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
