Generally, you infringe a patent if you infringe a single claim. There is no such term as "linked claims." There is something called a dependent claim, which is a claim that references another claim in a specific way. A dependent claim includes all the language of the claims it depends on.

Thus, if claim 5 says something like, "The system of claim 3, wherein" and claim 3 says something like "The system of claim 1, wherein" then claim 5 will probably be deemed to include all the language of claims 3 and 1. However, claim 1 does not necessarily include claims 3 and 5 and you can infringe claim 1, even if your system does not have any of the elements listed in claims 3 and 5.