- #1

RChristenk

- 49

- 4

- Homework Statement
- Solve ##\begin{cases} \sqrt{\dfrac{x}{y}}+\sqrt{\dfrac{y}{x}}=4\dfrac{1}{4}\\ \dfrac{x}{\sqrt{y}}+\dfrac{y}{\sqrt{x}}=16\dfrac{1}{4} \end{cases}##

- Relevant Equations
- Advanced Algebra manipulation skills and insights

##\Rightarrow \begin{cases} (\sqrt{\dfrac{x}{y}}+\sqrt{\dfrac{y}{x}})^2=(4\dfrac{1}{4})^2\\ (\dfrac{x}{\sqrt{y}}+\dfrac{y}{\sqrt{x}})^2=(16\dfrac{1}{4})^2 \end{cases}##

##\Leftrightarrow \begin{cases} \dfrac{x}{y}+\dfrac{y}{x}+2=\dfrac{289}{16}\\ \dfrac{x^2}{y}+\dfrac{y^2}{x} +\dfrac{2xy}{\sqrt{xy}}=\dfrac{4225}{16}\end{cases}##

##\Leftrightarrow \begin{cases}\dfrac{x^2+y^2}{xy}=\dfrac{257}{16}\\ \dfrac{x^3+y^3}{xy}+2\sqrt{xy} =\dfrac{4225}{16} \end{cases}##

And then I'm lost here. The presence of ##\sqrt{xy}## in the second equation makes me question if I'm even doing this problem correctly. Did I miss another pattern or simplification in the very first step?

##\Leftrightarrow \begin{cases} \dfrac{x}{y}+\dfrac{y}{x}+2=\dfrac{289}{16}\\ \dfrac{x^2}{y}+\dfrac{y^2}{x} +\dfrac{2xy}{\sqrt{xy}}=\dfrac{4225}{16}\end{cases}##

##\Leftrightarrow \begin{cases}\dfrac{x^2+y^2}{xy}=\dfrac{257}{16}\\ \dfrac{x^3+y^3}{xy}+2\sqrt{xy} =\dfrac{4225}{16} \end{cases}##

And then I'm lost here. The presence of ##\sqrt{xy}## in the second equation makes me question if I'm even doing this problem correctly. Did I miss another pattern or simplification in the very first step?