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The problem presented in the image requires solving the system of equations using **Cramer's rule**. The given system of equations is:

\[
\text{(1)} \quad 3x + 2y = 6
\]
\[
\text{(2)} \quad -x + y = 4
\]

### Step-by-Step Solution Using Cramer's Rule:

#### Step 1: Write the system in matrix form \( AX = B \)
The system of equations can be written in matrix form as:

\[
\begin{pmatrix}
3 & 2 \\
-1 & 1
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
6 \\
4
\end{pmatrix}
\]

Here, the coefficient matrix \( A \) is:

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

and the constant matrix \( B \) is:

\[
B = \begin{pmatrix}
6 \\
4
\end{pmatrix}
\]

#### Step 2: Calculate the determinant of matrix \( A \), \( \Delta \)
The determinant of \( A \) is:

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

#### Step 3: Calculate \( \Delta_x \) and \( \Delta_y \)
Now, calculate the determinants \( \Delta_x \) and \( \Delta_y \) by replacing the respective columns of matrix \( A \) with the constants from matrix \( B \).

- For \( \Delta_x \), replace the first column of \( A \) with \( B \):

\[
A_x = \begin{pmatrix} 6 & 2 \\ 4 & 1 \end{pmatrix}
\]

\[
\Delta_x = \text{det}(A_x) = \begin{vmatrix} 6 & 2 \\ 4 & 1 \end{vmatrix} = (6)(1) - (2)(4) = 6 - 8 = -2
\]

- For \( \Delta_y \), replace the second column of \( A \) with \( B \):

\[
A_y = \begin{pmatrix} 3 & 6 \\ -1 & 4 \end{pmatrix}
\]

\[
\Delta_y = \text{det}(A_y) = \begin{vmatrix} 3 & 6 \\ -1 & 4 \end{vmatrix} = (3)(4) - (6)(-1) = 12 + 6 = 18
\]

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

\[
x = \frac{\Delta_x}{\Delta} = \frac{-2}{5} = -\frac{2}{5}
\]

\[
y = \frac{\Delta_y}{\Delta} = \frac{18}{5} = \frac{18}{5}
\]

### Final Answer:
\[
x = -\frac{2}{5}, \quad y = \frac{18}{5}
\]

---

Would you like any further details or clarifications? Here are some related questions for you to consider:

1. How is Cramer's rule different from other methods of solving systems of equations?
2. Can you always use Cramer's rule for any system of equations?
3. What happens if the determinant \( \Delta \) is zero?
4. How would you solve this system of equations using substitution or elimination instead of Cramer's rule?
5. What are the advantages of using matrix methods like Cramer's rule in larger systems of equations?

**Tip:** Cramer's rule is particularly useful for small systems of linear equations (2x2 or 3x3) but becomes computationally expensive for larger systems.

أوجد حل النظام الآتي باستخدام كرامر: 
 3x + 2y = 6 
 -x + y = 4

Learn how to solve the system of linear equations 3x + 2y = 6 and -x + y = 4 using Cramer's rule. This step-by-step guide covers key concepts such as determinants and matrix algebra, suitable for high school students (Grades 10-12).

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