The following system follows from a maximum likelihood estimation:
$\alpha_1$ = $18c(\beta_2+\beta_3)^{-1}$
$\alpha_2$ = $5c(\beta_1+\beta_3)^{-1}$
$\alpha_3$ = $8c(\beta_1+\beta_2)^{-1}$
$\beta_1$ = $4c(\alpha_2+\alpha_3)^{-1}$
$\beta_2$ = $10c(\alpha_1+\alpha_3)^{-1}$
$\beta_3$ = $17c(\alpha_1+\alpha_2)^{-1}$
with the following constraint:
$\alpha_1 + \alpha_2 + \alpha_3 = \beta_1 + \beta_2 + \beta_3$
$c$ is a fixed known constant, and the $\alpha$'s and $\beta$'s are the unknowns. Hence, I now have 7 equations for 6 unknowns. I know there should be a solution to this problem, i just don't know how to find it properly.
How do I find this solution, possibly using R or Matlab.