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.)

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.