2015 AMC 10A Problem 16

Add and subtract the two equations separately

Since $x\ne y$, divide the difference equation by $x-y$

Adding the two equations gives us $$x^2 + y^2 - 4x - 4y + 8$$ $$= x + y + 8.$$ We can rearrange this equation to get $$x^2 + y^2 = 5(x + y).$$ We can then subtract them to get $$x^2 - y^2 - 4x + 4y = y - x.$$ Once again rearranging, we can find $$x^2 - y^2 = 3(x - y).$$ We have that $x \neq y,$ which means that we can divide both sides by $x - y.$ This gives us $$x + y = 3$$$$x^2 + y^2 = 15.$$

Thus, B is the correct answer.