We consider a set of S independent encoders that must transmit a set of correlated sources through a network of noisy, independent, broadcast channels to T receivers, with no interference at the receivers. For the general problem of sending correlated sources through broadcast networks, it is known that the source-channel separation theorem breaks down and the achievable rate region as well as the proper method of coding are unknown. For our scenario, however, we establish the optimal rate region using a form of joint source-channel coding. When the optimal channel input distribution from transmitter i to receiver j is independent of j, our result has a max-flow/min-cut interpretation. Specifically, in this case, our result implies that if it is possible to send the sources to each receiver separately while ignoring the others, then it is possible to send to all receivers simultaneously.