For what value of k does the system of equations

x + 2y = 3, 5x + ky + 7 = 0

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

Also, show that there is no value of k for which the given system of equations has infinitely many solutions.

(i) Given: x + 2y = 3 – eq 1

5x + ky + 7 = 0 – eq 2

Here,

a₁ = 1, b₁ = 2, c₁ = - 3

a₂ = 5, b₂ = k, c₂ = 7

Given systems of equations has a unique solution

∴ ≠

≠

k ≠10

∴ k ≠ 10

(ii) Given: x + 2y = 3 – eq 1

5x + ky + 7 = 0 – eq 2

Here,

a₁ = 1, b₁ = 2, c₁ = - 3

a₂ = 5, b₂ = k, c₂ = 7

Given that system of equations has no solution

∴ = ≠

Here,

=

Here,

k = 10

∴ k = 10

For the system of equations to have infinitely many solutions

= =

= = which is wrong.

That is, for any value of k the give system of equations cannot have infinitely many solutions.