I have system of 4 equation, this is part of work for experiment in fluid dynamics, where with two known variables I need to determine other unknown variables. Known variables are:
$n, m, \beta, K$
and equations are: $$\beta \varepsilon = \dfrac{\kappa M^2}{R}$$ $$\gamma \varepsilon^m = \kappa M^2$$ $$\eta \varepsilon^n = K$$ $$K = \sqrt{\dfrac{\pi \kappa}{2}} \dfrac{M}{R}$$ from these equations I can get next relationships, which are checking point for me,it means if I get some values from the first four equation I check it in next relationships: $$\gamma = \dfrac{\beta^2 \pi}{2 \eta^2}$$ $$R = \dfrac{\gamma}{\beta}\varepsilon^{m-1}$$ so it is necessary at least to determine $K, M, R$, and if it is possible determine and all other variables.
With known variables, for example $n = 0.25, m = 0.75, \beta = 1, K = 0.1$ I always get result as one variable is missing, it means I get all results in terms of one variable, for example $\varepsilon$. If I put one more variable as known, system is redefined, because results which I get from the first four equations are not the same when I check it with other two equations.
How can I solve this system?