To determine the system of linear equations represented by the given augmented matrix:
[tex]\[
\left[\begin{array}{rrr|r}
4 & 9 & -4 & -12 \\
9 & 2 & 7 & 6 \\
4 & -2 & 2 & -1
\end{array}\right]
\][/tex]
we need to convert each row of the matrix into its corresponding linear equation. Each row consists of the coefficients of the variables [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex], followed by the constant term after the vertical bar.
1. The first row, [tex]\([4, 9, -4, -12]\)[/tex], corresponds to the equation:
[tex]\[ 4x + 9y - 4z = -12 \][/tex]
2. The second row, [tex]\([9, 2, 7, 6]\)[/tex], corresponds to the equation:
[tex]\[ 9x + 2y + 7z = 6 \][/tex]
3. The third row, [tex]\([4, -2, 2, -1]\)[/tex], corresponds to the equation:
[tex]\[ 4x - 2y + 2z = -1 \][/tex]
Now, compiling these equations, we have the system of linear equations:
[tex]\[
\begin{cases}
4x + 9y - 4z = -12 \\
9x + 2y + 7z = 6 \\
4x - 2y + 2z = -1
\end{cases}
\][/tex]
Comparing this with the provided options:
- Option a:
[tex]\[
\begin{cases}
4x = -12 \\
2y = 6 \\
2z = -1
\end{cases}
\][/tex]
This does not match our system.
- Option b:
[tex]\[
\begin{cases}
4x + 9y + 4z = -12 \\
9x + 2y - 2z = 6 \\
-4x + 7y + 2z = -1
\end{cases}
\][/tex]
This does not match our system either.
- Option c:
[tex]\[
\begin{cases}
4x + 9y - 4z = -12 \\
9x + 2y + 7z = 6 \\
4x - 2y + 2z = -1
\end{cases}
\][/tex]
This matches our system of equations exactly.
- Option d:
[tex]\[
\begin{cases}
4x + 9y - 4z = 12 \\
9x + 2y + 7z = -6 \\
4x - 2y + 2z = 1
\end{cases}
\][/tex]
This does not match our system.
Therefore, the correct option is:
c. [tex]\(4x + 9y - 4z = -12\)[/tex], [tex]\(9x + 2y + 7z = 6\)[/tex], [tex]\(4x - 2y + 2z = -1\)[/tex].
