First slide

System of linear equations

Question

If the system of equations $x + y + z = 6, x + 2 y + λ z = 0, x + 2 y + 3 z = 10$ has no solution then $λ$ =

Moderate

A

2
B

3
C

4
D

5

Solution

$Augmented matrix is [\begin{array}{cccc} 1 & 1 & 1 & 6 \\ 1 & 2 & 3 & 10 \\ 1 & 2 & λ & 0 \end{array}]$
$\begin{matrix} R_{2} - R_{1} \\ R_{3} - R_{1} \\ ~ \end{matrix} [\begin{matrix} 1 & 1 & 1 & 6 \\ 0 & 1 & 2 & 4 \\ 0 & 1 & λ - 1 & - 6 \end{matrix}]$
$\begin{matrix} R_{3} - R_{2} \\ ~ \end{matrix} [\begin{matrix} 1 & 1 & 1 & 6 \\ 0 & 1 & 2 & 4 \\ 0 & 0 & λ - 3 & 10 \end{matrix}]$
$∴ Given system of equations has no solution \Rightarrow λ - 3 = 0 \Rightarrow λ = 3$

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