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

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 infinitely many solutions.

We know that,

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

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

i) The pair of linear equation is dependent and consistent

ii) The graph will be a pair of coincident lines. Each point on the lines will be a solution and so the pair of equations will have infinitely many solutions.

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 infinitely many solutions,

\(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)

λ = 1/λ = λ²

Case 1) λ = 1/λ

λ² = 1

So, λ = ±1

Case 2) λ = λ²

λ² - λ = 0

λ(λ - 1) = 0

So, λ = 0

λ - 1 = 0

λ = 1

So, λ = 1 satisfies both cases.

Therefore, for the value of λ = 1, the pair of linear equations have infinitely many solutions.

✦ Try This: For which value(s) of λ, do the pair of linear equations λ x + y = 2λ and x + λ y = 1 have infinitely many solutions.

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

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

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

Summary:

For the value of λ = 1, the pair of linear equations x + y = 2 and x + y = 1 have infinitely many solutions.

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