The values of a and b , for which the system of equations 2x + 3y +...

The given equation is

$2x + 3y + 6z = 8$

$x + 2y + az = 5$

$3x + 5y + 9z = b$

So, D = \left|\begin{cases}2 & 3 & 6 \\ 1 & 2 & a \\ 3 & 5 & 9\end{cases}\right|

$= 2\left(18 - 5a\right) - 3\left(9 - 3a\right) + 6\left(5 - 6\right)$

$= 36 - 10a - 27 + 9a + 30 - 36$

$= 3 - a$

{D}_{z} = \left|\begin{cases}2 & 3 & 8 \\ 1 & 2 & 5 \\ 3 & 5 & b\end{cases}\right|

$= 2\left(2b - 10\right) - 3\left(b - 15\right) + 8\left(5 - 6\right)$

$= 4b - 20 - 3b + 15 + 40 - 48$

$= b - 13$

The system of equation has no solution if $D = 0 \text{ and } {D}_{z} \ne 0$

So, If $a = 3, b \ne 13$, then the system of equation has no solution.