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To solve this system of equations using **Cramer's Rule**, we will compute the determinant of the coefficient matrix and the determinants of matrices obtained by replacing one column at a time with the constant column vector. Then, we will find the values of \(x\), \(y\), and \(z\).

Here is the system of equations again:

\[
\begin{aligned}
6x + y - 3z &= 5 \\
x + 3y - 2z &= 5 \\
2x + y + 4z &= 8
\end{aligned}
\]

Let's proceed step by step by applying Cramer's Rule.Using **Cramer's Rule**, the determinants and solutions are:

- Determinant of the coefficient matrix \( \text{det}(A) = 91 \)
- Determinant with the first column replaced \( \text{det}(A_x) = 91 \)
- Determinant with the second column replaced \( \text{det}(A_y) = 182 \)
- Determinant with the third column replaced \( \text{det}(A_z) = 91 \)

Thus, the solution is:

\[
x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{91}{91} = 1
\]
\[
y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{182}{91} = 2
\]
\[
z = \frac{\text{det}(A_z)}{\text{det}(A)} = \frac{91}{91} = 1
\]

Would you like further clarification on any of these steps?

Here are five related questions:
1. How does Cramer's Rule work for systems of linear equations?
2. What happens if the determinant of the coefficient matrix is zero?
3. How can you generalize Cramer's Rule for larger systems?
4. What are the limitations of using Cramer's Rule in practice?
5. How do we interpret the determinant in terms of geometry?

**Tip:** Cramer's Rule is computationally expensive for large systems; it's better suited for smaller systems like this one!

Learn how to solve a system of linear equations using Cramer's Rule with step-by-step explanations and examples. Understand the concept of determinants and their role in solving equations. Suitable for advanced high school students.

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