How do you find the eutectic point?

Raziel2701

I solved numerically the equations of the chemical potentials for a binary system and I have plotted them using matlab and so I can see where the eutectic is. But what if I want to corroborate that this point is correct? Is there a way to calculate by hand the eutectic point or are numerical methods the only way?

Also I can't look it up in a table because the binary system is an arbitrary one with its own set of melting temperature and omega values.