https://file.aiquickdraw.com/mathbot/file/1729299319405ihsvqa8w.jpg

The image shows a system of linear equations that needs to be written as a single matrix equation. The system of equations is:

\[
\begin{aligned}
7x - 3y + 6z &= -1 \\
4x + 8y - z &= -13 \\
y - z &= -12
\end{aligned}
\]

### Matrix Representation:

We can represent the system of equations in the form \( A \mathbf{x} = \mathbf{b} \), where:

- \( A \) is the coefficient matrix.
- \( \mathbf{x} \) is the column vector of variables.
- \( \mathbf{b} \) is the column vector of constants.

From the given system:

1. The coefficients of \( x \), \( y \), and \( z \) form the matrix \( A \):
   \[
   A = \begin{bmatrix}
   7 & -3 & 6 \\
   4 & 8 & -1 \\
   0 & 1 & -1
   \end{bmatrix}
   \]

2. The variables \( x \), \( y \), and \( z \) form the vector \( \mathbf{x} \):
   \[
   \mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}
   \]

3. The constants on the right-hand side form the vector \( \mathbf{b} \):
   \[
   \mathbf{b} = \begin{bmatrix} -1 \\ -13 \\ -12 \end{bmatrix}
   \]

### Matrix Equation:
\[
\begin{bmatrix}
7 & -3 & 6 \\
4 & 8 & -1 \\
0 & 1 & -1
\end{bmatrix}
\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} -1 \\ -13 \\ -12 \end{bmatrix}
\]

This is the required matrix equation.

Would you like further explanation or additional examples?

Here are 5 related questions to explore this topic further:
1. How can you solve a system of equations using matrix inversion?
2. What is the determinant of a matrix, and how does it relate to solving systems of equations?
3. How does Gaussian elimination work to solve a system of linear equations?
4. What are the advantages of using matrix methods to solve systems of equations over substitution or elimination methods?
5. How do you perform matrix multiplication in the context of a system of equations?

**Tip:** When working with matrix equations, always check that the matrix is square and non-singular (determinant not zero) if you plan to use matrix inversion to solve the system.

Write the following system of equations as a single matrix equation: 7x - 3y + 6z = -1, 4x + 8y - z = -13, y - z = -12

Learn how to convert the system of equations 7x - 3y + 6z = -1, 4x + 8y - z = -13, y - z = -12 into a matrix equation. Understand the matrix representation of equations and how to solve using matrix methods. Suitable for students in Grades 10-12.

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