Solutions to Problems 1271-1280

Q1271 Solve simultaneously
\begin{align*}
x^2 + xy + y^2 &= 189 \\
x - \sqrt{xy} + y &= 9.
\end{align*}
ANS: (suggested by Julius Guest, Victoria, and Keith Anker, Victoria)
Let $p=x+y$ and $q=\sqrt{xy}$. Then it follows from the given equations that $p-q=9$ and $p^2-q^2=189$, implying $p+q=21$.
Hence $p=15$ and $q=6$. From $x+y=15$ and $xy=36$ we deduce $x=3$ and $y=12$ or $x=12$ and $y=3$.