The ordered pair (a, b) , for which the system of linear equations...

Given system of linear equations are
$3x - 2y + z = b$

$5x - 8y + 9z = 3$

$2x + y + az = - 1$

For system to have no solution,

∆ = \left|\begin{cases}3 & - 2 & 1 \\ 5 & - 8 & 9 \\ 2 & 1 & a\end{cases}\right| = 0

$⇒ 3\left(- 8a - 9\right) + 2\left(5a - 18\right) + 1\left(21\right) = 0$

$⇒ a = - 3$

Also, {∆}_{2} = \left|\begin{cases}3 & - 2 & b \\ 5 & - 8 & 3 \\ 2 & 1 & - 1\end{cases}\right| = 3\left(5\right) + 2\left(- 11\right) + b\left(21\right) = 21b - 7

If $b = \dfrac{1}{3}$, ${∆}_{2} = 0$

So $b$ must be equal to $- \dfrac{1}{3}$