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Examine whether the following system of equation is consistent or unconsistent. If consistent find the complete solution.
$x + y + z = 6,$
$x - y + z = 2,$
$2 x - y + 3 z = 9$

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Detailed Solution

Given system of equations can be written as $A X = B \Rightarrow [\begin{array}{ccc} 1 & 1 & 1 \\ 1 & - 1 & 1 \\ 2 & - 1 & 3 \end{array}] [\begin{array}{l} x \\ y \\ z \end{array}] = [\begin{array}{l} 6 \\ 2 \\ 9 \end{array}]$

Augmented matrix $[A B] = [\begin{array}{cccc} 1 & 1 & 1 & 6 \\ 1 & - 1 & 1 & 2 \\ 2 & - 1 & 3 & 9 \end{array}]$

$| A | = |\begin{array}{ccc} 1 & 1 & 1 \\ 1 & - 1 & 1 \\ 2 & - 1 & 3 \end{array}| = 1 (- 3 + 1) - 1 (3 - 2) + 1 (- 1 + 2)$

$= 1 (- 2) - 1 (1) + 1 (1)$

$= - 2 - 1 + 1 = - 2$

$∴ Rank of (A) = Rank of [A B] = 3$

The system of equations are consistent and has a unique solution

$[A B] = [\begin{array}{cccc} 1 & 1 & 1 & 6 \\ 1 & - 1 & 1 & 2 \\ 2 & - 1 & 3 & 9 \end{array}]$

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

$[\begin{array}{cccc} 1 & 1 & 1 & 6 \\ 0 & - 2 & 0 & - 4 \\ 0 & - 3 & 1 & - 3 \end{array}]$

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

$[\begin{array}{cccc} 1 & 1 & 1 & 6 \\ 0 & 1 & 0 & 2 \\ 0 & - 3 & 1 & - 3 \end{array}]$

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

$[\begin{array}{llll} 1 0 1 4 \\ 0 1 0 2 \\ 0 0 1 3 \end{array}] R_{1} \to R_{1} - R_{3}$

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

$∴ x = 1, y = 2, z = 3$ in the solution

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