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The given system of linear equations is:

\[
\begin{aligned}
5x - y + 2z &= 22 \\
x + 6y - 3z &= -30 \\
2x - 2y + 7z &= 23
\end{aligned}
\]

### (a) Coefficient Matrix \( A \) and Constant Matrix \( B \):

The coefficient matrix \( A \) is obtained by taking the coefficients of \(x\), \(y\), and \(z\) from each equation, while the constant matrix \( B \) is obtained by the right-hand side constants of each equation.

1. **Coefficient Matrix \( A \):**
\[
A = \begin{pmatrix}
5 & -1 & 2 \\
1 & 6 & -3 \\
2 & -2 & 7
\end{pmatrix}
\]

2. **Constant Matrix \( B \):**
\[
B = \begin{pmatrix}
22 \\
-30 \\
23
\end{pmatrix}
\]

These matrices represent the system of equations in matrix form \( AX = B \), where \( X = \begin{pmatrix} x \\ y \\ z \end{pmatrix} \).

Would you like to proceed with solving the system, or need further details? Let me know!

Here are five related questions to expand on this:

1. How do you solve a system of linear equations using the matrix method?
2. What is the determinant of matrix \( A \), and why is it important for solving the system?
3. How can you apply Gaussian elimination to solve this system?
4. What are the conditions for a system of linear equations to have a unique solution, infinite solutions, or no solution?
5. How can you check if matrix \( A \) is invertible?

**Tip:** The inverse of a matrix can be used to solve systems of linear equations, provided the determinant is non-zero.

Given the system of linear equations as follows: 5x - y + 2z = 22, x + 6y - 3z = -30, 2x - 2y + 7z = 23. (a) Create a coefficient matrix, A, and a constant matrix B.

Learn how to create a coefficient matrix A and constant matrix B for the system of linear equations 5x - y + 2z = 22, x + 6y - 3z = -30, 2x - 2y + 7z = 23. This example uses concepts from linear algebra and is suitable for high school students in Grades 10-12.

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