“Life by” is an Old Testament expression. Paul quotes the “life by” statements in the Old Testament from time to time. Notice in the following that it means to be in a right standing before God, and therefore equivalent to “justified” (to be right with God). Words in capital letters are quotes from the Old Testament:
“man does not live by bread alone, but man lives by everything that proceeds out of the mouth of the LORD” (Deuteronomy 8:3)
Ezekiel 33:12 “a righteous man will not be able to live by his righteousness on the day when he commits sin”. 19 “But when the wicked turns from his wickedness and practices justice and righteousness, he will live by them”.
The soul of “the righteous”, who “live by his faith”, is “right within him” (Habakkuk 2:4).
“For Moses writes that the man who practices the righteousness which is based on law shall live by that righteousness” (Romans 10:5).
“The life which I now live in the flesh I live by faith in the Son of God” (Galatians 2:20). “Live by faith” therefore seems to be equivalent to “justified by faith”.
Galatians 3:11 quotes from Habakkuk 2:4 and states that “no one is justified by the Law before God … for, “HE RIGHTEOUS MAN SHALL LIVE BY FAITH”. “Justified by the Law” is therefore contrasted to “LIVE BY FAITH”, but elsewhere “justified by the Law” is contrasted to “justified … through faith in Christ Jesus” (2:16). In other words “LIVE BY FAITH” is equivalent to “justified … through faith in Christ Jesus” (2:16).
Galatians 3:12 quotes from Leviticus 18:5 and identifies the origin of the view that one is “justified by the Law” as a statement made by Moses, which said that “HE WHO PRACTICES THEM (the laws) SHALL LIVE BY THEM.”
Romans 1:17 also quotes the phrase “live by … faith” from Habakkuk 2:4 to substantiate Paul’s claim that “the good news of the Christ … is the power of God to salvation to every one who is believing” (YLT – Rom. 1:16). “Live by … faith” therefore means “salvation to every one who is believing”.
Gal. 5:25 “If we live by the Spirit, let us also walk by the Spirit”.
Hebrews 10:38 “MY RIGHTEOUS ONE SHALL LIVE BY FAITH”.