1. ## need help

we have u=y/x and v=xy

how to solve x and y in terms of u and v ?

2. Originally Posted by kittycat

we have u=y/x and v=xy

how to solve x and y in terms of u and v ?
$u = \frac yx$ ......................(1)
$v = xy$ .....................(2)

$\Rightarrow y = xu$ ................from equation (1)

$\Rightarrow y = \frac vy \cdot u$ .............solved for $x$ in equation (2) and plugged it in.

$\Rightarrow y^2 = uv$

$\Rightarrow y = \sqrt {uv}$

now solve for $x$ in a similar manner

3. Hello, kittycat!

Another approach . . .

Given: . $u \:=\:\frac{y}{x},\;\;\;v\:=\:xy$

Solve for $x$ and $y$ in terms of $u$ and $v.$

We have: . $\begin{array}{cccc}\frac{y}{x} & = & u & {\color{blue}[1]} \\ xy & = & v & {\color{blue}[2]} \end{array}$

Divide [1] by [2]: . $\frac{\frac{y}{x}}{xy} \:=\:\frac{u}{v}\quad\Rightarrow\quad\frac{1}{x^2} \:=\:\frac{u}{v}\quad\Rightarrow\quad x^2\:=\:\frac{v}{u}\quad\Rightarrow\quad\boxed{ x \:=\:\sqrt{\frac{v}{u}}}$

Multiply [1] and [2]: . $\frac{y}{x}\cdot xy \:=\:uv\quad\Rightarrow\quad y^2\:=\:uv\quad\Rightarrow\quad\boxed{ y \:=\:\sqrt{uv}}$