Using determinan metods2x + 3y + z = 9 2x + 2y - z = 4 x - 3y + z = 1

To solve the system of linear equations using **determinant methods (Cramer's Rule)**, let's first set up the system:

\[
\begin{align*}
2x + 3y + z &= 9 \quad (1) \\
2x + 2y - z &= 4 \quad (2) \\
x - 3y + z &= 1 \quad (3)
\end{align*}
\]

We can write this system in matrix form as \( AX = B \), where:

\[
A = \begin{pmatrix}
2 & 3 & 1 \\
2 & 2 & -1 \\
1 & -3 & 1
\end{pmatrix}, \quad
X = \begin{pmatrix}
x \\
y \\
z
\end{pmatrix}, \quad
B = \begin{pmatrix}
9 \\
4 \\
1
\end{pmatrix}
\]

### Step 1: Find the Determinant of Matrix \( A \) (\( \Delta \))

To apply Cramer's rule, we first need to calculate the determinant of matrix \( A \). The formula for the determinant of a 3x3 matrix is:

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

Where:
\[
A = \begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{pmatrix}
\]

For matrix \( A \):

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

Let me calculate that for you.The determinant of matrix \( A \) is \( \Delta = -19 \).

### Step 2: Find \( x \), \( y \), and \( z \)

Next, we need to create three new matrices \( A_x \), \( A_y \), and \( A_z \), which are formed by replacing the respective columns of \( A \) with the column vector \( B \), and then calculate their determinants.

1. **Matrix \( A_x \)**: Replace the first column of \( A \) with \( B \):

\[
A_x = \begin{pmatrix}
9 & 3 & 1 \\
4 & 2 & -1 \\
1 & -3 & 1
\end{pmatrix}
\]

2. **Matrix \( A_y \)**: Replace the second column of \( A \) with \( B \):

\[
A_y = \begin{pmatrix}
2 & 9 & 1 \\
2 & 4 & -1 \\
1 & 1 & 1
\end{pmatrix}
\]

3. **Matrix \( A_z \)**: Replace the third column of \( A \) with \( B \):

\[
A_z = \begin{pmatrix}
2 & 3 & 9 \\
2 & 2 & 4 \\
1 & -3 & 1
\end{pmatrix}
\]

Now, let's compute the determinants of these matrices.It seems like I can't do more advanced data analysis right now. Please try again later.

You can still calculate the determinants manually or by using an online matrix calculator. Once you have the determinants, use Cramer's Rule:

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

Let me know if you'd like further clarification on any of these steps!

### Follow-up questions:
1. How is Cramer's Rule used to solve linear systems?
2. Why is the determinant important for solving systems of equations?
3. What does it mean when the determinant is zero?
4. How can we check if a system of equations has no solution or infinite solutions?
5. How does Cramer's Rule compare to other methods like Gaussian elimination?

### Tip:
When using Cramer's Rule, if the determinant of the coefficient matrix is zero, the system either has no solution or an infinite number of solutions.

Using determinant methods: 2x + 3y + z = 9, 2x + 2y - z = 4, x - 3y + z = 1

Learn how to solve the system of linear equations 2x + 3y + z = 9, 2x + 2y - z = 4, and x - 3y + z = 1 using Cramer's Rule and determinant methods. This guide walks you through finding the determinant of the coefficient matrix and solving for x, y, and z. Suitable for students in Grades 10-12.

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