How do the Scriptures teach about righteousness by faith?
The teaching of righteousness by faith is found throughout the Scriptures, beginning with the first promises given to Adam and Eve in Genesis 3 of God defeating Satan. It continues from there with the promises given to the Patriarchs, enactments of God's care and deliverance of his people like the flood, in the stories of various Bible characters like Jacob when he is fleeing his home and is visited by God in the dream of the ladder stretching to heaven, in the progressive exodus journey of the children of Israel out of Egypt, the sanctuary in the wilderness, the prophecies of the Bible, more promises, the teachings of the prophets, the teachings of the Psalms, the teachings of Jesus, nature such as the vine and the branches, the writings of Paul, etc. There is no book of the Bible that does not present some aspect of righteousness by faith.
John 5:39 (NKJV) 39 "You search the Scriptures, for in them you think you have eternal life; and these are they which testify of Me."
Mark 12:10-11 (NKJV) 10 "Have you not even read this Scripture: The stone which the builders rejected Has become the chief cornerstone. 11 This was the LORD’s doing, And it is marvelous in our eyes’?”
Acts 8:32-35 (NKJV) 32 "The place in the Scripture which he read was this: “He was led as a sheep to the slaughter; And as a lamb before its shearer is silent, So He opened not His mouth. 33 In His humiliation His justice was taken away, And who will declare His generation? For His life is taken from the earth.” 34 So the eunuch answered Philip and said, “I ask you, of whom does the prophet say this, of himself or of some other man?” 35 Then Philip opened his mouth, and beginning at this Scripture, preached Jesus to him."
Acts 26:22-23 (NKJV) 22 Therefore, having obtained help from God, to this day I stand, witnessing both to small and great, saying no other things than those which the prophets and Moses said would come— 23 that the Christ would suffer, that He would be the first to rise from the dead, and would proclaim light to the Jewish people and to the Gentiles.”
Luke 24:27 (NKJV) 27 "And beginning at Moses and all the Prophets, He expounded to them in all the Scriptures the things concerning Himself."
Luke 24:44-48 (NKJV) 44 "Then He said to them, “These are the words which I spoke to you while I was still with you, that all things must be fulfilled which were written in the Law of Moses and the Prophets and the Psalms concerning Me.” 45 And He opened their understanding, that they might comprehend the Scriptures.
46 Then He said to them, “Thus it is written, and thus it was necessary for the Christ to suffer and to rise from the dead the third day, 47 and that repentance and remission of sins should be preached in His name to all nations, beginning at Jerusalem. 48 And you are witnesses of these things."
In the earthly sanctuary, first built in the wilderness hole the people were leaving Egypt to go the promises land, they had a visual representation of the righteousness by faith that continues to inform in our day. On this link you will find a power point of what it looked like, along with the sanctuaries of Solomon and Zerubbabel.