First slide

System of linear equations

Question

The solution of the system of equations whose augmented matrix $[\begin{matrix} 1 & 1 & 1 & 2 \\ 1 & 2 & 3 & 1 \\ 3 & 1 & - 5 & 4 \end{matrix}]$ is

Moderate

A

$x = 3, y = 1, z = 1$
B

$x = 4, y = - 3, z = 1$
C

$x = 1, y = 1, z = 1$
D

$x = 7, y = - 10, z = 4$

Solution

$[\begin{matrix} 1 & 1 & 1 & 2 \\ 1 & 2 & 3 & 1 \\ 3 & 1 & - 5 & 4 \end{matrix}] ~ [\begin{matrix} 1 & 1 & 1 & 2 \\ 0 & 1 & 2 & - 1 \\ 0 & - 2 & - 8 & - 2 \end{matrix}] (R_{2} \to R_{2} - R_{1}, R_{3} \to R_{3} - 3 R_{1}) ~ [\begin{matrix} 1 & 0 & - 1 & 3 \\ 0 & 1 & 2 & - 1 \\ 0 & 0 & - 4 & - 4 \end{matrix}] (R_{1} \to R_{1} - R_{2}, R_{3} \to R_{3} + 2 R_{2}) ~ [\begin{matrix} 4 & 0 & 0 & 16 \\ 0 & 2 & 0 & - 6 \\ 0 & 0 & - 4 & - 4 \end{matrix}] (R_{1} \to 4 R_{1} - R_{3}, R_{2} \to 2 R_{2} + R_{3}) \Rightarrow t h e g i v e n s y s t e m i s e q u i v a l e n t t o 4 x = 16, 2 y = - 6 a n d - 4 z = - 4 . \Rightarrow x = 4, y = - 3 a n d z = 1$

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