Question

Find the value of k for which the system of equations

8x + 5y = 9, kx + 10y = 15

has a non-zero solution.

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

8x + 5y = 9, kx + 10y = 15

Answer 1

The given system of equations:

8x + 5y = 9

8x + 5y – 9 = 0   ...(i)

kx + 10y = 15

kx + 10y – 15 = 0   ...(ii)

These equations are of the following form:

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

where, a₁ = 8, b₁ = 5, c₁ = –9 and a₂ = k, b₂ = 10, c₂ = –15

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., `8/k = 5/10 ≠ (-9)/(-15)`

i.e. `8/k = 1/2 ≠ 3/5`

`8/k = 1/2` and `8/k ≠ 3/5`

⇒ k = 16 and k ≠ `40/3`

Hence, the given system of equations has no solutions when k is equal to 16.