First slide

System of linear equations

Question

The system of linear equations $4 x + λ y + 6 z = 10, 2 λ x + 3 y + 5 z = 8, λ x + 2 y + 2 z = 5$ for $λ = 2$ , has

Moderate

A

Unique solution
B

Infinite solutions
C

No solutions
D

two solutions

Solution

$W e h a v e$
$Δ = |\begin{matrix} 4 & λ & 6 \\ 2 λ & 3 & 5 \\ λ & 2 & 2 \end{matrix}| = 4 (6 - 10) - λ (4 λ - 5 λ) + 6 (4 λ - 3 λ) = λ^{2} + 6 λ - 16 = (λ + 8) (λ - 2) I f λ = 2, t h e n Δ = 0 N o w f o r λ = 2, w e h a v e Δ_{1} = |\begin{matrix} 10 & 2 & 6 \\ 8 & 3 & 5 \\ 5 & 2 & 2 \end{matrix}| = 10 (6 - 10) - 2 (16 - 25) + 6 (16 - 15) = - 16 \neq 0 ∴ W e h a v e Δ = 0 a n d Δ_{1} \neq 0 \Rightarrow T h e s y s t e m h a s n o s o l u t i o n$

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