Salvation by ...

In the institution of the Levitical Law, its easy to see why a non-faithful person could read it as "salvation by works"

God said "You shall therefore keep my statutes and my rules; if a person does them, he shall live by them: I am the LORD." (Leviticus 18:5)

But King David helps to see behind the law: "You do not delight in sacrifice, or I would bring it; you do not take pleasure in burnt offerings. The sacrifices of God are a broken spirit; a broken and contrite heart, O God, you will not despise." (Psalm 51:16-17)

"but the righteous shall live by his faith" (Habakkuk 2:4b)

The obedience of the law SHOULD stem from the broken and contrite heart. IF the broken spirit is the source of the obedience, then the person is righteous.

"For by works of the law no human being will be justified in his sight, since through the law comes knowledge of sin. But now the righteousness of God has been manifested apart from the law, although the Law and the Prophets bear witness to it — the righteousness of God through faith in Jesus Christ for all who believe. For there is no distinction: for all have sinned and fall short of the glory of God" (Romans 3:20-22)

"[that I may gain Christ] and be found in him, not having a righteousness of my own that comes from the law, but that which comes through faith in Christ, the righteousness from God that depends on faith" (Philippians 3:9)