Question

If the system of equations: $$ \begin{aligned} 3x + y + \beta z &= 3 \\2x + \alpha y + z &= 2 \\x + 2y + z &= 4 \end{aligned} $$ has infinitely many solutions, then the value of \( 22\beta - 9\alpha \) is:

• A. 49
• B. 31
• C. 43
• D. 37

Hint:

Infinitely many solutions occur when rank(A) = rank(A|B)<number of variables.

Answer 1

The Correct Option is B

For infinitely many solutions, the rank of the coefficient matrix and the augmented matrix must be equal and less than the number of variables. Using determinant conditions: 
\[ \Delta = \begin{vmatrix} 3 & 1 & \beta 2 & \alpha & 1 \\ 1 & 2 & 1 \end{vmatrix} = 0 \quad \text{and} \quad \Delta_3 = \begin{vmatrix} 3 & 1 & 3\\ 2 & \alpha & 2 \\1 & 2 & 4 \end{vmatrix} = 0 \] Solving gives \( \alpha = \frac{19}{9}, \beta = \frac{6}{11} \), so: \[ 22\beta - 9\alpha = 31 \]