Behold, his soul which is lifted up is not upright in him: but the just shall live by his faith. (Habakkuk 2:4 KJV)
It is written, "The just shall live by faith (Habakkuk 2:4, Romans 1:7, Galatians 3:11, Hebrews 10:38)". This is repeated four times in the Bible, once by Habakkuk in Old Testament, and quoted three times by Paul in the New Testament.
In reality, everyone lives by faith. We all have faith in the usual. We just take for granted that everything will just happen as per normal. Nothing extraordinary will happen and nature will take its course. When we take our next step, we never fear that the ground will give way and swallow us up. When we sleep, we assume we will wake up. When we sit on a chair, we do not expect the chair to give way.
We cannot live without faith because we are not all powerful (omnipotent), all knowing (omniscience) and all-present (omnipresent). We have no time and no means to prove everything. We just have to trust that the information we have is true.
The Bible never teaches that doing good will save you. The Bible teaches that you are saved by believing in the truth, the saving truth, the historical event that Christ had died for our sin.
What you do will not save you. What you put your faith in will determine whether you have eternal life or eternal death.
The just shall live by faith. The wicked shall die by faith.
And without faith it is impossible to please God, because anyone who comes to him must believe that he exists and that he rewards those who earnestly seek him. (Hebrews 11:6 NIV)