Use the Gauss-Jordan method to solve the following system of equations. \[ \begin{aligned} 7 x-4 y+4 z & =6 \\ 7 x+3 y-z & =11 \\ 21 x-5 y+7 z & =23 \end{aligned} \] Write the augmented matrix for the corresponding system of equations. Select the correct choice below and fill the answer boxes to complete your choice. A. $\left[\begin{array}{ccc|c}7 & -4 & 4 & 6 \\ 7 & 3 & -1 & 11 \\ 21 & -5 & 7 & 23\end{array}\right]$ Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution is ( , , in the order $x, y, z$. (Simplify your answers.) B. There is an infinite number of solutions. The solution is $\square, \square, z$, where $z$ is any real number. (Simplify your answers. Use integers or fractions for any numbers in the expressions.) C. There is no solution.

Step 1 ：First, we write the system of equations as an augmented matrix: \[\left[\begin{array}{ccc|c}7 & -4 & 4 & 6 \\ 7 & 3 & -1 & 11 \\ 21 & -5 & 7 & 23\end{array}\right]\]

Step 2 ：Next, we subtract the first row from the second and third rows to eliminate the x-coefficients in the second and third rows: \[\left[\begin{array}{ccc|c}7 & -4 & 4 & 6 \\ 0 & 7 & -5 & 5 \\ 0 & -1 & 3 & 17\end{array}\right]\]

Step 3 ：Then, we multiply the second row by 1/7 and add the second row to the third row to eliminate the y-coefficient in the third row: \[\left[\begin{array}{ccc|c}7 & -4 & 4 & 6 \\ 0 & 1 & -5/7 & 5/7 \\ 0 & 0 & 16/7 & 22/7\end{array}\right]\]

Step 4 ：We multiply the third row by 7/16 to make the z-coefficient in the third row equal to 1: \[\left[\begin{array}{ccc|c}7 & -4 & 4 & 6 \\ 0 & 1 & -5/7 & 5/7 \\ 0 & 0 & 1 & 11/8\end{array}\right]\]

Step 5 ：We add 4 times the third row to the first row and add 5/7 times the third row to the second row to eliminate the z-coefficients in the first and second rows: \[\left[\begin{array}{ccc|c}7 & -4 & 0 & 2 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 11/8\end{array}\right]\]

Step 6 ：We add 4 times the second row to the first row to eliminate the y-coefficient in the first row: \[\left[\begin{array}{ccc|c}7 & 0 & 0 & 6 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 11/8\end{array}\right]\]

Step 7 ：We divide the first row by 7 to make the x-coefficient in the first row equal to 1: \[\left[\begin{array}{ccc|c}1 & 0 & 0 & 6/7 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 11/8\end{array}\right]\]

Step 8 ：Finally, we read off the solutions from the last column of the matrix. The solution is \(\boxed{(6/7, 1, 11/8)}\) in the order \(x, y, z\).