Question

The system of equations \( 2x + 3y + 5z = 9 \); \( 7x + 3y - 2z = 8 \); \( 2x + 3y + \lambda z = h \) have a unique solution ____ .

• A. For all values of \( \lambda \)
• B. For all values of \( \lambda \) except \( \lambda = 5 \)
• C. Only at \( \lambda = 5 \)
• D. Does not depend on \( \lambda \)

Hint:

For systems of linear equations, always check the determinant of the coefficient matrix to determine if a unique solution exists.

Answer 1

The Correct Option is B

The system of equations will have a unique solution as long as the determinant of the coefficient matrix is non-zero. To find this, we calculate the determinant of the coefficient matrix: \[ \text{det} \begin{bmatrix} 2 & 3 & 5
7 & 3 & -2
2 & 3 & \lambda \end{bmatrix} \] Expanding the determinant, we get: \[ \text{det} = 2 \left( 3 \cdot \lambda - (-2 \cdot 3) \right) - 3 \left( 7 \cdot \lambda - (-2 \cdot 2) \right) + 5 \left( 7 \cdot 3 - 3 \cdot 2 \right) \] This simplifies to: \[ \text{det} = 2 (3\lambda + 6) - 3 (7\lambda + 4) + 5 (21 - 6) \] Simplifying further: \[ \text{det} = 6\lambda + 12 - 21\lambda - 12 + 75 = -15\lambda + 75 \] For the system to have a unique solution, the determinant must be non-zero. Therefore, we set: \[ -15\lambda + 75 \neq 0 \] Solving for \( \lambda \): \[ \lambda \neq 5 \] Thus, the system has a unique solution for all values of \( \lambda \) except when \( \lambda = 5 \).