Non-linear system of two equations (Method of Moments for Generalized Pareto) In a Method of Moments Estimation problem involving the generalized Pareto Distribution, the following system of 2 non-linear equations arise
\begin{align*}
\bar{X} &= \frac{\alpha \beta}{\alpha - 1}. \\
\frac{1}{n}\sum_{i=1}^{n} X^2_i  &= \frac{\alpha \beta^2}{\alpha - 2}.
\end{align*}
The problem is to solve the system for $\alpha$ and $\beta$ in terms of $\bar{X} $ and $\frac{1}{n}\sum_{i=1}^{n} X^2_i $. The thing is, it's easy to isolate $\alpha$ or $\beta$ from any of the two questions, but after this has been done you cannot isolate the other parameter in the other equation. Using Mathematica you can get two solutions but I cannot just say "According to Mathematica, the solution is ..."

$$
\begin{cases}
\frac{a b}{a-1}=x\\
\frac{a b^2}{a-2}=y\\
\end{cases}
$$
multiply the first equation by $b$
$$
\begin{cases}
a b^2=(a-1)bx\\
a b^2=(a-2)y\\
\end{cases}
$$
thus
$$(a-1)bx=(a-2)y\to b=\frac{(a-2) y}{(a-1) x} $$
Now plug this in the first, the very first equation
$$ab=x(a-1)\to a\cdot \frac{(a-2) y}{(a-1) x}=x(a-1)$$
You get a quadratic equation in $a$ which gives
$$a=\frac{x^2-y\pm\sqrt{y^2-x^2 y}}{x^2-y}$$
plug back to get $b$
$$b_1=\frac{y \left(\frac{x^2-y-\sqrt{y^2-x^2 y}}{x^2-y}-2\right)}{x \left(\frac{x^2-y-\sqrt{y^2-x^2 y}}{x^2-y}-1\right)}=\frac{y}{x}\cdot \frac{x^2-y-\sqrt{y \left(y-x^2\right)}}{\sqrt{y \left(y-x^2\right)}}=\frac{y-\sqrt{y \left(y-x^2\right)}}{x}$$
and
$$b_2=\frac{y+\sqrt{y \left(y-x^2\right)}}{x}$$
