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

By Ritesh|Updated : November 14th, 2022

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

Step 1: Equations should be changed to standard form.

Considering two equations,

λx + y = λ2 and x + λy = 1

We know that the standard linear equation is given as,

ax + by + c = 0

In the conventional form, the two equations are thus given as,

λx + y = λ2 → λx + y - λ2 = 0

x + λy = 1 → x + λy - 1 = 0

Step 2: Compare the equations to the standard equation.

Using the standard equation we define to compare the two equations,

a1/a2 = b1/b2 = c1/c2

Thus

λ/1 = 1/λ = -λ2/12

Step 3: Solve for λ

Take a look at the first and last components of the equation.

λ = λ2 and λ2 = 1

λ (λ - 1) = 0 and λ = ± 1

λ = 1

As a result, the set of equations has an infinite number of solutions when = 1.

Solution of a Linear Equation -

The points where the lines representing the intersection of two linear equations intersect are referred to as the solution of a linear equation. In other words, the set of all feasible values for the variables that satisfy the specified linear equation constitutes the solution set of the system of linear equations.

Summary:

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

For λ = 1, the pair of linear equations λx + y = λ2 and x + λy = 1 have infinitely many solutions. An equation is a mathematical assertion in which the algebraic expression is separated by an equal sign (=). Equations of degree 1 are linear equations.

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