Question

Solution of \[ x - y + z = 4, \quad x - 2y + 2z = 9, \quad 2x + y + 3z = 1 \] is

• A. \( x = 3, \, y = 6, \, z = 9 \)
• B. \( x = -4, \, y = -3, \, z = 2 \)
• C. \( x = -1, \, y = -3, \, z = 2 \)
• D. \( x = 2, \, y = 4, \, z = 6 \)

Hint:

When solving a system of linear equations, you can use substitution or elimination to reduce the number of variables and solve the system step by step.

Answer 1

The Correct Option is C

We are given the system of equations: 1. \( x - y + z = 4 \) 2. \( x - 2y + 2z = 9 \) 3. \( 2x + y + 3z = 1 \) We will solve this system using substitution or elimination. Step 1: Solve the first two equations Start with the first and second equations: \[ x - y + z = 4 \quad \text{(Equation 1)} \] \[ x - 2y + 2z = 9 \quad \text{(Equation 2)} \] We subtract Equation 1 from Equation 2 to eliminate \( x \): \[ (x - 2y + 2z) - (x - y + z) = 9 - 4 \] \[ -x + 2y - 2z + x - y - z = 5 \] \[ -y + z = 5 \quad \text{(Equation 4)} \] Step 2: Solve for \( y \) and \( z \) Now solve Equation 4 for \( y \) in terms of \( z \): \[ y = z - 5 \] Substitute this value of \( y \) into Equation 1: \[ x - (z - 5) + z = 4 \] \[ x - z + 5 + z = 4 \] \[ x = -1 \] Step 3: Substitute \( x = -1 \) into the equations Now that we know \( x = -1 \), substitute it into Equation 4: \[ y = z - 5 \] Step 4: Solve the third equation Substitute \( x = -1 \) and \( y = z - 5 \) into the third equation: \[ 2x + y + 3z = 1 \] \[ 2(-1) + (z - 5) + 3z = 1 \] \[ -2 + z - 5 + 3z = 1 \] \[ 4z - 7 = 1 \] \[ 4z = 8 \] \[ z = 2 \] Step 5: Find \( y \) Now substitute \( z = 2 \) into \( y = z - 5 \): \[ y = 2 - 5 = -3 \] Thus, the solution is \( x = -1 \), \( y = -3 \), and \( z = 2 \). Therefore, the correct answer is \( \boxed{x = -1, \, y = -3, \, z = 2} \).