Examine the consistency of the system of equations: $x+3y=5$ and $2x+6y=8$.

If $AX=B$,where $A=\begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 4 \\ 1 & 3 & 4 \end{bmatrix}$,$X=\begin{bmatrix} x \\ y \\ z \end{bmatrix}$ and $B=\begin{bmatrix} 12 \\ 15 \\ 13 \end{bmatrix}$,then $x^{2}+y^{2}+z^{2}=$

The system of equations ${x_1} + 2{x_2} + 3{x_3} = a$,$2{x_1} + 3{x_2} + {x_3} = b$,and $3{x_1} + {x_2} + 2{x_3} = c$ has:

If $S$ is the set of distinct values of $b$ for which the following system of linear equations $x + y + z = 1$,$x + ay + z = 1$,and $ax + by + z = 0$ has no solution,then $S$ is:

For the system of linear equations
$2x + 4y + 2az = b$
$x + 2y + 3z = 4$
$2x - 5y + 2z = 8$
which of the following is $NOT$ correct?

If the augmented matrix corresponding to the system of equations $x+y-z=1$, $2x+4y-z=0$ and $3x+4y+5z=18$ is transformed to $\left[\begin{array}{cccc}1 & a & 0 & -1 \\ 0 & 2 & 1 & b \\ 0 & 0 & c & 32\end{array}\right]$, then $\sqrt{a+b+c}=$