The incorporation of a price elasticity in your regression requires that your dependent variable, quantity, be logged as well.
Take an example of a basic demand side equation including two independent variables
$$Q_d=\beta_0+\beta_1 P+\beta_2 S+\mu$$
where $Q_d$ is quantity demanded, $P$ is the price of the good in question and $S$ is the price of a substitute.
To incorporate price elasticity in this regression you have to log your dependent $Q_d$ and then log price $P$ as well.
$$\ln(Q_d)=\beta_0+\beta_1 \ln(P)+\beta_2 S+\mu$$
Only via this framework can we calculate elasticity, given by $\beta_1$.
In terms of calculating equilibrium we would require the use of simultaneous equations where supply $Q_s$ is logged.
However this calculation of percentage change in $Q$ can be transformed easily by specific equilibrium quantity by using basic algebra such as:
1.$$Q_d=Q_s=Q^*$$
2.$$\ln(Q_d)=\ln(Q_s)=\ln(Q^*)$$
3.$$\mathrm e^{\ln(Q_d)}=\mathrm e^{\ln(Q_s)}=\mathrm e^{\ln(Q^*)}$$
4.$$Q_d=Q_s=Q^*$$
The same steps can be followed when calculating $P^*$.
Hope this helps.