Question

Write the values of k for which the system of equations x + ky = 0, 2x – y = 0 has unique solution.

Answer 1

Given: x + ky = 0, 2x – y = 0

Step-wise calculation:

1. For a unique solution the coefficient matrix must be non-singular (its determinant ≠ 0). This is the standard criterion: `a_1/a_2 ≠ b_1/b_2` for two linear equations.

2. Form the determinant:

`|(1, k),(2, -1)| = (1)(-1) - (2)(k)` 

= –1 – 2k

3. Require –1 – 2k ≠ 0 

⇒ 2k + 1 ≠ 0

⇒ `k ≠ (-1)/2`

The system has a unique solution for all real k except k = `(-1)/2`.