From the very beginning faith directed toward Messiah was taught in the Old Testament. After Adam and Eve fell God promised the Seed to come would destroy satan but would Himself be wounded. (Gen. 3:15). Adam believed, (faith), and declared Eve was the Mother of all the living. (Gen. 3:20). Eve evidenced her faith in that same promise though it was misdirected. (Gen. 4:1) Faith in the promised Seed to come who would destroy satan.
Abraham also was saved by faith. (Gen. 15:6). Abraham had exercised faith previously when he left Ur of the Chaldees. (Gen. 12:4). But it is here in (Gen. 15) that Abraham's faith results in imputed righteousness. Why? Because it pertained to the Seed promise, which would result in the Messiah.
That the people were not aware of all that this faith entailed is immaterial. It works because God says it works. He imputes righteousness at His direction. And unless righteousness is imputed, there is no salvation.
When Israel was commanded to place the blood of the Passover lamb on the doorpost's of their houses, they were exercising faith in God in doing so, and that faith was directed toward the Messiah who would also be the Lamb of God. Though they didn't understand all of that at that time, is immaterial. They exercised faith in God toward that which pertained to Messiah.
Don't know which Old Testament passages you are referring to when you say 'the prophets clearly teach'. Israel would never be saved by obedience to the Law. She clearly demonstrates she cannot keep the Law. No one is ever saved by keeping the Law. But with the demands of the Law came the sacrifices. And the one who brought the sacrifices evidenced a faith towards God. And those sacrifices pointed to the Sacrifice of the Lamb of God, Jesus Christ.
Good-Ole-Rebel
The "seed" of Eve is plural, referring to humanity. Not Jesus. It is not a messianic passage. And it says nothing, NOTHING about having faith in a messiah.
Abraham was saved by a faith that was exhibited by his obedience -- the two are inseparable from each other. And what does it mean that he was saved? I don't think it means that he was washed in the blood of the lamb, as Christians assume.
As for the blood of the lamb being painted on the doorposts of the homes, many other things were done as well. But you find no religious meaning in these other things. You basically cherry pick the one thing that you want.
I never said, obedience saves. I stay away from the word saves because basically the Tanakh has a very different meaning for Salvation than your New Testament. Your Christian Scriptures mean salvation from sins, from eternal damnation. In the Tanakh, salvation is a very earthly thing, salvation from our enemies, etc. What obedience does is keeps us in God's good graces.
And what I specifically said was that the remnant of Israel were those who were obedience.
And that that Tanakh never EVER talk about having faith in the Messiah as the way to escape eternal damnation. EVER.
Which is quite a problem for Christianity. I mean, it's like this big gaping hole.