Solve the following system of linear equations using Matrices as well as Cramer’s rule.

2x – y + z = 5 4x +2y + 3z = 8 3x – 4y – z = 3

### System of Linear Equations:
We are given the system:

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

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

\[
A \cdot X = B
\]

Where:
\[
A = \begin{pmatrix} 
2 & -1 & 1 \\
4 & 2 & 3 \\
3 & -4 & -1 
\end{pmatrix}, 
\quad X = \begin{pmatrix} 
x \\
y \\
z 
\end{pmatrix}, 
\quad B = \begin{pmatrix} 
5 \\
8 \\
3 
\end{pmatrix}
\]

### Solution using Matrices (Inverse Method):
To solve using matrices, we need to find \( X = A^{-1} \cdot B \), where \( A^{-1} \) is the inverse of matrix \( A \).

1. **Find the determinant of \( A \)**.
2. **Find the inverse of matrix \( A \)**.
3. **Multiply the inverse matrix \( A^{-1} \) by matrix \( B \)**.

Let's proceed to calculate these steps.

#### Step 2: Calculate the determinant of matrix \( A \).

The determinant of a 3x3 matrix \( A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \) is:

\[
\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
\]

For our matrix \( A \):

\[
A = \begin{pmatrix} 
2 & -1 & 1 \\
4 & 2 & 3 \\
3 & -4 & -1 
\end{pmatrix}
\]

Let's compute this determinant.

\[
\text{det}(A) = 2 \left( (2 \cdot -1) - (3 \cdot -4) \right) - (-1) \left( (4 \cdot -1) - (3 \cdot 3) \right) + 1 \left( (4 \cdot -4) - (2 \cdot 3) \right)
\]

Now, solving for the determinant.

#### Step 3: Solve using Cramer's Rule.

Cramer's Rule states that the solution to a system of linear equations can be found using determinants. The solution for \( x, y, z \) is:

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

Where \( A_x, A_y, A_z \) are matrices formed by replacing the columns of \( A \) with matrix \( B \). I'll calculate these steps.

Let me proceed to compute the necessary determinants.### Solution Using Matrix Inverse:

The determinant of matrix \( A \) is:

\[
\text{det}(A) = -15
\]

Since the determinant is non-zero, we can compute the inverse of \( A \). The solution for the system using the inverse matrix is:

\[
\begin{pmatrix}
x \\
y \\
z
\end{pmatrix}
= \begin{pmatrix}
0.333 \\
-1.267 \\
3.067
\end{pmatrix}
\]

Thus, the solution is:
\[
x = 0.333, \quad y = -1.267, \quad z = 3.067
\]

### Solution Using Cramer's Rule:
The determinants for Cramer's rule are:

\[
\text{det}(A_x) = 5, \quad \text{det}(A_y) = 19, \quad \text{det}(A_z) = -46
\]

Using Cramer's rule:

\[
x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{5}{-15} = 0.333
\]
\[
y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{19}{-15} = -1.267
\]
\[
z = \frac{\text{det}(A_z)}{\text{det}(A)} = \frac{-46}{-15} = 3.067
\]

The solutions for both methods match:
\[
x = 0.333, \quad y = -1.267, \quad z = 3.067
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you calculate the inverse of a 3x3 matrix?
2. What is the significance of a non-zero determinant in a matrix?
3. Can you explain how to derive Cramer's rule for 3 variables?
4. What would happen if the determinant of \( A \) was zero?
5. How does Cramer's rule apply to systems with more than three variables?

### Tip:
When the determinant of a matrix is zero, the system of equations either has no solution or infinitely many solutions, making the system singular.

Solve the following system of linear equations using Matrices as well as Cramer’s rule. 2x – y + z = 5, 4x + 2y + 3z = 8, 3x – 4y – z = 3

Learn how to solve the system of linear equations 2x – y + z = 5, 4x + 2y + 3z = 8, 3x – 4y – z = 3 using matrix inversion and Cramer's Rule. This detailed solution covers key concepts from linear algebra, including matrix form, determinants, and Cramer's Rule. Suitable for students in Grades 10-12.

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