[tex]\left\{\begin{array}{ccc}-3x-y+z=6\\-3x-y+3z=0\\x-3z=3&|\text{add 3z to both sides}\end{array}\right\\\\x=3+3z\\\\\text{Substitute it to the first and the second equation:}\\\\\left\{\begin{array}{ccc}-3(3+3z)-y+z=6\\-3(3+3z)-y+3z=0\end{array}\right\\\\\text{use distributive property}\\\\\left\{\begin{array}{ccc}-9-9z-y+z=6&|\text{add 9 to both sides}\\-9-9z-y+3z=0&|\text{add 9 to both sides}\end{array}\right\\\\\text{combine like terms}[/tex]
[tex]\left\{\begin{array}{ccc}-8z-y=15\\-6z-y=9&|\text{change the signs}\end{array}\right\\\\\underline{+\left\{\begin{array}{ccc}-8z-y=15\\6z+y=-9\end{array}\right}\qquad\text{add both sides of the equations}\\.\qquad-2z=6\qquad\text{divide both sides by (-2)}\\.\qquad\boxed{z=-3}\\\\\text{Put the value of z to the second equation:}\\\\6(-3)+y=-9\\-18+y=-9\qquad\text{add 18 to both sides}\\\boxed{y=9}\\\\\text{Put the value of z to the equation}\ x=3+3z:\\\\x=3+3(-3)\\x=3-9\\\boxed{x=-6}[/tex]
[tex]Answer:\ \boxed{x=-6,\ y=9,\ z=-3\to(-6,\ 9,\ -3)}[/tex]
