First slide

System of linear equations

Question

$T h e s y s t e m o f e q u a t i o n s 3 x - y + 4 z = 3, x + 2 y - 3 z = - 2 a n d 6 x + 5 y + λ z = - 3 h a s a t l e a s t o n e s o l u t i o n w h e n$

Moderate

A

$λ = - 5$
B

$λ = 3$
C

$λ = 5$
D

$λ = 6$

Solution

Augmented matrix = $[\begin{matrix} 3 & - 1 & 4 & 3 \\ 1 & 2 & - 3 & - 2 \\ 6 & 5 & λ & - 3 \end{matrix}]$ $\overset{\begin{array}{l} 3 R_{2} - R_{1}, \\ R_{3} - 2 R_{1} \end{array}}{~} [\begin{matrix} 3 & - 1 & 4 & 3 \\ 0 & 7 & - 13 & - 9 \\ 0 & 7 & λ - 8 & - 9 \end{matrix}]$
$\overset{R_{3} - R_{2}}{~} [\begin{matrix} 3 & - 1 & 4 & 3 \\ 0 & 7 & - 13 & - 9 \\ 0 & 0 & λ + 5 & 0 \end{matrix}]$
The system of equations has at least one solution $\Rightarrow λ + 5 = 0$
$\Rightarrow λ = - 5$

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