<s>[INST] If $x^2+y^2-xy-x-y+1=0$ ($x,y$ real) then calculate $x+y$ If $x^2+y^2-xy-x-y+1=0$ ($x,y$ real) then calculate $x+y$
Ideas for solution include factorizing the expression into a multiple of $x+y$ and expressing the left hand side as a sum of some perfect square expressions.  
 [/INST] We have
$$2(x^2+y^2-xy-x-y+1)=(x-y)^2+(x-1)^2+(y-1)^2.$$ The right-hand side (for real $x$ and $y$) is equal to $0$ if and only if $x=y=1$.
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