In[2]:= Reduce[{x^2 + y^2 + 2 x y/(x + y) == 1,
x^2 + Sqrt[2010] + Sqrt[2011] y - Sqrt[x + y] == 1}, {x, y}]
During evaluation of In[2]:= Reduce::useq: The answer found by Reduce contains unsolved equation(s) RowBox[{"{", RowBox[{\(0 == \(\(-2\) + \@2010 + \@2011 - \(\@2011\ \(Root[\(\(\(4021 + \(Times[\(<< 2 >>\)]\) + \(Times[\(<< 2 >>\)]\) + \(Times[\(<< 3 >>\)]\) + \(Times[\(<< 2 >>\)]\) + \(Times[\(<< 2 >>\)]\) + \(Times[\(<< 3 >>\)]\) + \(Power[\(<< 2 >>\)]\)\) &\), 1\)]\)\) + \(Root[\(\(\(4021 + \(Times[\(<< 2 >>\)]\) + \(Times[\(<< 2 >>\)]\) + \(Times[\(<< 3 >>\)]\) + \(Times[\(<< 2 >>\)]\) + \(Times[\(<< 2 >>\)]\) + \(Times[\(<< 3 >>\)]\) + \(Power[\(<< 2 >>\)]\)\) &\), 1\)]\)\^2\)\), ",", \(0 == \(\(-2\) + \@2010 + \@2011 - \(\@2011\ \(Root[\(\(\(4021 + \(<< 6 >>\) + \(Power[\(<< 2 >>\)]\)\) &\), 2\)]\)\) + \(Root[\(\(\(4021 + \(<< 6 >>\) + \(Power[\(<< 2 >>\)]\)\) &\), 2\)]\)\^2\)\), ",", \(<< 8 >>\), ",", RowBox[{"0", "==", FractionBox[\(<< 1 >>\), RowBox[{"", \(<< 66 >>\), ""}]]}], ",", \(0 == \(<< 1 >>\)\/735705629768612331158718611476175537815974496201716851830936735778\)}], "}"}]. A likely reason for this is that the solution set depends on branch cuts of Wolfram Language functions. >>
that branch cut thing brings the Reals into mind
In[3]:= Reduce[{x^2 + y^2 + 2 x y/(x + y) == 1,
x^2 + Sqrt[2010] + Sqrt[2011] y - Sqrt[x + y] == 1}, {x, y}, Reals]
Out[3]= (x ==
Root[{-2010 + #1^2 &, -2011 + #2^2 &,
4021 - 2 #1 - 2 #2 + 2 #1 #2 - 4022 #3 + 2 #2 #3 - 2 #1 #2 #3 +
2009 #3^2 + 2 #1 #3^2 + 2 #2 #3^2 - 2 #2 #3^3 + #3^4 &}, {2,
2, 1}] &&
y == Root[{-2010 + #1^2 &, -2011 + #2^2 &,
4021 - 2 #1 - 2 #2 + 2 #1 #2 - 4022 #3 + 2 #2 #3 - 2 #1 #2 #3 +
2009 #3^2 + 2 #1 #3^2 + 2 #2 #3^2 -
2 #2 #3^3 + #3^4 &, -1 + #3 + #4 &}, {2, 2, 1, 1}]) || (x ==
Root[{-2010 + #1^2 &, -2011 + #2^2 &,
4021 - 2 #1 - 2 #2 + 2 #1 #2 - 4022 #3 + 2 #2 #3 - 2 #1 #2 #3 +
2009 #3^2 + 2 #1 #3^2 + 2 #2 #3^2 - 2 #2 #3^3 + #3^4 &}, {2,
2, 4}] &&
y == Root[{-2010 + #1^2 &, -2011 + #2^2 &,
4021 - 2 #1 - 2 #2 + 2 #1 #2 - 4022 #3 + 2 #2 #3 - 2 #1 #2 #3 +
2009 #3^2 + 2 #1 #3^2 + 2 #2 #3^2 -
2 #2 #3^3 + #3^4 &, -1 + #3 + #4 &}, {2, 2, 4, 1}])
In[4]:= Reduce[{x^2 + y^2 + 2 x y/(x + y) == 1,
x^2 + Sqrt[2010] + Sqrt[2011] y - Sqrt[x + y] == 1}, {x, y},
Reals] // N
Out[4]= (x == 2.04875 && y == -1.04875) || (x == 42.7954 && y == -41.7954)
It could be rewarding if you would pose problems instead of throwing them. Here nobody knows about the area of definition for x and y. It could be interesting (people call this problem posing) whether a Riemann Surface or a point solution is searched for.
Why my computer don't run.
Your computer did not only not run, it did in the opposite run enormously, especially it runs out of resources! So either you deliver more resources to it or you pose the problem to the community.
