Righteousness is by faith, simply because righteousness is by Christ. Christ is the Christians righteousness. Christians do not possess a righteousness that God will accept. Our salvation is dependent upon the righteousness of Christ. "To declare, I say, at this time his righteousness: that he might be just and the justifier of him that believes in Jesus" Romans 1:26. We are saved and justified by the righteousness of Christ and not our own.
Righteousness is not by works or obedience to the law. "But to him that does NO WORKS, but believes upon him that justifies the UNGODLY, his faith is counted for righteousness" Romans 4:5. If one is trusting in their works or in their obedience to the law, they will not be justified. Jesus justifies ungodly sinners that have no righteousness of their own because they have been born after Adam. "Wherefore as by one man (Adam) sin entered into the world, and death by sin; and so death passed upon all men, for that all have sinned" Romans 5:1.
It is not possible to become righteous by the law, because the law has been abolished, Ephesians 2:15. This is why Paul wrote, "For Christ is the end of the law for righteousness to everyone that believes" Romans 10:4. Jesus nailed the law to his cross, Colossians 2:14. Jesus fulfilled the law and then he abolished it. There are no laws, rules or religion for Christians. The Old Mosiac law is history for Christians. Under the New Covenant God's people live by faith and not by laws, rules or religion. This is why Paul wrote, "The Just (justified) shall live by faith" Romans 1:17.
Living by laws, rules and religion is a denial of the Gospel and justification by faith. In the Gospel, Jesus fulfills all of the demands of God's Holy Law. He does this in our name and on our behalf, just like we did it ourselves. In the Gospel, Jesus atones for our sins and the sins of the whole world, 1 John 2:2. We can now say with all certainty that, "We are complete in him" Colossians 2:10. All that believe this Gospel have, "Entered into his rest, he has also ceased from his own works (religion) as God did from his" Hebrews 4:10.