Sin[π]

AudiencePoints = N[{{10,15,2},{-1,5,7},{11,15,18},{10,9,1},{7,1.1,2},{π,π^2,π^3},{-4,-8,-3.8},{0,1,-1},{5,6,7},{Sqrt[2],(E^1),π}}]

AudiencePlot = ListPointPlot3D[AudiencePoints,PlotStyle->{Black,PointSize[Medium]}, PlotRange->{{-32,32},{-32,32},{-32,32}}]

f1 = x;
f2 = y;
f3 = 1;

AudiencePoints//MatrixForm

ZVals = Transpose[{AudiencePoints[[All,3]]}]

(AMatrix =Table[ {AudiencePoints[[i,1]],AudiencePoints[[i,2]],1},{i,1,Length[AudiencePoints]}])//MatrixForm

Soln = Inverse[Transpose[AMatrix ].AMatrix ].(Transpose[AMatrix ].ZVals)

PlaneFit = -1.2538967923356237`*x+1.5537227052467297`*y+3.163120229604907`*1

PlanePlot = Plot3D[PlaneFit,{x,-32,32},{y,-32,32}, MeshStyle->None, PlotStyle->{Blue,Opacity[0.5]}]

Show[PlanePlot,AudiencePlot, PlotRange->{{-32,32},{-32,32},{-32,32}}]

(AMatrix1 =Table[ {AudiencePoints[[i,1]],AudiencePoints[[i,2]],AudiencePoints[[i,1]]*AudiencePoints[[i,2]],1},{i,1,Length[AudiencePoints]}])//MatrixForm

Soln1 = Inverse[Transpose[AMatrix1 ].AMatrix1].(Transpose[AMatrix1].ZVals)

NewSurface = -0.9631400210285463`*x+1.6931672939447688`*y-0.04765709282384281`x*y+3.5398404881064547*1

NewSurfacePlot = Plot3D[NewSurface,{x,-32,32},{y,-32,32}, MeshStyle->None, PlotStyle->{Yellow,Opacity[0.5]}]

Show[NewSurfacePlot ,PlanePlot,AudiencePlot, PlotRange->{{-32,32},{-32,32},{-32,32}}]

Norm[AMatrix1.Soln1-ZVals]

Norm[AMatrix.Soln-ZVals]