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Solve the following equations by using Gauss Jordan method
$x + y + z = 1$
$2 x + 2 y + 3 z = 6$
$x + 4 y + 9 z = 3$

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Detailed Solution

The given system of equation can be written as $A X = B$

Where $A = [\begin{array}{lll} 1 1 1 \\ 2 2 3 \\ 1 4 9 \end{array}], X = [\begin{array}{l} x \\ y \\ z \end{array}], B = [\begin{array}{l} 1 \\ 6 \\ 3 \end{array}]$

The augmented matrix $[A B] = [\begin{array}{llll} 1 1 1 1 \\ 2 2 3 6 \\ 1 4 9 3 \end{array}]$

$R_{2} \to R_{2} - 2 R_{1}, R_{3} \to R_{3} - R_{1}$

$~ [\begin{array}{llll} 1 1 1 1 \\ 0 0 1 4 \\ 0 3 8 2 \end{array}]$

$R_{2} \leftrightarrow R_{3}$

$~ [\begin{array}{llll} 1 1 1 1 \\ 0 3 8 2 \\ 0 0 1 4 \end{array}]$

$R_{2} \to R_{2} - 8 R_{3}$

$~ [\begin{array}{cccc} 1 & 1 & 1 & 1 \\ 0 & 3 & 0 & - 30 \\ 0 & 0 & 1 & 4 \end{array}]$

$R_{2} \to R_{2} / 3$

$~ [\begin{array}{cccc} 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & - 10 \\ 0 & 0 & 1 & 4 \end{array}]$

$R_{1} \to R_{1} - R_{2}$

$~ [\begin{array}{cccc} 1 & 0 & 1 & 11 \\ 0 & 1 & 0 & - 10 \\ 0 & 0 & 1 & 4 \end{array}]$

$R_{1} \to R_{1} - R_{3}$

$~ [\begin{array}{cccc} 1 & 0 & 0 & 7 \\ 0 & 1 & 0 & - 10 \\ 0 & 0 & 1 & 4 \end{array}]$

$∴ x = 7, y = - 10, z = 4$ is the solution for given system of equations

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