imc

Note that we can add the two equations to yield the equation $x^2+y^2-4x-4y+8=x+y+8.$ Moving terms gives the equation $x^2+y^2=5\left(x+y\right).$ We can also subtract the two equations to yield the equation $x^2-y^2-4x+4y=y-x.$ Moving terms gives the equation $x^2-y^2=(x+y)(x-y)=3x-3y=3(x-y).$ Because $x\ne y,$ we can divide both sides of the equation by $x-y$ to yield the equation $x+y=3.$ Substituting this into the equation for $x^2+y^2$ that we derived earlier gives $x^2+y^2=5\left(x+y\right)=5\left(3\right)=\boxed{15}$