Atypical system of equations.

Question

The system is $$x+\sqrt{xy}+y=14$$ $$x^2+xy+y^2=84$$

The set of solutions is $\{(2,8),(8,2)\}$.

How do we get to this set of solutions?

Answer 1

Hint:

Let $a=\sqrt{xy}$ and $b=x+y$.

Then we get $a+b=14$ and $b^2-a^2=84$.

Can you take it from here?

Answer 2

Writing $$\sqrt{xy}=14-(x+y)$$ after squaring we get $$xy=196+(x+y)^2-28(x+y)$$ or using the second equation $$0=196+84-28(x+y)$$ solve this equation for $x$ or $y$ and plug this in the second one you will get an equation in only one variable.