Question

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

${\displaystyle \frac{a_1}{a_2}=\frac{\lambda }{1}}$

${\displaystyle \frac{b_1}{b_2}=\frac{1}{\lambda }}$

${\displaystyle \frac{c_1}{c_2}={\lambda }^2}$ 

For a unique solution,

${\displaystyle \frac{a_1}{a_2}\ne \frac{b_1}{b_2}}$

So λ ≠ ${\displaystyle \frac{1}{\lambda }}$

Hence, λ² ≠ 1

λ ≠ ±1

So, all real values of λ except ±1.