Question

Find the value of k for which the following system of equations has no solution:

3x – y – 5 = 0, 6x – 2y + k = 0 (k ≠ 0)

Answer 1

The given system of equations:

3x – y – 5 = 0   ...(i)

And 6x – 2y + k = 0   ...(ii)

These equations are of the following form:

a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0

where, a₁ = 3, b₁ = –1, c₁ = –5 and a₂ = 6, b₂ = –2, c₂ = k

In order that the given system has no solution, we must have:

`(a_1)/(a_2) = (b_1)/(b_2) ≠ (c_1)/(c_2)`

i.e., `3/6 = (-1)/(-2) ≠ (-5)/k`

⇒ `(-1)/(-2) ≠ (-5)/k` 

⇒ k ≠ –10

Hence, equations (i) and (ii) will have no solution if k ≠ –10.