Righteousness through faith

"Yet we know that a person is not justified by works of the law but through faith in Jesus Christ, so we also have believed in Christ Jesus, in order to be justified by faith in Christ and not by works of the law, because by works of the law, no one will be justified." - Galatians 2:16
"... in order that I might be found in Him, not having a righteousnesss of my OWN that comes for the law, but that which comes through faith in Christ, the righteousness from God that depends on faith." -Phil 3:9
"For what does Scripture say? "Abraham believed God, and it was counted to him as righteousness." Now to the one who works, his wages are not counted as a gift but as his due. And to the one who does not work but believes in Him who justifies the ungodly, his faith is counted as righteousness." - Romans 4:3-5
"Yet we know that a person is not justified by works of the law, but through faith in Jesus Christ, so we also have believed in Christ Jesus, in order to be justified by faith in Christ and not by works of the law, because by works of the law no one will be justified.... I do not set set aside the grace of God, for if righteousness were through the law, then Christ died in vain." - Galatians 4:16, 21
We are not righteous because we obey the law, We are righteous because we live by faith. We believe that Jesus paid our debt and abolished sin and death. Through His sacrifice, we are free from sin and thus are clean and righteous. We continue to walk in faith: we continue to believe that, through Christ, we are free and blameless. And that's what righteousness through faith means: our faith in Jesus Christ makes us righteous.