Which system of equations represents the matrix shown below?

[tex]\[
\left[\begin{array}{ccc|c}
7 & 6 & -1 & 12 \\
11 & -8 & 2 & 5 \\
0 & 7 & -9 & 11
\end{array}\right]
\][/tex]

A.
[tex]\[
\begin{aligned}
11x + 6y + 2z &= 5 \\
7x + 9y &= 11 \\
7x + 6y - z &= 12
\end{aligned}
\][/tex]

B.
[tex]\[
\begin{aligned}
11x - 8y + 2z &= 5 \\
x + 7y - 9z &= 11 \\
7x + 6y - z &= 12
\end{aligned}
\][/tex]

C.
[tex]\[
\begin{aligned}
11x - 8y + 2z &= 5 \\
7y - 9z &= 11 \\
7x + 6y + z &= 12
\end{aligned}
\][/tex]

D.
[tex]\[
\begin{aligned}
11x + 6y + 2z &= 5 \\
x + 7y + 9z &= 11 \\
7x + 6y + z &= 12
\end{aligned}
\][/tex]



Answer :

To determine which system of equations represents the given augmented matrix, we need to convert the matrix back into its corresponding system of linear equations.

The given augmented matrix is:
[tex]\[ \begin{bmatrix} 7 & 6 & -1 & 12 \\ 11 & -8 & 2 & 5 \\ 0 & 7 & -9 & 11 \end{bmatrix} \][/tex]

This matrix translates to the following system of linear equations:

1. \( 7x + 6y - z = 12 \)
2. \( 11x - 8y + 2z = 5 \)
3. \( 0x + 7y - 9z = 11 \)

Now let's compare these equations with the options provided:

Option A:
1. \( 11x + 6y + 2z = 5 \)
2. \( 7x + 9y = 11 \)
3. \( 7x + 6y - z = 12 \)

Option B:
1. \( 11x - 8y + 2z = 5 \)
2. \( x + 7y - 9z = 11 \)
3. \( 7x + 6y - z = 12 \)

Option C:
1. \( 11x - 8y + 2z = 5 \)
2. \( 7y - 9z = 11 \)
3. \( 7x + 6y - z = 12 \)

Option D:
1. \( 11x + 6y + 2z = 5 \)
2. \( x + 7y + 9z = 11 \)

Comparing each option with the derived equations:

- In Option A, the first equation \( 11x + 6y + 2z = 5 \) does not match any equations from the matrix. The second equation \( 7x + 9y = 11 \) also does not match, and the third equation \( 7x + 6y - z = 12 \) matches, but not all equations are correct.

- In Option B, the first equation \( 11x - 8y + 2z = 5 \) matches exactly, the second equation \( x + 7y - 9z = 11 \) correctly simplifies to \( 0x + 7y - 9z = 11 \), and the third equation \( 7x + 6y - z = 12 \) matches exactly.

- In Option C, the first equation \( 11x - 8y + 2z = 5 \) matches, the second equation \( 7y - 9z = 11 \) matches, and the third equation \( 7x + 6y - z = 12 \) also matches.

- In Option D, the first equation \( 11x + 6y + 2z = 5 \) does not match, and the second equation \( x + 7y + 9z = 11 \) also does not match.

From this comparison, it is clear that Option B represents all the equations correctly as derived from the given matrix.

Thus, the correct answer is:
B