Number of solutions of the simultaneous equation
$3 x + 4 y + 9 z = 6$
$2 x + 4 y + 6 z = 7$
$12 x + 24 y + 60 z = 48$

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Solution

Consider the Augumented Matrix
$⎡ ⎢ ⎣ \begin{matrix} 3 & 4 & 9 & 6 2 & 4 & 6 & 7 12 & 24 & 60 & 48 \end{matrix} ⎤ ⎥ ⎦$

$R_{3} \to \frac{R_{3}}{12}$

$⎡ ⎢ ⎣ \begin{matrix} 3 & 4 & 9 & 6 2 & 4 & 6 & 7 1 & 2 & 5 & 4 \end{matrix} ⎤ ⎥ ⎦$

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

$⎡ ⎢ ⎣ \begin{matrix} 3 & 4 & 9 & 6 0 & 4 & 0 & 9 0 & 2 & 6 & 6 \end{matrix} ⎤ ⎥ ⎦$

$R_{3} \to 2 R_{3} - R_{2}$
$⎡ ⎢ ⎣ \begin{matrix} 3 & 4 & 9 & 6 0 & 4 & 0 & 9 0 & 0 & 12 & 3 \end{matrix} ⎤ ⎥ ⎦$

so,the rank of $A =$ rank of $A B = 3$
$⟹$ system has unique solution

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