First slide

System of linear equations

Question

The system of equations $x + y + z = 3, x + 2 y + 3 z = 4, x + 4 y + 9 z = 6$ has

Difficult

A

$x = 1, y = 2, z = 0$
B

$x = 2, y = 1, z = 0$
C

$x = 0, y = 2, z = 1$
D

None f these

Solution

The system of equations can be written as $[\begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 4 & 9 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 3 \\ 4 \\ 6 \end{matrix}]$
Let $A = [\begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 4 & 9 \end{matrix}] \Rightarrow | A | = 2$ and $a d j A = [\begin{matrix} 6 & - 5 & 1 \\ - 6 & 8 & - 2 \\ 2 & - 3 & 1 \end{matrix}]$
Also, $A^{- 1} = \frac{a d j A}{| A |} = [\begin{matrix} 3 & - \frac{5}{2} & \frac{1}{2} \\ - 3 & 4 & - 1 \\ 1 & - \frac{3}{2} & \frac{1}{2} \end{matrix}]$
Hence, the solution of the given system is obtained from
$[\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 3 & - \frac{5}{2} & \frac{1}{2} \\ - 3 & 4 & - 1 \\ 1 & - \frac{3}{2} & \frac{1}{2} \end{matrix}] [\begin{matrix} 3 \\ 4 \\ 6 \end{matrix}]$
$\Rightarrow x = 2, y = 1, z = 0$

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