Faith / Answer across Scripture
Romans 1:17 connects the gospel, God's righteousness, and the line that the righteous shall live by faith.
That the righteous live by faith means life before God is received and lived by trusting him, not by earning a righteous standing. Romans 1:17 connects the gospel with God's righteousness being revealed from faith for faith and then supports the point with the line, 'The righteous shall live by faith.' In Romans, faith is not bare opinion; it is trust in God's saving action.
Romans 1:16-17 functions as a thesis for the letter. Paul introduces the gospel, God's righteousness, and life by faith before unfolding how that promise reaches Jews and Gentiles in Christ.
Romans 1:16-17 is the central passage for this answer. Romans 4 later develops the same faith theme through Abraham, showing that trust in God's promise is central to Paul's argument.
Key passages
16For I am not ashamed of the gospel of Christ, for it is the power of God to salvation to everyone who believes; to the Jew first, and also to the Greek.
17For therein is revealed a righteousness of God from faith to faith. As it is written, "But the righteous shall live by faith."
A common misunderstanding is to treat faith as a vague optimistic attitude. In Romans, faith is directed toward God and his gospel.