First slide

System of linear equations

Question

Let $λ$ be a real number for which the system of linear equations $x + y + z = 6, 4 x + λ y - λ z = λ - 2$ and $3 x + 2 y - 4 z = - 5$ has infinitely many solutions . Then is a root of the quadratic equation

Moderate

A

$λ^{2} - 3 λ - 4 = 0$
B

$λ^{2} + 3 λ - 4 = 0$
C

$λ^{2} - λ - 6 = 0$
D

$λ^{2} + λ - 6 = 0$

Solution

Given, system of linear equations
$\begin{aligned} x + y + z = 6 \\ 4 x + λ y - λ z = λ - 2 \end{aligned}$
$And 3 x + 2 y - 4 z = - 5$
$Has infinitely manv solutions then Δ = 0$
$\begin{array}{l} \Rightarrow |\begin{array}{ccc} 1 & 1 & 1 \\ 4 & λ & - λ \\ 3 & 2 & - 4 \end{array}| = 0 \\ \Rightarrow 1 (- 4 λ + 2 λ) - 1 (- 16 + 3 λ) + 1 (8 - 3 λ) = 0 \\ \Rightarrow - 8 λ + 24 = 0 \Rightarrow λ = 3 \end{array}$
$From the options λ = 3 . satisfy the quadratic equation λ^{2} - λ - 6 = 0$

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