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Supposing I had two equations: $\alpha/\beta=\mu$ and $\alpha/\beta^{2}=\sigma^{2}$, where $\alpha$, $\beta$, $\sigma$ and $\mu \in \mathbb R$. How can I solve for $\alpha$ and $\beta$?

Furthermore, is there a general method for this kind of system?

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Divide. We get $\beta=\frac{\sigma^2}{\mu}$. Square the first expression and divide. We get $\alpha=\frac{\sigma^2}{\mu^2}$. – André Nicolas Oct 9 '12 at 14:24

Solving one equation for $\alpha$ gives p. e. $\alpha = \beta\mu$. Plug this into the other equation, giving $\beta\mu/\beta^2 = \sigma^2 \iff \beta = \mu/\sigma^2$. So $\alpha = \mu^2/\sigma^2$.

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