Question

Find the value of \( k \) for which the equations \( 2x + ky = 1 \) and \( 3x - 5y = 7 \) have a unique solution.

Hint:

For a system of linear equations to have a unique solution, the determinant of the coefficient matrix must be non-zero.

Answer 1

We are given two linear equations: 1. \( 2x + ky = 1 \) 2. \( 3x - 5y = 7 \) For the system of linear equations to have a unique solution, the determinant of the coefficients must be non-zero. The determinant \( D \) of the coefficient matrix is given by: \[ D = \begin{vmatrix} 2 & k \\ 3 & -5 \end{vmatrix}. \] The determinant is calculated as: \[ D = (2)(-5) - (3)(k) = -10 - 3k. \] For the system to have a unique solution, \( D \neq 0 \). Therefore, we require: \[ -10 - 3k \neq 0. \] Solving for \( k \): \[ -3k \neq 10 \quad \implies \quad k \neq -\frac{10}{3}. \]
Conclusion:
The system will have a unique solution if \( k \neq -\frac{10}{3} \).