Question

Solution of the system of equations \( 3x + y + 2z = 3 \), \( 2x - 3y - z = -3 \), \( x + 2y + z = 4 \)

• A. \( (1, 2, 1)
• B. \( (1, 2, -1)
• C. \( (2, 3, 1)
• D. \( (4, 3, 2)

Hint:

For systems of linear equations, matrix methods like Gaussian elimination are useful to find the solution efficiently.

Answer 1

The Correct Option is B

We are given the system of equations:

\[ 3x + y + 2z = 3 \quad \text{(1)} \] \[ 2x - 3y - z = -3 \quad \text{(2)} \] \[ x + 2y + z = 4 \quad \text{(3)} \]

Step 1: Use Equation (3) to express \( z \) in terms of \( x \) and \( y \):

\[ z = 4 - x - 2y \quad \text{(4)} \]

Step 2: Substitute \( z = 4 - x - 2y \) into Equations (1) and (2):

Substitute into Equation (1):

\[ 3x + y + 2(4 - x - 2y) = 3 \]

Simplify:

\[ 3x + y + 8 - 2x - 4y = 3 \]

\[ x - 3y = -5 \quad \text{(5)} \]

Substitute into Equation (2):

\[ 2x - 3y - (4 - x - 2y) = -3 \]

Simplify:

\[ 2x - 3y - 4 + x + 2y = -3 \]

\[ 3x - y = 1 \quad \text{(6)} \]

Step 3: Solve the system of two equations (5) and (6):

From Equation (5): \( x = 3y - 5 \), substitute this into Equation (6):

\[ 3(3y - 5) - y = 1 \]

\[ 9y - 15 - y = 1 \]

\[ 8y = 16 \]

\[ y = 2 \]

Step 4: Substitute \( y = 2 \) into \( x = 3y - 5 \) from Equation (5):

\[ x = 3(2) - 5 = 6 - 5 = 1 \]

Step 5: Substitute \( x = 1 \) and \( y = 2 \) into Equation (4) to find \( z \):

\[ z = 4 - 1 - 2(2) = 4 - 1 - 4 = -1 \]

Thus, the solution to the system of equations is \( (x, y, z) = (1, 2, -1) \), which corresponds to option (B).