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Answer:
To more directly answer your question I am going to focus on the Jewish concept and expectations of the messiah and what Jewish tradition generally accepts as messianic prophecy. In light of the life, death, resurrection, and teachings of Jesus, Christians came to see more prophecy that spoke of Jesus.
The concept of the messiah is a gradual one in the Old Testament. The Torah does not explicitly foretell a messiah. However, it does speak of the end days which later Jewish generations would consider an implicit reference to the messianic era. The explicit references to a coming messiah are found in the words of the prophets:
See, days are coming—oracle of the LORD—when I will raise up a righteous branch for David; As king he shall reign and govern wisely, he shall do what is just and right in the land. In his days Judah shall be saved, Israel shall dwell in security. (Jer. 23:5)
In those days, at that time, I will make a just shoot spring up for David; he shall do what is right and just in the land. (Jer. 33:15)
The spirit of the LORD shall rest upon him: a spirit of wisdom and of understanding, a spirit of counsel and of strength, a spirit of knowledge and of fear of the LORD, and his delight shall be the fear of the LORD. Not by appearance shall he judge, nor by hearsay shall he decide, But he shall judge the poor with justice, and decide fairly for the land’s afflicted. He shall strike the ruthless with the rod of his mouth, and with the breath of his lips he shall slay the wicked. Justice shall be the band around his waist, and faithfulness a belt upon his hips. (Is. 11:2-5)
For the Israelites will remain many days without king or prince, Without sacrifice or sacred pillar, without ephod or household gods. Afterward the Israelites will turn back and seek the LORD, their God, and David, their king; They will come trembling to the LORD and to his bounty, in the last days. (Hos. 3:4-5)
The Old Testament prophets also spoke of a coming messianic era (Is. 2:4; 11:6-9; Jer. 23:8; 30:3; Mic. 4:2-3; Zec. 14:9).
The Jewish people expected the messiah to be from the line of David who would free Israel to properly worship God and allow them to live in abundance and justice. This expectation was based on God’s promise to David that the kingdom of his descendants shall not end (2 Sam. 7:13).