F[x_,y_,z_]:=-x^2-y^2-z^2;
G[x_,y_,z_]:=x+y+z;
Solve[{D[F[x,y,z],x]== L*D[G[x,y,z],x],
       D[F[x,y,z],y]== L*D[G[x,y,z],y],
       D[F[x,y,z],z]== L*D[G[x,y,z],z],
       G[x,y,z]==1},{x,y,z,L}]
