LORD God 

Beginning with Genesis chapter 2 verse 5 the phrase LORD God is used as the name for God throughout the rest of the second chapter and through the third chapter.  The word "LORD" is in all capitals in the King James translation of the bible.  Wherever, this use of all capitals in spelling the word, lord, is found it has reference to the title "Jehovah" to describe God.  Jehovah was not spelled out in the Old Testament, but only the consonants were used JHVH or YHWH which when the vowels are added make the names Jehovah or Yahweh.  This has reference to God as the covenant making covenant keeping God.  In chapter 2, God had made with man the covenant of the law of sin and death.  Likewise, he had made with the man and woman the covenant of marriage.

 When the word, LORD, is appended with the word, God, it brings together the description of God as both the creator of all things or the Almighty and God as the covenant making covenant keeping God.  As the creator of all things, God has the right and authority and ability to enforce the covenants that he establishes.  God made the covenant of the law of sin and death in chapter 2 and he comes to enforce this covenant in chapter 3. 

Eve the Mother of all living 

 Gen. 3:20 "And Adam called his wife's name Eve; because she was the mother of all living."  The name Eve literally means mother.  She was given this name because she is the mother of all living.  There has been much speculation in scientific communities about their being multiple sources or lines of genealogy back to the fountainheads of mankind.  However, even science has finally come to recognize that through DNA studies that all mankind can be traced back to a single set of parents. 

Further, people have speculated in Genesis chapter 6 when the scriptures speak of the sons of God and the daughters of men, that perhaps angels had cohabitated with men and brought forth children.  However, our verse above completely rules out this idea as a possibility.  We will deal more with Genesis chapter 6 when we get there.  

God provided redemption vs. the impossibility of man redeeming himself

 Gen. 3:21 "Unto Adam also and to his wife did the LORD God make coats of skins, and clothed them. 22 And the LORD God said, Behold, the man is become as one of us, to know good and evil: and now, lest he put forth his hand, and take also of the tree of life, and eat, and live for ever: 23 Therefore the LORD God sent him forth from the garden of Eden, to till the ground from whence he was taken. 24 So he drove out the man; and he placed at the east of the garden of Eden Cherubims, and a flaming sword which turned every way, to keep the way of the tree of life."

Previously Adam and Eve had sewed fig leaves together and made themselves aprons to hide the shame of their nakedness (sin).  This, however, was not an appropriate covering.  Their efforts to cover their sin were totally ineffective.  Only God can cover sin.  In the passage above we find that God made coats of skins and clothed Adam and Eve.  This symbolizes to us that for a suitable covering of sin there must be a sacrifice.  Further, this sacrifice must be provided by God.  Truly our works (sewed fig leaves) cannot save us from our sins: 2 Tim. 1:9 "Who hath saved us, and called us with an holy calling, not according to our works, but according to his own purpose and grace, which was given us in Christ Jesus before the world began."  The Lord Jesus Christ came as the Lamb of God to save his people from their sins (Matt. 1:21).  God provided the only sacrifice that was suitable to cover us from our sins.

Before Adam and Eve ate of the fruit of the tree of the knowledge of good and evil in the midst of the Garden, they were free to eat of every tree of the Garden but of this one.  Also in the midst of the Garden was the tree of life.  They had been free to eat of that tree according to the covenant God had made with Adam in the Garden.  Had they eaten of that tree they would have lived forever in their innocent state.  Once, however, they had eaten of the tree of the knowledge of good and evil, Adam and Eve lost the privilege and ability to eat of the tree of life.  God "drove out the man; and he placed at the east of the garden of Eden Cherubims, and a flaming sword which turned every way, to keep the way of the tree of life."  No matter how much man may have wanted to come back and eat of the tree of life, he had no authority or ability to come back as God made it impossible for man to come back. 

Similarly, we read that the Lord said that it was impossible for man to do anything to get eternal life: Matt. 19:25 "When his disciples heard it, they were exceedingly amazed, saying, Who then can be saved? 26 But Jesus beheld them, and said unto them, With men this is impossible; but with God all things are possible."  Eternal life is the gift of God and is given to the elect.  Man cannot do anything to get eternal life.  Like Adam and Eve we are clothed with the righteousness of the Lord Jesus Christ thru his sacrificial atonement: 2 Cor. 5:21 "For he hath made him to be sin for us, who knew no sin; that we might be made the righteousness of God in him."