Question:
If the system of linear equations
$2 x+y-z=3$
$x-y-z=\alpha$
$3 x+3 y+\beta z=3$
has infinitely many solution, then $\alpha+\beta-\alpha \beta$ is equal to
Solution:
$2 \times(\mathrm{i})-(\mathrm{ii})-(\mathrm{iii})$ gives :
$-(1+\beta) z=3-\alpha$
For infinitely many solution
$\beta+1=0=3-\alpha \Rightarrow(\alpha, \beta)=(3,-1)$
Hence, $\alpha+\beta-\alpha \beta=5$