The reduced row echelon form of the augmented matrix of a system of equations is given. Find the solutions of the system. $$ \left[\begin{array}{rrrr|r} 1 & 0 & 0 & 9 & 13 \ 0 & 1 & 0 & 1 & -8 \ 0 & 0 & 1 & 0 & -1 \ 0 & 0 & 0 & 0 & 0 \end{array}\right] $$ Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The system has exactly one solution. The solution is $$ \square $$ , , , ). (Type integers or fractions.) B. The system has infinitely many solutions of the form $$ \square $$ , $$ \square $$ ,w), where $$ w $$ is any real number. (Type integers or fractions.)

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UpStudy AI · Accepted Answer

To analyze the given reduced row echelon form of the augmented matrix, we can interpret it in terms of a system of equations. The matrix represents the following system:

1. $$ x_1 + 9x_4 = 13 $$
2. $$ x_2 + x_4 = -8 $$
3. $$ x_3 = -1 $$

The fourth row, which consists entirely of zeros, indicates that there are no additional constraints on the system, meaning that the system is consistent.

### Step 1: Express the variables in terms of free variables

From the equations, we can express $$ x_1 $$ and $$ x_2 $$ in terms of the free variable $$ x_4 $$:

1. From the first equation:
   $$
   x_1 = 13 - 9x_4
   $$

2. From the second equation:
   $$
   x_2 = -8 - x_4
   $$

3. The third equation gives us:
   $$
   x_3 = -1
   $$

### Step 2: Identify the free variable

Here, $$ x_4 $$ is a free variable, meaning it can take any real number value. We can denote it as $$ w $$.

### Step 3: Write the solution in terms of $$ w $$

Substituting $$ w $$ for $$ x_4 $$, we can express the solution as follows:

- $$ x_1 = 13 - 9w $$
- $$ x_2 = -8 - w $$
- $$ x_3 = -1 $$
- $$ x_4 = w $$

Thus, the solution can be written in the form:
$$
(x_1, x_2, x_3, x_4) = (13 - 9w, -8 - w, -1, w)
$$

### Conclusion

Since $$ w $$ can take any real number value, the system has infinitely many solutions.

The correct choice is:
B. The system has infinitely many solutions of the form $$ (13 - 9w, -8 - w, -1, w) $$, where $$ w $$ is any real number.
The system has infinitely many solutions of the form $$ (13 - 9w, -8 - w, -1, w) $$, where $$ w $$ is any real number.

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