Question

For which value(s) of λ, do the pair of linear equations λx + y = λ² and x + λy = 1 have infinitely many solutions?

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 = λ^2` 

For infinitely many solutions,

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

i.e. λ = `1/λ` = λ²

So λ = `1/λ` gives λ = ±1

λ = λ² gives λ = 1, 0

Hence satisfying both the equations

λ = 1 is the answer.