The system of equations $x - 2y + 3z = 5$,$2x - 2y + z = 0$,and $-x + 2y - 3z = 6$ has

Consider the system of linear equations:
$-x+y+2z=0$
$3x-ay+5z=1$
$2x-2y-az=7$
Let $S_{1}$ be the set of all $a \in \mathbb{R}$ for which the system is inconsistent and $S_{2}$ be the set of all $a \in \mathbb{R}$ for which the system has infinitely many solutions. If $n(S_{1})$ and $n(S_{2})$ denote the number of elements in $S_{1}$ and $S_{2}$ respectively,then:

If the system of equations $x + 5y + 6z = 4$, $2x + 3y + 4z = 7$, and $x + 6y + az = b$ has infinitely many solutions, then the point $(a, b)$ lies on the line:

If the system of equations $x + y + z = 6$,$x + 2y + 3z = 10$,and $x + 2y + \lambda z = 0$ has a unique solution,then $\lambda$ is not equal to:

If $AX=B$,where $A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & -1 & 0 \\ 3 & 3 & -4\end{array}\right]$,$B=\left[\begin{array}{l}1 \\ 1 \\ 2\end{array}\right]$ and $X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right]$,then $x+y+z=$

If the system of linear equations given by $x+y+z=3$,$2x+2y-z=3$,and $x+y-z=1$ is consistent and if $(x_0, y_0, z_0)$ is a solution,then $2x_0+2y_0+z_0=$