The matrix associated with the solution to a system of linear equations in \( x, y \), and \( z \) is given. Write the solution to the system, if it exists. \( \left[\begin{array}{rrr|r}1 & 0 & -6 & 5 \\ 0 & 1 & 4 & 5 \\ 0 & 0 & 0 & 0\end{array}\right] \) Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. \( x=\square, y=\square \), and \( z=\square \) (Simplify your answers.) B. There are infinitely many solutions. If \( z \) is allowed to be any real number, then \( x=\square \) and \( y=\square \). (Type expressions usinaz as the variable.)

To solve the system of linear equations represented by the augmented matrix \[ \left[\begin{array}{rrr|r} 1 & 0 & -6 & 5 \\ 0 & 1 & 4 & 5 \\ 0 & 0 & 0 & 0 \end{array}\right], \] we can interpret this matrix as a system of equations: 1. \( x - 6z = 5 \) 2. \( y + 4z = 5 \) The third row, which consists entirely of zeros, indicates that there are no additional constraints on the variables, suggesting that the system has infinitely many solutions. To express the solutions in terms of \( z \), we can solve for \( x \) and \( y \): From the first equation: \[ x = 5 + 6z \] From the second equation: \[ y = 5 - 4z \] Thus, we can express the solution in terms of \( z \): - \( x = 5 + 6z \) - \( y = 5 - 4z \) - \( z = z \) (where \( z \) can be any real number) Therefore, the correct choice is: B. There are infinitely many solutions. If \( z \) is allowed to be any real number, then \[ x = 5 + 6z \quad \text{and} \quad y = 5 - 4z. \]