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To solve this system of linear equations using **Cramer's Rule**, we will follow these steps:

### Cramer's Rule Overview:
- Cramer's Rule states that for a system of linear equations \( A \mathbf{x} = \mathbf{b} \), where \( A \) is an invertible \( n \times n \) matrix, the solution can be given as:
  \[
  x_i = \frac{\det(A_i)}{\det(A)}
  \]
  where:
  - \( A \) is the coefficient matrix.
  - \( A_i \) is the matrix obtained by replacing the \( i \)-th column of \( A \) with the vector \( \mathbf{b} \).
  - \( \det(A) \) is the determinant of matrix \( A \).
  
### Given System:
The coefficient matrix \( A \) and the vector \( \mathbf{b} \) from the image are:

\[
A = \begin{pmatrix}
3 & 1 & 2 & 0 \\
0 & 1 & 3 & 2 \\
1 & 2 & 0 & 3 \\
1 & 0 & 1 & 0
\end{pmatrix}, \quad
\mathbf{b} = \begin{pmatrix} 2 \\ 1 \\ 1 \\ 2 \end{pmatrix}
\]

Let me calculate the determinant of matrix \( A \) and the determinants of the modified matrices \( A_x, A_y, A_z, A_t \) for solving the system using Cramer's Rule. I will proceed with the calculations now.We have calculated the following determinants:

- \( \det(A) = -10 \)
- \( \det(A_x) = -19 \)
- \( \det(A_y) = 39 \)
- \( \det(A_z) = -1 \)
- \( \det(A_t) = -23 \)

### Applying Cramer's Rule:
Using the formula \( x_i = \frac{\det(A_i)}{\det(A)} \), we can now find the values of \( x, y, z, t \):

\[
x = \frac{\det(A_x)}{\det(A)} = \frac{-19}{-10} = 1.9
\]
\[
y = \frac{\det(A_y)}{\det(A)} = \frac{39}{-10} = -3.9
\]
\[
z = \frac{\det(A_z)}{\det(A)} = \frac{-1}{-10} = 0.1
\]
\[
t = \frac{\det(A_t)}{\det(A)} = \frac{-23}{-10} = 2.3
\]

### Final Solution:
\[
x = 1.9, \quad y = -3.9, \quad z = 0.1, \quad t = 2.3
\]

Would you like further details on any step or additional clarifications?

Here are some related questions to expand this topic:
1. How does Cramer's Rule change for systems with non-invertible matrices?
2. What are some alternative methods to solve this system of equations?
3. How do you calculate the determinant of larger matrices efficiently?
4. What are the properties of determinants that make them useful in solving linear systems?
5. How does Cramer's Rule work for systems with more or fewer variables?

**Tip:** When dealing with larger systems, matrix factorizations like LU decomposition can offer a more computationally efficient alternative to Cramer's Rule.

Solve the system of equations using Cramer's Rule.

Learn how to solve a system of 4 linear equations using Cramer's Rule. This example covers key concepts in linear algebra, including determinants and matrix modifications. Suitable for advanced algebra or linear algebra students.

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