Let's break down the problem step-by-step, starting with identifying the equations represented by the given matrix and then determining the consistency of the system:
1. Identify the Equations from the Matrix:
The matrix provided is:
[tex]\[
\left[\begin{array}{rrrr|r}
1 & 0 & 0 & 2 & -9 \\
0 & 1 & 0 & -4 & -3 \\
0 & 0 & 1 & -9 & 5
\end{array}\right]
\][/tex]
This matrix corresponds to the following system of equations:
- The first row: [tex]\(1 \cdot w + 0 \cdot x + 0 \cdot y + 2 \cdot z = -9\)[/tex], which simplifies to:
[tex]\[
w + 2z = -9
\][/tex]
- The second row: [tex]\(0 \cdot w + 1 \cdot x + 0 \cdot y - 4 \cdot z = -3\)[/tex], which simplifies to:
[tex]\[
x - 4z = -3
\][/tex]
- The third row: [tex]\(0 \cdot w + 0 \cdot x + 1 \cdot y - 9 \cdot z = 5\)[/tex], which simplifies to:
[tex]\[
y - 9z = 5
\][/tex]
2. Determine the Consistency of the System:
To determine if the system is consistent, we need to check for any contradictions in the equations. In this case, there are no contradictory statements, so the system is consistent. As each equation expresses one of the variables in terms of [tex]\(z\)[/tex], the system has infinitely many solutions.
3. Solve the Equations in Terms of [tex]\(z\)[/tex]:
- From the first equation: [tex]\(w + 2z = -9\)[/tex]
[tex]\[
w = -9 - 2z
\][/tex]
- From the second equation: [tex]\(x - 4z = -3\)[/tex]
[tex]\[
x = -3 + 4z
\][/tex]
- From the third equation: [tex]\(y - 9z = 5\)[/tex]
[tex]\[
y = 5 + 9z
\][/tex]
4. Choose the Correct Answer:
Given that the system is consistent and has infinitely many solutions where each variable is expressed in terms of [tex]\(z\)[/tex], the correct choice is:
B. There are infinitely many solutions. The solution can be written as [tex]\((w, x, y, z) \mid w = -9 - 2z, x = -3 + 4z, y = 5 + 9z, z\)[/tex] is any real number.
Hence, the answer in the correct form is:
[tex]\[
(w, x, y, z) \mid w = -9 - 2z, x = -3 + 4z, y = 5 + 9z, \text{ and } z \text{ is any real number}
\][/tex]
