If the system of linear equations : $x+y+2z=6$,$2x+3y+az=a+1$,$-x-3y+bz=2b$ where $a, b \in R$,has infinitely many solutions,then $7a+3b$ is equal to :

The solution of the linear system of equations $\begin{bmatrix} 2 & 2 & 3 \\ 7 & 1 & 1 \\ 0 & 6 & 5 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 3 y + 11 \\ 6 z - 1 \\ 5 y + 11 \end{bmatrix} + \begin{bmatrix} x \\ x \\ 4 z \end{bmatrix} + \begin{bmatrix} z \\ 3 x \\ 4 y \end{bmatrix}$ is

If the matrix equation is given,then $x=$ . . . . . . and $y=$ . . . . . . .

Consider the system of linear equations $a_1x + b_1y + c_1z + d_1 = 0$,$a_2x + b_2y + c_2z + d_2 = 0$ and $a_3x + b_3y + c_3z + d_3 = 0$. Let us denote by $\Delta (a,b,c)$ the determinant $\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}$. If $\Delta (a,b,c) \neq 0$,then the value of $x$ in the unique solution of the above equations is:

For $\alpha, \beta \in R$,suppose the system of linear equations $x-y+z=5$,$2x+2y+\alpha z=8$,and $3x-y+4z=\beta$ has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of:

The positive value of $a$ for which the system of linear homogeneous equations $x+ay+z=0$, $ax+2y-z=0$, and $2x+3y+z=0$ has non-trivial solutions is