The two equations, then, are
\epsilon'= \epsilon_\infty+ \frac{\epsilon_s- \epsilon_\infty}{1+ (\omega \tau)^2}
and
\epsilon"= \frac{(\epsilon_s- \epsilon_\infty)\omega\tau}{1+ (\omega \tau)^2}}
Of course, since there are only two equations in 3 unknown values, you cannot solve for all three. What you could do is solve for two of them in terms of the third.
You might, for example, multiply the first equation by \omega\tau, so that the two fractions are the same, and then subtract one equation from the other, eliminating the fractions:
\epsilon'- \epsilon"= \epsilon_\infty (\omega\tau)
Then
\epsilon_\infty= \frac{\epsilon'-\epsilon"}{\omega\tau}
so you have solved for \epsilon_\infty in terms of \tau.
Replace \epsilon_\infty by that in either of the two equations, and you can then also solve for \epsilon_\infty in terms of \tau.
