. Next,
multiply the numbers on the three downward diagonals, and add these
products together. Multiply the numbers on the upward diagonals, and
add these products together. Then subtract the sum of the
products of the upward diagonals from the sum of the product of the
downward diagonals (subtract the second number from the first
number):

A =

Example: Find the determinant of:

Solution:

Step 1

Step 2

Step 3

Step 4

10 - 80 = -70. detA = - 70.

Cramer's Rule

Recall the general 3×4 matrix used to solve systems of three
equations:

This matrix will be used to solve systems by Cramer's Rule. We
divide it into four separate 3×3 matrices:

D =

D_{x} =

D_{y} =

D_{z} =

D is the 3×3 coefficient matrix, and D_{x}, D_{y}, and D_{z}
are each the result of substituting the constant column for one of the
coefficient columns in D.

Cramer's Rule states that:

x = y = z =

Thus, to solve a system of three equations with three variables using
Cramer's Rule,

Note: If detD = 0 and detD_{x}, detD_{y}, or detD_{z}≠ 0, the system is inconsistent. If detD = 0 and detD_{x} = detD_{y} = detD_{z} = 0, the system has multiple solutions.