Question

For real numbers $\alpha$ and $\beta ,$ consider the following system of linear equations :
$x+y-z=2,x+2y+\alpha z=1,2x-y+z=\beta$. If the system has infinite solutions, then $\alpha +\beta$ is equal to

Answer 1

$\Delta =\left|\begin{array}{ccc}1& 1& -1\\ 1& 2& \alpha \\ 2& -1& 1\end{array}\right|=0$
$\Rightarrow 1(2+\alpha )-1(1-2\alpha )-1(-1-4)=0$
$\Rightarrow 2+\alpha -1+2\alpha +5=0$
$\therefore \alpha =-2$
For the system to have infinite solutions, we have
$\Delta ={\Delta }_1={\Delta }_2={\Delta }_3=0$
Now,
${\Delta }_2=\left|\begin{array}{ccc}1& 2& -1\\ 1& 1& -2\\ 2& \beta & 1\end{array}\right|=0$
$\Rightarrow 1(1+2\beta )-2(1+4)-1(\beta -2)=0$
$\Rightarrow 1+2\beta -10-\beta +2=0$
$\therefore \beta =7$
Hence, $\alpha +\beta =5$