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:

detA = a₁b₂ - a₂b₁.
For example,

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):

Example: Find the determinant of:

Solution:

Step 1

Step 2

Step 3

Step 4

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

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

1. Arrange the system in the following form:
   
   a₁x + b₁y + c₁z = d₁
   a₂x + b₂y + c₂z = d₂
   a₃x + b₃y + c₃z = d₃
   
2. Create D, D_x, D_y, and D_z.
3. Find detD, detD_x, detD_y, and detD_z.
4. x = , y = , and z = .

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.

Example:Solve the following system:

1. Rearrange the system:
   
   8x - 7y + 10z = 15
   2x + 3y + 8z = 7
   -4x + 5y - 2z = - 9
   
2. Create the matrices:
   

   D =

   
   
   
   

   Dx =

   
   

   Dy =

   
   

   Dz =

   
   

3. Find the determinants:
   
   detD = (- 48 + 224 + 100) - (- 120 + 320 + 28) = 276 - 228 = 48
   detD_x = (- 90 + 504 + 350) - (- 270 + 600 + 98) = 764 - 428 = 336
   detD_y = (- 112 - 480 - 180) - (- 280 - 576 - 60) = - 772 - (- 916) = 144
   detD_z = (- 216 + 196 + 150) - (- 180 + 280 + 126) = 130 - 226 = - 96
   
   
4. x = = 7. y = = 3. z = = - 2.

Thus, (x, y, z) = (7, 3, - 2).