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If x + sqrt(xy) + y = 9 and x^2 + xy + y^2 = 27, find x - sqrt(xy) + y.

x + sqrt (xy) + y  =  9

x + y  =  9 - sqrt (xy)           square both sides

x^2 + 2xy + y^2  =  81 - 18sqrt (xy) + xy     simplify

x^2 +xy + y^2  =  81 -  18 sqrt (xy)

27  =  81 - 81sqrt (xy)

-54  = -18 sqrt (xy)

3  =  sqrt (xy)

And

x + sqrt (xy) + y  =  9

x + (3 + y)  = 9

x + y  = 6

So

x - sqrt (xy) + y  =

x + y - sqrt (xy)  =

6   - 3   =   

3

Notice that

\(\displaystyle (x+\sqrt{xy}+y)(x-\sqrt{xy}+y)=x^{2}+xy+y^{2},\\ \text{ so }\\9(x-\sqrt{xy}+y)=27,\\ \text{and therefore}\\ x-\sqrt{xy}+y = 3.\)