Solve the following system of equations by using the inverse of the coefficient matrix. 

negative 8−8xminus−yplus+66z

equals=negative 33−33

88yplus+55z

equals=1919

22xplus+33yplus+66z

equals=99

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Part 1

Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

A.

The solution of the system is 

enter your response here.

​(Simplify your answer. Type an ordered​ triple.)

B.

There are infinitely many solutions. The solutions are 

xequals=enter your response here 

and 

yequals=enter your response here​, 

where z is any real number.

​(Use integers or fractions for any numbers in the​ expression.)

C.

There is no solution.

To solve the system of equations using the inverse of the coefficient matrix, let's first represent the system in matrix form:

Given system of equations:

\[
\begin{aligned}
-8x - y + 6z &= -33 \\
8y + 5z &= 19 \\
2x + 3y + 6z &= 9
\end{aligned}
\]

We can represent this system as \( A \cdot X = B \), where:

- \( A \) is the coefficient matrix,
- \( X \) is the column vector of variables \([x, y, z]\),
- \( B \) is the column vector of constants.

### Step 1: Coefficient Matrix \(A\)
The coefficient matrix \( A \) is:

\[
A = \begin{pmatrix}
-8 & -1 & 6 \\
0 & 8 & 5 \\
2 & 3 & 6
\end{pmatrix}
\]

### Step 2: Column Vector of Variables \(X\)
The column vector of variables \(X\) is:

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

### Step 3: Column Vector of Constants \(B\)
The column vector of constants \(B\) is:

\[
B = \begin{pmatrix} -33 \\ 19 \\ 9 \end{pmatrix}
\]

### Step 4: Solving using the inverse of \(A\)

We can find the solution \(X = A^{-1} \cdot B\), but first we need to calculate \(A^{-1}\), the inverse of the coefficient matrix.

Let's compute \(A^{-1}\) and then multiply it by \(B\).

The solution to the system of equations is:

\[
x = 3, \quad y = 3, \quad z = -1
\]

Thus, the correct choice is **A**: The solution of the system is \( (3, 3, -1) \).

Would you like more details on how the inverse matrix was calculated or any other part of the process?

Here are 5 related questions for further practice:
1. How do you calculate the inverse of a 3x3 matrix?
2. What are other methods to solve a system of linear equations besides using the inverse matrix?
3. Can all systems of equations be solved using matrix inversion?
4. What does it mean if the determinant of a matrix is zero?
5. How can you verify the solution to a system of equations?

**Tip:** When solving systems of equations, if the determinant of the coefficient matrix is zero, the system may have no solution or infinitely many solutions.

Solve the following system of equations by using the inverse of the coefficient matrix: 
-8x - y + 6z = -33 
8y + 5z = 19 
2x + 3y + 6z = 9

Learn how to solve a system of linear equations using the inverse of the coefficient matrix. This problem involves three variables and covers important concepts like matrix inversion and matrix multiplication. Suitable for high school students in Grades 10-12.

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