To solve the given problem, we need to interpret the reduced row echelon form matrix and determine the system of equations it represents.
The matrix given is:
[tex]\[
\left[\begin{array}{lll|l}
1 & 0 & 0 & -7 \\
0 & 1 & 0 & -7 \\
0 & 0 & 0 & -3
\end{array}\right]
\][/tex]
Each row in this matrix represents an equation. Let’s break it down row by row:
1. First Row:
The first row of the matrix is [tex]\([1 \quad 0 \quad 0 \mid -7]\)[/tex]. This represents the equation:
[tex]\[
1x + 0y + 0z = -7
\][/tex]
Simplifying this, we get:
[tex]\[
x = -7
\][/tex]
2. Second Row:
The second row of the matrix is [tex]\([0 \quad 1 \quad 0 \mid -7]\)[/tex]. This represents the equation:
[tex]\[
0x + 1y + 0z = -7
\][/tex]
Simplifying this, we get:
[tex]\[
y = -7
\][/tex]
3. Third Row:
The third row of the matrix is [tex]\([0 \quad 0 \quad 0 \mid -3]\)[/tex]. This represents the equation:
[tex]\[
0x + 0y + 0z = -3
\][/tex]
Simplifying this, we get:
[tex]\[
0 = -3
\][/tex]
This third equation [tex]\(0 = -3\)[/tex] is a contradiction because it states that 0 is equal to -3, which is clearly false.
### Consistency Check:
A system of linear equations is consistent if there is at least one solution and inconsistent if there are no solutions.
Since we have encountered the equation [tex]\(0 = -3\)[/tex], which is a contradiction, this means the system of equations does not have any solutions.
### Answer:
The system is inconsistent. Therefore, we choose:
D. The system is inconsistent.
