Plane intersection

For planes \(\alpha: x+y+z-10=0\), \(\beta: x+y-z-4=0\), \(\gamma: -x+y+z-6=0\), and \(\delta: x-y+z-8=0\), determine the intersections:

Variant 1
\(\beta \cap \gamma \cap \delta \)
Resolution
This yields a system of equations. In this particular case we may add all equations together to get \(x+y+z-18=0\) and this subtract from othres.
Result
The intersection has coordinates: \((x,y,z)=(6{,}5,7)\).
Variant 2
\(\alpha \cap \gamma \cap \delta \)
Result
\((2{,}1,7)\)
Variant 3
\(\alpha \cap \beta \cap \delta \)
Result
\((6{,}1,3)\)
Variant 4
\(\alpha \cap \beta \cap \gamma \)
Result
\((2{,}5,3)\)