Question

Solve the system of equations: \[ 2x + 3y + z = 0 \] \[ x + y = 0 \] \[ y + z = 0 \] The given system of equations has:

• A. a unique solution
• B. infinitely many solutions
• C. no solution
• D. a finite number of solutions

Hint:

For systems of linear equations, check for consistency. If the equations represent planes that intersect along a line or a point, there will be infinitely many solutions.

Answer 1

The Correct Option is B

We are given the following system of equations: \[ 2x + 3y + z = 0 \quad {(1)} \] \[ x + y = 0 \quad {(2)} \] \[ y + z = 0 \quad {(3)} \] From equation (2), \( x = -y \). Substitute this in equation (1): \[ 2(-y) + 3y + z = 0 \quad \Rightarrow \quad -2y + 3y + z = 0 \quad \Rightarrow \quad y + z = 0 \] Equation (3) is \( y + z = 0 \), which is identical to the result we got above. Hence, the system is consistent and has infinitely many solutions.