Linear Systems

P={{8,-5,-2,-2},{-2,4,-4,1},{-2,1,4,4},{4,1,3,10}};
Q={35,-8,-11,10};
Solve[P.{x1,x2,x3,x4}==Q,{x1,x2,x3,x4}]
{LinearSolve[P,Q],Inverse[P].Q}
{MapThread[Append,{P,Q}]//MatrixForm,
RowReduce[MapThread[Append,{P,Q}]][[All,5]]}