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To solve the first system using determinants (Cramer's Rule), let's first rewrite the system of equations in standard form. The system is:

\[
3y + 2x - z = 1
\]
\[
3x + 2z = 8 - 5y
\]
\[
3z - 1 = x - 2y
\]

Rearrange each equation:

1. \(2x + 3y - z = 1\)
2. \(3x + 5y + 2z = 8\)
3. \(-x + 2y + 3z = 1\)

Now, we have the system in the form \(Ax = B\), where \(A\) is the matrix of coefficients, \(x\) is the vector of variables, and \(B\) is the constant matrix.

The coefficient matrix \(A\), variable vector \(X\), and constant vector \(B\) are:

\[
A = \begin{pmatrix} 2 & 3 & -1 \\ 3 & 5 & 2 \\ -1 & 2 & 3 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad B = \begin{pmatrix} 1 \\ 8 \\ 1 \end{pmatrix}
\]

To find the values of \(x\), \(y\), and \(z\) using Cramer's Rule, we calculate the determinant of \(A\) and the determinants of matrices obtained by replacing each column of \(A\) with \(B\).

1. **Compute \( \det(A) \):**

   \[
   \det(A) = \begin{vmatrix} 2 & 3 & -1 \\ 3 & 5 & 2 \\ -1 & 2 & 3 \end{vmatrix}
   \]

2. **Compute \( \det(A_x) \):**
   Replace the first column of \(A\) with \(B\).

   \[
   A_x = \begin{pmatrix} 1 & 3 & -1 \\ 8 & 5 & 2 \\ 1 & 2 & 3 \end{pmatrix}
   \]

3. **Compute \( \det(A_y) \):**
   Replace the second column of \(A\) with \(B\).

   \[
   A_y = \begin{pmatrix} 2 & 1 & -1 \\ 3 & 8 & 2 \\ -1 & 1 & 3 \end{pmatrix}
   \]

4. **Compute \( \det(A_z) \):**
   Replace the third column of \(A\) with \(B\).

   \[
   A_z = \begin{pmatrix} 2 & 3 & 1 \\ 3 & 5 & 8 \\ -1 & 2 & 1 \end{pmatrix}
   \]

Once we have each determinant, apply Cramer's Rule:

\[
x = \frac{\det(A_x)}{\det(A)}, \quad y = \frac{\det(A_y)}{\det(A)}, \quad z = \frac{\det(A_z)}{\det(A)}
\]

I will calculate these determinants and provide the solution values for \(x\), \(y\), and \(z\).The solution to the system using Cramer's Rule is:

\[
x = 3, \quad y = -1, \quad z = 2
\]

Would you like further details on any step or have any questions?

Here are five related questions to deepen your understanding:

1. How does Cramer's Rule work with a 3x3 system in terms of determinant replacement?
2. What would happen if the determinant of \(A\) were zero?
3. Can Cramer's Rule be used for systems larger than 3x3, and what challenges might arise?
4. How can we verify the solution obtained from Cramer's Rule by substituting back into the original equations?
5. Are there alternative methods to solve this system of equations?

**Tip:** Cramer's Rule is only applicable when the determinant of the coefficient matrix \(A\) is non-zero, ensuring a unique solution.

Solve the system using determinants (Cramer's Rule).

Learn how to solve a 3x3 system of linear equations using determinants and Cramer's Rule. This example walks through each step of calculating the determinants for the coefficient matrix and modified matrices to find the values of x, y, and z. Suitable for advanced high school and college students studying linear algebra.

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