If the system of equations x - 2y + 3z = 9 2x + y + z = b x - 7y +...

For infinitely many solutions $D = {D}_{1} = {D}_{2} = {D}_{3} = 0$

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

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

$⇒ - 5 - 35 + 5a = 0$

$⇒ a = 8$

{D}_{1} = \left|$\begin{array}{ll}9 - 23 \\ b11 \\ 24 - 7a\end{array}$\right| = 0

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

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

$⇒ - 5b + 15 = 0$

$⇒ b = 3$

similarly ${D}_{2} = {D}_{3} = 0$ for $a = 8, b = 3$

$∴$ $a - b = 8 - 3 = 5$