Question

Investigate for what values of λ and μ the equations
2x + 3y + 5z = 9
7x + 3y - 2z = 8
2x + 3y + λz = μ have
A. No solutions
B. Unique solutions
C. An infinite number of solutions.

Answer 1

Consider the system of equation of
2x + 3y + 5z = 9
7x + 3y - 2z = 8
2x + 3y + λz = μ
The above system is given as Ax=B

`[(7,3,-2),(2,3,5),(2,3,lamda)] [(x),(y),(z)]=[(8),(9),(mu)]`

Where A = `[(7,3,-2),(2,3,5),(2,3,lamda)]`     x = `[(x),(y),(z)]`

And B = `[(8),(9),(mu)]`

R₃ – R₂ 

`[(7,3,-2),(2,3,5),(2,3,lamda-5)] [(x),(y),(z)]=[(8),(9),(mu-9)]`

(A) For no solution,
𝜌(𝐴) ≠ 𝜌(𝐴 𝐵)
∴ λ - 5 = 0 and μ - 9 ≠ 0
∴ λ = 5 and μ ≠ 9
(B) For a unique solution
𝜌(𝐴) = 𝜌(𝐴 𝐵) = 3
∴ λ - 5 ≠ 0 and μ may be anything
∴ λ ≠ 5 for all values of μ
(C) For infinite solutions,
𝜌(𝐴) = 𝜌(𝐴 𝐵) < 3
∴ λ - 5 = 0 and μ - 9 = 0
∴ λ = 5 and μ = 9