Two firms, $1$ and $2$, producing one product, with a quantity of $Q_1$ and $Q_2$ respectively.
Price of the product: $P=a−bQ.$
Profit function for each firm, $i=1,2$ given by $\pi i=Q_i(P−C_i).$
$ Qi = \dfrac{a−Ci−bQj}{2b} $
Find the equilibrium production quantity of each firm, $Q_i$*
I know the stationary point of each firm’s revenue function can be expressed by the stationary point of the other firm’s.
Hence
$ Q1 = \dfrac{a−C1−bQ2}{2b} $ and $ Q2 = \dfrac{a−C2−bQ1}{2b} $
Would I solve this via simultaneous equation? How would i go about this?
That's ok. You have a system of two equations in $Q_1$ and $Q_2$ and you have to find the solution to that system, that is, a pair $(Q_1^*,Q_2^*)$ such that both equations are simultaneously verified. Why is that the solution?
Assuming that the two firms won't collude, this represents a static game in which utility is represented by the profit function of each firm and these two equations give each firm's strategy, that is each firm's optimum decision (output level in this case) given the other firm's decision.
That is, if firm $1$ produces $\hat Q_1$ units, the best decision for firm $2$ (that is, the one that maximizes its profit among those reachable profits, bearing in mind the fact that firm one is producing $\hat Q_1$, like it or not) is to produce $$\hat Q_2= \frac {a-C_2 -b\hat Q_1}{2b}.$$ Other decision wouldn't maximize the profit of that firm.
But if it happened at the same time that $$\hat Q_1 \neq \frac {a-C_1 -b\hat Q_2}{2b},$$ that would say that firm $1$ is not maximizing profit given the decision of firm $2$, and then firm $1$ wouldn't really choose that level of production.
The only reasonable solutions in these circumstances are those combinations of decisions such that no firm would desire to unilaterally change the choice it made, unless the other would change it too: such a combination of decisions is called a Nash equilibrium, and this model of a duopoly is called Cournot model. Other models offer solutions assuming other conditions, as for instance the existence of collusion between both firms, which gives different equilibria and profits (in general, higher prices and profits, and lower total output).