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The reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex] as variables. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.

[tex]\[
\left[\begin{array}{lll|l}
1 & 0 & 0 & -7 \\
0 & 1 & 0 & -7 \\
0 & 0 & 0 & -3
\end{array}\right]
\][/tex]

1. What equation does the first row represent?
[tex]\[ \square \][/tex] (Type an equation.)

2. What equation does the second row represent?
[tex]\[ \square \][/tex] (Type an equation.)

3. What equation does the third row represent?
[tex]\[ \square \][/tex] (Type an equation.)

4. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution. Select the correct choice below and, if necessary, fill in the answer box(es) within your choice.

A. The system is consistent. The solution is [tex]\[ \square \][/tex] [tex]\[ \square \][/tex] [tex]\[ \square \][/tex].
(Simplify your answers.)

B. There are infinitely many solutions. The solution can be written as [tex]\(\{(x, y, z) \mid x= \square, y= \square, z \text{ is any real number}\}\)[/tex].
(Simplify your answers. Type expressions using [tex]\(z\)[/tex] as the variable.)

C. There are infinitely many solutions. The solution can be written as [tex]\(\{(x, y, z) \mid x= \square, y \text{ is any real number}, z \text{ is any real number}\}\)[/tex].
(Simplify your answer. Type an expression using [tex]\(y\)[/tex] and [tex]\(z\)[/tex] as the variables.)

D. The system is inconsistent.



Answer :

GinnyAnswer
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.

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