For real numbers α and β, consider the following system of linear...

Given system of linear equations are $x + y - z = 2, x + 2y + \alphaz = 1, 2x - y + z = \beta$

Using Cramer Rule, we can say for infinite solutions

$\Delta = {\Delta}_{1} = {\Delta}_{2} = {\Delta}_{3} = 0$

Here, \Delta = \left|\begin{cases}1 & 1 & - 1 \\ 1 & 2 & \alpha \\ 2 & - 1 & 1\end{cases}\right| = 0

Apply ${R}_{1} ↔ {R}_{1} + {R}_{3}$

⇒ \left|\begin{cases}3 & 0 & 0 \\ 1 & 2 & \alpha \\ 2 & - 1 & 1\end{cases}\right| = 0

$⇒ 3\left(2 + \alpha\right) = 0$

$⇒ \alpha = - 2$

And {\Delta}_{2} = \left|\begin{cases}1 & 2 & - 1 \\ 1 & 1 & - 2 \\ 2 & \beta & 1\end{cases}\right| = 0

$⇒ 1\left(1 + 2\beta\right) - 2\left(1 + 4\right) - \left(\beta - 2\right) = 0$

$⇒ \beta - 7 = 0$

$⇒ \beta = 7$

$∴ \alpha + \beta = 5$