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Question 7 of 8

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]\( w, x, y, \)[/tex] and [tex]\( z \)[/tex] as variables.

Determine whether the system is consistent or inconsistent. If it is consistent, give the solution.

[tex]\[
\left[\begin{array}{rrrr|r}
1 & 0 & 0 & 2 & -9 \\
0 & 1 & 0 & -4 & -3 \\
0 & 0 & 1 & -9 & 5
\end{array}\right]
\][/tex]

1. What equation does the first row represent?
[tex]\[ w + 2z = -9 \][/tex]

2. What equation does the second row represent?
[tex]\[ x - 4z = -3 \][/tex]

3. What equation does the third row represent?
[tex]\[ y - 9z = 5 \][/tex]

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]\(\left( \square, \square, \square, \square \right) \)[/tex]. (Simplify your answers.)

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

D. The system is inconsistent.

Sagot :

GinnyAnswer
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]
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