Solve the following system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent. Select the correct choice below and, if necessary, fill in the answer box(es) in your choice. A. The solution is ( (Simplify your answers.) B. There are infinitely many solutions. The solution can be written as \( \{(x, y, z) \mid x=\square, y=\square, ~ \) (Simplify your answers. Type expressions using \( z \) as the variable.) C. There are infinitely many solutions. The solution can be written as \( \{(x, y, z) \mid x=\square, y \) is any real number \( \} \), (Simplify your answer. Type an expression using \( y \) and \( z \) as the variables.) D. The system is inconsistent.

Let's say we have the following system of equations: 1. \( x + 2y + z = 4 \) 2. \( 2x + y - z = 1 \) 3. \( 3x + 4y + 5z = 9 \) To solve the system using matrices, we can create the augmented matrix: \[ \begin{bmatrix} 1 & 2 & 1 & | & 4 \\ 2 & 1 & -1 & | & 1 \\ 3 & 4 & 5 & | & 9 \end{bmatrix} \] Now we perform row operations to reduce this matrix to row echelon form or reduced row echelon form. Performing the row operations, we can simplify the matrix step by step to isolate the variables. After a series of operations, if we reach a row indicating a contradiction (for example, a row that simplifies to \( 0 = 5 \)), the system is inconsistent, indicating no solutions. However, if we find free variables (often represented by parameters), we could express the solutions in terms of those parameters to indicate infinitely many solutions. Let’s assume we conducted the operations and found that the final matrix leads us to the conclusion of an inconsistency. In that case, we would select: D. The system is inconsistent.