What does Acts 15:15 mean?

The church in Jerusalem is tasked with determining whether Gentiles must be circumcised and bound to the Mosaic law to fully follow Jesus (Acts 15:1–6). James has leaned on the testimony of Peter as evidence that God never intended Gentiles to become Jews (Acts 15:7–14). Now, he quotes "the prophets," giving scriptural weight to Peter's life experiences (Acts 15:16–18). James quotes the Septuagint translation of Amos 9:11–12, borrowing phrasing from Jeremiah 12:15 and Isaiah 45:21, and switching around a couple of lines.

Despite James' looseness with the translation, he says nothing that isn't found elsewhere in the Old Testament. These differences have no effect on the inerrancy of Scripture. It's important to remember that Luke is recording what James said, not what Amos recorded God saying. In Acts 7:43, Stephen also slightly alters Amos' words, using different, more familiar names of old pagan gods. In Acts 1:20, Peter uses Psalm 109:8, an imprecatory plea that David's enemies be destroyed, to convince the Jesus-followers they must find a replacement for Judas. Luke didn't destroy the integrity of Scripture by recording their words. Likewise, Peter, Stephen, and James did not misuse Scripture when they were led by the Holy Spirit to use and quote Old Testament passages as they did.

The fact that James can combine three different prophets into one thought proves the prophets do, indeed, agree.