If the system of linear equations $x - 4y + 7z = g$,$3y - 5z = h$,and $-2x + 5y - 9z = k$ is consistent,then:

If $(\alpha, \beta, \gamma)$ is the solution of the system of simultaneous linear equations given by $3x + 4y - 5z = -6$,$2x + 3y - 4z = -7$,and $4x - 2y + z = 9$,then find the value of $\alpha + 3\beta - 2\gamma$.

Let $A = \begin{bmatrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 4 \\ 0 \\ 2 \end{bmatrix}$ such that $AX = B$,then $X =$

If the system of linear equations
$2x + y - z = 3$
$x - y - z = \alpha$
$3x + 3y + \beta z = 3$
has infinitely many solutions,then $\alpha + \beta - \alpha \beta$ is equal to .... .

The system $x+2y+3z=4$,$4x+5y+3z=5$,$3x+4y+3z=\lambda$ is consistent and $3\lambda=n+100$,then $n=$

Let $AX=D$ be a system of three linear non-homogeneous equations. If $|A|=0$ and $\operatorname{rank}(A)=\operatorname{rank}([AD])=\alpha$, then