The following system of linear equations is given: $2x + 3y + 2z = 9$,$3x + 2y + 2z = 9$,and $x - y + 4z = 8$. Which of the following statements is true?

For the system of linear equations $a x+y+z=1$,$x+a y+z=1$,$x+y+a z=\beta$,which one of the following statements is $NOT$ correct?

Give the correct order of initials $T$ or $F$ for following statements. Use $T$ if statement is true and $F$ if it is false.
Statement $-1$ : If the graphs of two linear equations in two variables are neither parallel nor the same,then there is a unique solution to the system.
Statement $-2$ : If the system of equations $ax + by = 0, cx + dy = 0$ has a non-zero solution,then it has infinitely many solutions.
Statement $-3$ : The system $x + y + z = 1, x = y, y = 1 + z$ is inconsistent.
Statement $-4$ : If two of the equations in a system of three linear equations are inconsistent,then the whole system is inconsistent.

Let $\alpha, \beta$ and $\gamma$ be real numbers such that the system of linear equations
$x+2y+3z=\alpha$
$4x+5y+6z=\beta$
$7x+8y+9z=\gamma$
is consistent. Let $|M|$ represent the determinant of the matrix
$M=\begin{bmatrix} \alpha & 2 & \gamma \\ \beta & 1 & 0 \\ -1 & 0 & 1 \end{bmatrix}$
Let $P$ be the plane containing all those $(\alpha, \beta, \gamma)$ for which the above system of linear equations is consistent,and $D$ be the square of the distance of the point $(0,1,0)$ from the plane $P$.
$(1)$ The value of $|M|$ is
$(2)$ The value of $D$ is

Let the system of linear equations $4x + \lambda y + 2z = 0$,$2x - y + z = 0$,and $\mu x + 2y + 3z = 0$ (where $\lambda, \mu \in R$) have a non-trivial solution. Then which of the following is true?

The system of equations $x + y + z = 2$,$3x - y + 2z = 6$ and $3x + y + z = -18$ has