The number of solutions of the equations $x + 4y - z = 0,$ $3x - 4y - z = 0,$ and $x - 3y + z = 0$ is

If $\begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 4 \\ 1 & 3 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 12 \\ 15 \\ 13 \end{bmatrix}$,then the value of $x^2 + y^2 + z^2 =$

If $\begin{bmatrix} 1 & 1 & 1 \\ 1 & -2 & -2 \\ 1 & 3 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 0 \\ 3 \\ 4 \end{bmatrix}$,then $2x - y + z = $

If the system of equations $2x - y + z = 4$,$5x + \lambda y + 3z = 12$,and $100x - 47y + \mu z = 212$ has infinitely many solutions,then $\mu - 2\lambda$ is equal to

If the system of linear equations $3x + y + \beta z = 3$,$2x + \alpha y - z = -3$,and $x + 2y + z = 4$ has infinitely many solutions,then the value of $22\beta - 9\alpha$ is:

Consider the system of equations in $x, y$ and $z$:
$12x + by + cz = 0$
$ax + 24y + cz = 0$
$ax + by + 36z = 0$
(where $a, b, c$ are real numbers,$a \ne 12, b \ne 24, c \ne 36$).
If the system of equations has a non-trivial solution $(z \ne 0)$,then the value of $\frac{1}{a - 12} + \frac{2}{b - 24} + \frac{3}{c - 36}$ is: