Question

Write the value of k for which the system of equations 3x – 2y = 0 and kx + 5y = 0 has infinitely many solutions.

Answer 1

Given: 3x – 2y = 0 and kx + 5y = 0

For a homogeneous pair a₁x + b₁y = 0 and a₂x + b₂y = 0 to have infinitely many (nonzero) solutions, the coefficient ratios must satisfy `a_1/a_2 = b_1/b_2`. 

Here a₁ = 3, b₁ = –2, a₂ = k, b₂ = 5, so `3/k = (-2)/5` 

Cross-multiply: 3 × 5 = k × (–2) 

⇒ 15 = –2k 

⇒ `k = (-15)/2`

`k = (-15)/2` the two equations become proportional, so the system has infinitely many solutions.