For what value of k does the system of equations

x + 2y = 5, 3x + ky + 15 = 0

have (i) a unique solution, (ii) no solution?

(i) Given: x + 2y = 5 – eq 1

3x + ky + 15 = 0 – eq 2

Here,

a₁ = 1, b₁ = 2, c₁ = - 5

a₂ = 3, b₂ = k, c₂ = 15

Given systems of equations has a unique solution

∴ ≠

≠

k ≠6

∴ k ≠ 6

(ii) Given: x + 2y = 5 – eq 1

3x + ky + 15 = 0 – eq 2

Here,

a₁ = 1, b₁ = 2, c₁ = - 5

a₂ = 3, b₂ = k, c₂ = 15

Given that systems of equations has no solution

∴ = ≠

Here,

=

Here,

k = 6

∴ k = 6