Question

For what value of ‘m’, pair of equations \( x - 2y = 3 \) and \( 3x + my = 1 \) will have a unique solution?

• A. \( m = -6 \)
• B. \( m = 0 \) only
• C. \( m \neq -6 \)
• D. \( m \neq 0 \)

Hint:

For a system of two linear equations to have a unique solution, the determinant of the coefficient matrix must not be zero.

Answer 1

The Correct Option is C

Step 1: Understand the condition for a unique solution.
To determine when a system of linear equations has a unique solution, we need to check the condition on the determinant of the coefficient matrix. The system of equations given is: \[ x - 2y = 3 \quad \text{(1)} \] \[ 3x + my = 1 \quad \text{(2)} \] For a system to have a unique solution, the coefficient matrix: \[ \begin{bmatrix} 1 & -2 \\ 3 & m \end{bmatrix} \] must have a non-zero determinant. The determinant \( D \) of a 2x2 matrix is given by: \[ D = \begin{vmatrix} 1 & -2 \\ 3 & m \end{vmatrix} \]
Step 2: Calculate the determinant.
The determinant of the matrix is: \[ D = (1)(m) - (3)(-2) = m + 6 \]
Step 3: Set the condition for a unique solution.
For a unique solution to exist, the determinant must be non-zero: \[ m + 6 \neq 0 \] \[ m \neq -6 \]
Step 4: Conclusion.
Thus, for the pair of equations to have a unique solution, the value of \( m \) must not be \( -6 \). Therefore, the correct answer is (C).