To solve a system of linear equations using Cramer's rule, the system must be square, meaning it should have the same number of equations as unknowns. The general form of a system of linear equations is:

\[
\begin{align*}
a_1x + b_1y + c_1z &= d_1 \\
a_2x + b_2y + c_2z &= d_2 \\
a_3x + b_3y + c_3z &= d_3
\end{align*}
\]

To apply Cramer's rule:

1. **Determine the determinant of the coefficient matrix \( D \):**

   \[
   D = \begin{vmatrix} 
   a_1 & b_1 & c_1 \\
   a_2 & b_2 & c_2 \\
   a_3 & b_3 & c_3 
   \end{vmatrix}
   \]

2. **Determine the determinants of the matrices \( D_x \), \( D_y \), and \( D_z \) by replacing the corresponding column of the coefficients with the constants on the right-hand side of the equations.**

   \[
   D_x = \begin{vmatrix} 
   d_1 & b_1 & c_1 \\
   d_2 & b_2 & c_2 \\
   d_3 & b_3 & c_3 
   \end{vmatrix}
   \]

   \[
   D_y = \begin{vmatrix} 
   a_1 & d_1 & c_1 \\
   a_2 & d_2 & c_2 \\
   a_3 & d_3 & c_3 
   \end{vmatrix}
   \]

   \[
   D_z = \begin{vmatrix} 
   a_1 & b_1 & d_1 \\
   a_2 & b_2 & d_2 \\
   a_3 & b_3 & d_3 
   \end{vmatrix}
   \]

3. **Solve for \( x \), \( y \), and \( z \) using the following formulas:**

   \[
   x = \frac{D_x}{D}, \quad y = \frac{D_y}{D}, \quad z = \frac{D_z}{D}
   \]

Please provide the specific system of equations so I can help you solve it using Cramer's rule.

Would you like to see more details on how to calculate determinants or any other related concepts? Here are some related questions you might find interesting:

1. How do you calculate the determinant of a 3x3 matrix?
2. Can Cramer's rule be used if the determinant of the coefficient matrix is zero?
3. How does Cramer's rule apply to systems of equations with more than three variables?
4. What is the geometric interpretation of the determinant?
5. How is Cramer's rule related to the inverse of a matrix?

**Tip:** When using Cramer's rule, if the determinant \( D \) of the coefficient matrix is zero, the system either has no solution or an infinite number of solutions.

Learn how to solve a system of linear equations using Cramer's Rule. This detailed guide walks through the step-by-step process of calculating determinants for a 3x3 system and solving for variables x, y, and z. Suitable for high school students in Grades 10-12.

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