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The image contains a system of linear equations that needs to be solved using the determinant method (Cramer's Rule). The system of equations is:

\[
\begin{aligned}
5x - 6y + 3z &= -9 \quad \text{(Equation 1)}\\
2x - 3y + 2z &= -5 \quad \text{(Equation 2)}\\
3x - 7y + 5z &= -16 \quad \text{(Equation 3)}
\end{aligned}
\]

### Step 1: Write the system in matrix form
This system can be written as:

\[
A \cdot \mathbf{x} = \mathbf{b}
\]

where \( A \) is the coefficient matrix, \( \mathbf{x} \) is the vector of variables, and \( \mathbf{b} \) is the constant vector:

\[
A = \begin{pmatrix}
5 & -6 & 3 \\
2 & -3 & 2 \\
3 & -7 & 5
\end{pmatrix}, \quad
\mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad
\mathbf{b} = \begin{pmatrix} -9 \\ -5 \\ -16 \end{pmatrix}
\]

### Step 2: Compute the determinant of matrix \( A \)
The determinant of matrix \( A \), denoted as \( \det(A) \), is calculated as follows:

\[
\det(A) = \begin{vmatrix}
5 & -6 & 3 \\
2 & -3 & 2 \\
3 & -7 & 5
\end{vmatrix}
\]

Let's calculate this determinant.

### Step 3: Calculate determinants for \( x \), \( y \), and \( z \)
- **Determinant \( \Delta_x \)**: Replace the first column of \( A \) with \( \mathbf{b} \):

\[
\Delta_x = \begin{vmatrix}
-9 & -6 & 3 \\
-5 & -3 & 2 \\
-16 & -7 & 5
\end{vmatrix}
\]

- **Determinant \( \Delta_y \)**: Replace the second column of \( A \) with \( \mathbf{b} \):

\[
\Delta_y = \begin{vmatrix}
5 & -9 & 3 \\
2 & -5 & 2 \\
3 & -16 & 5
\end{vmatrix}
\]

- **Determinant \( \Delta_z \)**: Replace the third column of \( A \) with \( \mathbf{b} \):

\[
\Delta_z = \begin{vmatrix}
5 & -6 & -9 \\
2 & -3 & -5 \\
3 & -7 & -16
\end{vmatrix}
\]

### Step 4: Solve for \( x \), \( y \), and \( z \)
Using Cramer's Rule:

\[
x = \frac{\Delta_x}{\det(A)}, \quad y = \frac{\Delta_y}{\det(A)}, \quad z = \frac{\Delta_z}{\det(A)}
\]

I'll calculate these determinants and the solutions for \( x \), \( y \), and \( z \) now.The determinant of the coefficient matrix \( A \) is \( \det(A) = 4.0 \).

Using Cramer's Rule, the solutions for the variables are:

\[
\begin{aligned}
x &= 3.0 \\
y &= 5.0 \\
z &= 2.0
\end{aligned}
\]

Thus, the solution to the system of equations is:

\[
(x, y, z) = (3, 5, 2)
\]

Would you like more details on the solution or have any other questions?

Here are 5 related questions that might interest you:
1. How is Cramer's Rule derived from the determinant properties?
2. What are the conditions under which Cramer's Rule is applicable?
3. How do you solve a system of equations if the determinant of the coefficient matrix is zero?
4. Can you explain how the determinant of a matrix is calculated?
5. What are some other methods to solve systems of linear equations?

**Tip:** When using determinants to solve a system of equations, always check the determinant of the coefficient matrix first. If it is zero, Cramer's Rule cannot be applied, and the system may be either dependent or inconsistent.

Learn how to solve a system of linear equations using determinants, specifically through Cramer's Rule. This problem involves calculating the determinant of a 3x3 matrix and applying the rule to find the values of x, y, and z. Suitable for students in Grades 10-12.

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