https://file.aiquickdraw.com/mathbot/file/1728188577233awh8rn7e.jpg

The system of equations provided in the image is:

\[
\begin{aligned}
3x - 8y + 4z &= -1 \\
8x + 5y &= 6 \\
6y - z &= -9
\end{aligned}
\]

To write this system of equations as a single matrix equation, we express it in the form:

\[
A \mathbf{x} = \mathbf{b}
\]

where:
- \( A \) is the matrix of coefficients,
- \( \mathbf{x} \) is the column vector of variables \([x, y, z]\),
- \( \mathbf{b} \) is the column vector of constants on the right-hand side.

### Step 1: Form the coefficient matrix \( A \)

The coefficient matrix is formed by taking the coefficients of \(x\), \(y\), and \(z\) from each equation.

\[
A = \begin{pmatrix}
3 & -8 & 4 \\
8 & 5 & 0 \\
0 & 6 & -1
\end{pmatrix}
\]

### Step 2: Form the variable vector \( \mathbf{x} \)

\[
\mathbf{x} = \begin{pmatrix}
x \\
y \\
z
\end{pmatrix}
\]

### Step 3: Form the constant vector \( \mathbf{b} \)

\[
\mathbf{b} = \begin{pmatrix}
-1 \\
6 \\
-9
\end{pmatrix}
\]

### Final Matrix Equation

\[
\begin{pmatrix}
3 & -8 & 4 \\
8 & 5 & 0 \\
0 & 6 & -1
\end{pmatrix}
\begin{pmatrix}
x \\
y \\
z
\end{pmatrix}
=
\begin{pmatrix}
-1 \\
6 \\
-9
\end{pmatrix}
\]

This is the single matrix equation representing the given system of equations.

Would you like further details or have any questions?

Here are five related questions for you to explore:
1. How can this matrix equation be solved using Gaussian elimination?
2. What is the determinant of the coefficient matrix, and what does it signify?
3. How would you solve this matrix equation using the inverse of the coefficient matrix?
4. How does this system behave if additional equations or variables are added?
5. What does it mean for a system of equations if the determinant of the coefficient matrix is zero?

**Tip:** When solving systems of equations, always check if the determinant of the coefficient matrix is non-zero. If it is zero, the system might have no unique solution or infinitely many solutions.

Write the following system of equations as a single matrix equation: 3x - 8y + 4z = -1, 8x + 5y = 6, 6y - z = -9

Learn how to represent the system of equations 3x - 8y + 4z = -1, 8x + 5y = 6, and 6y - z = -9 as a single matrix equation. This problem covers key concepts in linear algebra, including matrix representation of systems and column vectors. Suitable for undergraduate students.

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