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

Solution:

Given, the pair of linear equations are

λx + y = λ²

x + λy = 1

We have to determine the value of λ for which the pair of linear equations have no solution.

We know that,

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

If \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}\neq \frac{c_{1}}{c_{2}}\), then the graph will be a pair of parallel lines and so the pair of equations will have no solution.

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 no solution,

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

So, λ = 1/λ ≠ λ²

Case 1) λ² = 1

λ = ±1

Case 2) λ² ≠ λ

λ ≠ 1

So, λ = -1

Therefore, for the value of λ = -1, the pair of linear equations has no 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 no solution

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

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

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

Summary:

For the value of λ = -1, the pair of linear equations x + y = 2 and x + y = 1 has no solution.

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