The following system of linear equations
Question:

The following system of linear equations

$3 x+3 y+2 z=9$

$3 x+2 y+2 z=9$

$x-y+4 z=8$

1. (1) does not have any solution

2. (2) has a unique solution

3. (3) has a solution $(\alpha, \beta, \gamma)$ satisfying $\alpha+\beta^{2}+\gamma^{3}=12$

4. (4) has infinitely many solutions

Correct Option: , 2

Solution:

$\Delta=\left|\begin{array}{ccc}2 & 3 & 2 \\ 3 & 2 & 2 \\ 1 & -1 & 4\end{array}\right|=-20 \neq 0 \quad \therefore$ unique solution

$\Delta_{x}=\left|\begin{array}{ccc}9 & 3 & 2 \\ 9 & 2 & 2 \\ 8 & -1 & 4\end{array}\right|=0$

$\Delta_{y}=\left|\begin{array}{lll}2 & 9 & 2 \\ 3 & 9 & 2 \\ 1 & 8 & 4\end{array}\right|=-20$

$\left|\begin{array}{ccc}2 & 3 & 9 \\ 3 & 2 & 9 \\ 1 & -1 & 8\end{array}\right|=-40$

$\therefore \quad x=\frac{\Delta_{\mathrm{X}}}{\Delta}=0$

$y=\frac{\Delta_{y}}{\Delta}=1$

$z=\frac{\Delta_{x}}{\Delta}=2$

Unique solution: $(0,1,2)$