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

For all real values of 𝜆 except 𝜆 ≠ 1, the pair of linear equations x + y = 2 and x + y = 1 have a unique solution.

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

Solution:

Given, the pair of linear equations is

λx + y = λ²

x + λy = 1

We have to determine the value of λ for which the pair of linear equations have a unique solution.

We know that,

For a pair of linear equations in two variables be a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0,

If \(\frac{a_{1}}{a_{2}}\neq \frac{b_{1}}{b_{2}}\), then

i) the pair of linear equations is consistent

ii) the graph will be a pair of lines intersecting at a unique point, which is the solution of the pair of equations.

Here, a₁ = λ, b₁ = 1, c₁ = λ²

a₂ = 1, b₂ = λ, c₂ = 1

So, a₁/a₂ = λ/1 = λ

b₁/b₂ = 1/λ

c₁/c₂ = λ²/1 = λ²

For unique solution,

\(\frac{a_{1}}{a_{2}}\neq \frac{b_{1}}{b_{2}}\)

So, λ ≠ 1/λ

λ² ≠ 1

λ ≠ +1

So, all real values of λ except +1

Therefore, for all real values of λ except λ ≠ 1, the pair of linear equations have a unique solution.

✦ Try This: For which value(s) of λ, do the pair of linear equations λ x + y = 2λ and x + λ y = 1 have a unique solution

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 3

NCERT Exemplar Class 10 Maths Exercise 3.3 Problem 1 (iii)

Summary:

For all real values of λ except λ ≠ 1, the pair of linear equations x + y = 2 and x + y = 1 have a unique solution.

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