Use a matrix method to solve: 9y + 2z = 18 3x + 2y + z = 1 x - y = - 3

To solve the system of linear equations using a matrix method, we can use the method of **matrix inversion** or **Gaussian elimination**. Let's start by expressing the system in matrix form:

Given system:
\[
\begin{aligned}
9y + 2z &= 18 \quad \text{(Equation 1)} \\
3x + 2y + z &= 1 \quad \text{(Equation 2)} \\
x - y &= -3 \quad \text{(Equation 3)}
\end{aligned}
\]

### Step 1: Express the system in matrix form
We write the system as \(AX = B\), where \(A\) is the coefficient matrix, \(X\) is the vector of variables, and \(B\) is the constant matrix:

\[
A = \begin{pmatrix}
0 & 9 & 2 \\
3 & 2 & 1 \\
1 & -1 & 0
\end{pmatrix},
\quad
X = \begin{pmatrix}
x \\
y \\
z
\end{pmatrix},
\quad
B = \begin{pmatrix}
18 \\
1 \\
-3
\end{pmatrix}
\]

So, the system becomes:

\[
\begin{pmatrix}
0 & 9 & 2 \\
3 & 2 & 1 \\
1 & -1 & 0
\end{pmatrix}
\begin{pmatrix}
x \\
y \\
z
\end{pmatrix}
=
\begin{pmatrix}
18 \\
1 \\
-3
\end{pmatrix}
\]

### Step 2: Solve using matrix inversion (if the inverse exists)

To solve for \(X\), we compute:

\[
X = A^{-1}B
\]

Let's calculate the inverse of matrix \(A\).

\[
A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A)
\]

Where \(\text{det}(A)\) is the determinant of \(A\) and \(\text{adj}(A)\) is the adjugate of \(A\).

Let's compute this step by step.

### Step 3: Calculate the determinant of \(A\)

\[
\text{det}(A) = 0\cdot(2\cdot0 - 1\cdot(-1)) - 9\cdot(3\cdot0 - 1\cdot1) + 2\cdot(3\cdot(-1) - 2\cdot1)
\]
\[
\text{det}(A) = 0 - 9\cdot(-1) + 2\cdot(-3 - 2)
\]
\[
\text{det}(A) = 9 + 2\cdot(-5) = 9 - 10 = -1
\]

### Step 4: Find the adjugate matrix \( \text{adj}(A) \)

This involves finding the cofactor matrix of \( A \) and then transposing it.

Let's compute it.

### Step 5: Multiply \( A^{-1} \) with \( B \) to find \( X \)

\[
X = A^{-1}B
\]

Now I'll calculate each step and find the solution.### Step 6: Results

1. **Determinant of \(A\)**: The determinant of matrix \(A\) is \(-1\), which is non-zero, so \(A\) is invertible.

2. **Inverse of \(A\)**:
   \[
   A^{-1} = \begin{pmatrix}
   -1 & 2 & -5 \\
   -1 & 2 & -6 \\
   5 & -9 & 27
   \end{pmatrix}
   \]

3. **Solution vector \(X\)**:
   \[
   X = \begin{pmatrix}
   x \\
   y \\
   z
   \end{pmatrix}
   =
   \begin{pmatrix}
   -1 \\
   2 \\
   0
   \end{pmatrix}
   \]

### Final Answer:
- \( x = -1 \)
- \( y = 2 \)
- \( z = 0 \)

Would you like more details on this method or have any other questions? 

Here are 5 additional questions you might find helpful:
1. How do you interpret the determinant of a matrix in the context of linear equations?
2. What is the significance of a matrix being invertible?
3. How can you verify the solution using substitution?
4. Can you solve the system using Gaussian elimination instead of matrix inversion?
5. What are some real-world applications of solving linear systems with matrices?

**Tip:** Always check the determinant first before attempting to find the inverse of a matrix, as a zero determinant indicates that the matrix is singular and does not have an inverse.

Learn how to solve a system of linear equations using matrix methods with detailed steps and examples. This problem involves matrix operations, including matrix inversion and Gaussian elimination, to find solutions for variables x, y, and z. Suitable for advanced high school and college-level students.

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