First slide

System of linear equations

Question

The system of linear equations $x - 4 y + 7 z = g, 3 y - 5 z = h, - 2 x + 5 y - 9 z = k$ is consistent, then

Moderate

A

$g + h + k = 0$
B

$2 g + h + k = 0$
C

$g + 2 h + k = 0$
D

$g + h + 2 k = 0$

Solution

Augmented matrix $[A D] = [\begin{matrix} 1 & - 4 & 7 \\ 0 & 3 & - 5 \\ - 2 & 5 & - 9 \end{matrix} \begin{matrix} g \\ h \\ k \end{matrix}] R_{3} \to R_{3} + 2 R_{1}$
$\begin{array}{l} ~ [\begin{matrix} 1 & - 4 & 7 & g \\ 0 & 3 & - 5 & h \\ 0 & - 3 & 5 & 2 g + k \end{matrix}] R_{3} \to R_{3} + R_{2} \\ ~ [\begin{matrix} 1 & - 4 & 7 & g \\ 0 & 3 & - 5 & h \\ 0 & 0 & 0 & 2 g + h + k \end{matrix}] \end{array}$
The system of equations is consistent $\Rightarrow 2 g + h + k = 0$

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