In the book of 2 Kings, the kingdoms of Judah and Israel were separate.
In Amaziah's (Judah) 15th year, Jeroboam (Israel) began to reign. In Jeroboam's (Israel) 27th year, Azariah (Judah) began to reign. That is, Azariah (Judah) began to reign 15+27=42 years after Amaziah (Judah) began to reign. However, Amaziah (Judah) only reigned 29 years. I've tried to illustrate this below:
ISRAEL: Jeroboam (reigned from the 15th year of Amaziah) JUDAH: Amaziah (reigned 29y) --> Azariah (reigned from the 27th year of Jeroboam)
Similarly, Ahaz began to reign in Pekah's 17th year. Hoshea began to reign in Ahaz' 12th year. But Pekah only reigned 20 years (leaving a gap of 9 years).
ISRAEL: Pekah (reigned 20y) --> Hoshea (began to reign in Ahaz' 12th year) JUDAH: Ahaz (began to reign in Pekah's 17th year)
I know that many have tried to make the chronologies fit by including co-regencies (two kings reigning at the same time). But the word מָלַךְ is the same for all of these kings - wouldn't all of the מָלַךְs either refer to the beginning of a co-regency, or the beginning of a reign alone? And if not: how do we differentiate the beginning of a co-regency from the beginning of a reign alone, based on the words used or the context?