If the system of equations x - 2   y + 3 z = 9 2 x + y + z = b x - 7 y + a z = 24 , has infinitely many…

For infinitely many solutions $D=D_1=D_2=D_3=0$

$D=\left|\begin{array}{l}1-23\\ 211\\ 1-7a\end{array}\right|=0$

$\Rightarrow 1\left(-2-3\right)+7\left(1-6\right)+a\left(1+4\right)=0$

$\Rightarrow -5-35+5a=0$

$\Rightarrow a=8$

$D_1=\left|\begin{array}{l}9-23\\ b11\\ 24-7a\end{array}\right|=0$

$\Rightarrow \left|\begin{array}{l}9-23\\ b11\\ 24-78\end{array}\right|=0$

$\Rightarrow -b\left(-16+21\right)+1\left(72-72\right)-1\left(-63+48\right)=0$

$\Rightarrow -5b+15=0$

$\Rightarrow b=3$

similarly $D_2=D_3=0$ for $a=8,b=3$

$\therefore$    $a-b=8-3=5$