Rome came into conflict with Carthage because they were both major powers in the same geographic area. As the two empires grew, they started to have conflict over who would get what in areas where the empires touched.
The first such conflict arose over the island of Sicily. Both empires worried that the other would gain an advantage if it could take control of this island because of its strategic location. This might allow the empire that controlled Sicily to dominate or even destroy the other empire. The First Punic War started because of this issue.
So, Rome and Carthage came into conflict because they were both expanding and each side worried that the other would become too powerful.
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