For the system of linear equations 2x - y + 3z = 5 3x + 2y - z = 7...

Given,

For the system of linear equations

$2x - y + 3z = 5$

$3x + 2y - z = 7$

$4x + 5y + \alphaz = \beta$,

Now finding,

D = \left|\begin{cases}2 & - 1 & 3 \\ 3 & 2 & - 1 \\ 4 & 5 & \alpha\end{cases}\right|

$⇒ D = 2\left(2\alpha + 5\right) + \left(3\alpha + 4\right) + 3\left(7\right)$

$⇒ D = 7\alpha + 35$

Now we know that, for unique solution$D \ne 0 ⇒ \alpha \ne – 5$

So, in option \left(B\right)\alpha = – 6 corresponds to infinitely many solution, which is completely wrong hence, this is answer.