First slide

System of linear equations

Question

For which of the following ordered pairs $(k, λ)$ ,the system of linear equations $x + 2 y + 3 z = k, 3 x + 4 y + 5 z = 4, 4 x + 4 y + 4 z = λ$ is inconsistent

Moderate

A

$(1, 6)$
B

$(1, 3)$
C

$(3, 2)$
D

$(4, 0)$

Solution

Augment matrix = $[\begin{matrix} 1 \\ 3 \\ 4 \end{matrix} \begin{matrix} 2 & 3 & k \\ 4 & 5 & 4 \\ 4 & 4 & λ \end{matrix}]$ apply: $\begin{array}{l} R_{2} \to R_{2} - 3 R_{1} \\ R_{3} \to R_{3} - 4 R_{1} \end{array}$
$~ [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix} \begin{matrix} 2 & 3 & k \\ - 2 & - 4 & 4 - 3 k \\ - 4 & - 8 & λ - 4 k \end{matrix}]$ apply: $R_{3} \to R_{3} - 2 R_{2}$
$~ [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix} \begin{matrix} 2 & 3 & k \\ - 2 & - 4 & 4 - 3 k \\ 0 & 0 & λ + 2 k - 8 \end{matrix}]$
System is Inconstistant , if $λ + 2 k - 8 \neq 0$ satisfy by (1,3) in the given options

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