Solve the system 2x +3y + z = 9, 4x + y = 7, x - 3y - 7z = 6

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Solution

Given AX = B

$\Rightarrow ⎡ ⎢ ⎣ \begin{matrix} 1 & - 3 & - 7 4 & 1 & 0 2 & 3 & 1 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} x y z \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 679 \end{matrix} ⎤ ⎥ ⎦$

$R_{2} \to R_{2} - 4 R_{1} : R_{3} \to R_{3} - 2 R_{1}$

$⎡ ⎢ ⎣ \begin{matrix} 1 & - 3 & - 7 0 & 13 & 28 0 & 9 & 15 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} x y z \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 6 - 17 - 3 \end{matrix} ⎤ ⎥ ⎦$

$R_{[} 3] \to 13 R_{3} - 9 R_{2}$

$⎡ ⎢ ⎣ \begin{matrix} 1 & - 3 & - 7 0 & 13 & 28 0 & 0 & - 57 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} x y z \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 6 - 17 114 \end{matrix} ⎤ ⎥ ⎦$ (Echelon form)

$\Rightarrow$ -57z = 114, 13y + 28z = -17 and x - 3y - 7z = 6

On solving, z = -2, y = 3 & x = 1

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