Step 1: Concept Explanation:
The provided matrix is an augmented matrix representing a system of three linear equations with three variables \( (x, y, z) \). The goal is to solve for these variables. This can be achieved by converting the matrix back into equations and using substitution or elimination methods, or by applying row operations to transform the matrix into row-echelon form.
Step 2: Core Method:
The primary approach involves converting the augmented matrix into a system of linear equations and solving for the unknowns. The augmented matrix \( [A | B] \) represents the system \( AX = B \).
Step 3: Solution Process:
The given augmented matrix is:
\[ \left[ \begin{array}{ccc|c} 1 & 1 & 1 & 3 \\ 0 & -2 & -2 & 4 \\ 1 & -5 & 0 & 5 \end{array} \right] \]
This corresponds to the following system of equations:
\begin{align}
x + y + z &= 3 &(1) \\\
-2y - 2z &= 4 &(2) \\\
x - 5y &= 5 &(3)
\end{align}
Simplifying equation (2) by dividing by -2 yields:
\[ y + z = -2 \implies z = -2 - y (4) \]
From equation (3), \( x \) can be expressed in terms of \( y \):
\[ x = 5 + 5y \quad (5) \]
Substituting equations (4) and (5) into equation (1):
\[ (5 + 5y) + y + (-2 - y) = 3 \]
Simplifying and solving for \( y \):
\[ 5 + 5y + y - 2 - y = 3 \]
\[ 5y + 3 = 3 \]
\[ 5y = 0 \]
\[ y = 0 \]
With \( y = 0 \), we can now determine \( x \) and \( z \).
Substituting \( y = 0 \) into equation (5):
\[ x = 5 + 5(0) \implies x = 5 \]
Substituting \( y = 0 \) into equation (4):
\[ z = -2 - 0 \implies z = -2 \]
Step 4: Solution:
The solution to the system of equations is \( x = 5 \), \( y = 0 \), and \( z = -2 \), corresponding to option (C).