For real numbers $\alpha$ and $\beta$, consider the following system of linear equations:
$x+y-z=2, x+2 y+\alpha z=1,2 x-y+z=\beta$
If the system has infinite solutions, then $\alpha+\beta$ is equal to
For infinite solutions
$\Delta=\Delta_{1}=\Delta_{2}=\Delta_{3}=0$
$\Delta=\left|\begin{array}{ccc}1 & 1 & -1 \\ 1 & 2 & \alpha \\ 2 & -1 & 1\end{array}\right|=0$
$\Delta=\left|\begin{array}{ccc}3 & 0 & 0 \\ 1 & 2 & \alpha \\ 2 & -1 & 1\end{array}\right|=0$
$\Delta=3(2+\alpha)=0$
$\Rightarrow \alpha=-2$
$\Delta_{2}=\left|\begin{array}{ccc}1 & 2 & -1 \\ 1 & 1 & -2 \\ 2 & \beta & 1\end{array}\right|=0$
$1(1+2 \beta)-2(1+4)-(\beta-2)=0$
$\beta-7=0$
$\beta=7$
$\therefore \alpha+\beta=5$ Ans.
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