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U[x_, y_] := x y + 2 x ;

B[Px_, x_, Py_, y_, Y_] := Y - Px x - Py y ;

L[Px_, x_, Py_, y_, Y_, \[Lambda]_] :=

U[x, y] + \[Lambda] B[Px, x, Py, y, Y];

MUx = \!\(

\*SubscriptBox[\(\[PartialD]\), \(x\)]\(U[x, y]\)\)

MUy = \!\(

\*SubscriptBox[\(\[PartialD]\), \(y\)]\(U[x, y]\)\)

dL[Px_, x_, Py_, y_, Y_, \[Lambda]_] :=

D[L[Px, x, Py, y, Y, \[Lambda]] == 0, {{x, y, \[Lambda]}, 1}]

dL[Px, x, Py, y, Y, \[Lambda]] // MatrixForm

d2L[Px_, x_, Py_, y_, Y_, \[Lambda]_] :=

D[L[Px, x, Py, y, Y, \[Lambda]], {{x, y, \[Lambda]}, 2}]

d2L[Px, x, Py, y, Y, \[Lambda]] // MatrixForm

sol[Px_, Py_, Y_] :=

Flatten[Solve[dL[Px, x, Py, y, Y, \[Lambda]], {x, y, \[Lambda]}]]

xs[Px_, Py_, Y_] := {x} /. sol[Px, Py, Y][[1]]

ys[Px_, Py_, Y_] := {y} /. sol[Px, Py, Y][[2]]

\[Lambda]s [Px_, Py_, Y_] := {\[Lambda]} /. sol[Px, Py, Y][[3]]