Question

For which value(s) of λ, do the pair of linear equations λx + y = λ² and x + λy = 1 have no solution?

Answer 1

The given pair of linear equations is

λx + y = λ² and x + λy = 1

a₁ = λ, b₁ = 1, c₁ = – λ²

a₂ =1, b₂ = λ, c₂ = –1

The given equations are

λx + y – λ² = 0

x + λy – 1 = 0

Comparing the above equations with ax + by + c = 0

We get,

a₁ = λ, b₁ = 1, c₁ = – λ²

a₂ = 1, b₂ = λ, c₂ = – 1

`a_1/a_2 = λ/1`

`b_1/b_2 = 1/λ`

`c_1/c_2` =  λ²

For no solution,

`a_1/a_2 = b_1/b_2 = c_1/c_2`

i.e. λ = `1/λ ≠ λ^2`

So, λ² = 1

And λ² ≠ λ

Here, we take only λ = – 1

Since the system of linear equations has no solution.