Solving for inverse transformation in change of variables This is something probably silly, but I don't seem to see how this works. I have 2 independent random variables $X \sim Gamma(\alpha, \beta_x)$ and $Y\sim Gamma(\alpha, \beta_y)$, and I need to show that $\frac{X}{Y}$ and $X+Y$ are independent.

To show this, I let $U = X+Y$ and $V = \frac{X}{Y}$ and tried to express $X$ and $Y$ in terms of $U$ and $V$. I know that the solution is $X = \frac{U+V}{1+V}$ and $Y = \frac{U}{1+V}$, but I have no idea how to algebraically manipulate $U$ and $V$ to get to this point. Is there a systematic way to solve for inverse transformation?
For the ratio distribution of $X, Y,$ the distribution $  U=X/Y $ is

$$f_U(u) = \int_{-\infty}^\infty f_{X,Y}(uy,y)|y|\ dy$$

For the summation distribution of  $X, Y,$ the distribution $  V=X+Y $ is

$$f_V(v) =\int_{-\infty}^\infty f_{X,Y}(x,v-x)\ dx = Conv(f_X(x),f_Y(y))$$
The x solution is incorrect.

Solutions  should  be   x = uv/(1+v)   and   y = u/(1+v)

*Although post is a year old, others may benefit from knowing the correct answer. I couldn't post image of solution and I am too lazy to type it in this post. Click here for solution.