Question

The system of equations given by \( \begin{bmatrix} 1 & 1 & 1 & : & 3 \\ 0 & -2 & -2 & : & 4 \\ 1 & -5 & 0 & : & 5 \end{bmatrix} \) has the solution:

• A. \(x = 1, y = 4, z = 0\)
• B. \(x = 3, y = 4, z = 5\)
• C. \(x = 5, y = 0, z = -2\)
• D. \(x = 1, y = 4, z = -1\)

Hint:

For systems of equations, you can quickly check the correct option by substituting the given values of \(x, y, z\) into the original equations. For option (C): (1) \(5 + 0 + (-2) = 3\) (Correct) (2) \(-2(0) - 2(-2) = 4\) (Correct) (3) \(5 - 5(0) = 5\) (Correct) This method is often faster in a multiple-choice exam than solving the system from scratch.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The given matrix is an augmented matrix representing a system of three linear equations in three variables (\(x, y, z\)). We need to solve this system. We can convert the matrix back into equations and solve using substitution or elimination, or perform row operations on the matrix to get it into row-echelon form.

Step 2: Key Formula or Approach:
Convert the augmented matrix into a system of linear equations and solve. The augmented matrix \( [A | B] \) corresponds to the system \(AX = B\).

Step 3: Detailed Explanation:
The augmented matrix is: \[ \left[ \begin{array}{ccc|c} 1 & 1 & 1 & 3 \\ 0 & -2 & -2 & 4 \\ 1 & -5 & 0 & 5 \end{array} \right] \] This translates to the following system of equations: \begin{align} x + y + z &= 3 &(1) \\ -2y - 2z &= 4 &(2) \\ x - 5y &= 5 &(3) \end{align} From equation (2), we can simplify by dividing by -2: \[ y + z = -2 \implies z = -2 - y (4) \] From equation (3), we can express \(x\) in terms of \(y\): \[ x = 5 + 5y \quad (5) \] Now, substitute equations (4) and (5) into equation (1): \[ (5 + 5y) + y + (-2 - y) = 3 \] Simplify the equation to solve for \(y\): \[ 5 + 5y + y - 2 - y = 3 \] \[ 5y + 3 = 3 \] \[ 5y = 0 \] \[ y = 0 \] Now that we have \(y = 0\), we can find \(x\) and \(z\).
Substitute \(y=0\) into equation (5): \[ x = 5 + 5(0) \implies x = 5 \] Substitute \(y=0\) into equation (4): \[ z = -2 - 0 \implies z = -2 \]
Step 4: Final Answer:
The solution to the system of equations is \(x = 5\), \(y = 0\), and \(z = -2\), which corresponds to option (C).