Without God's spirit living in and through us then it truly would be impossible to keep God's commandments in spirit, although we may be able to keep them physically. Nobody would have been saved except for the sacrifice of Christ:
Mat 19:25 When his disciples heard it, they were exceedingly amazed, saying, Who then can be saved?
Mat 19:26 But Jesus beheld them, and said unto them, With men this is impossible; but with God all things are possible.
With the new covenant, God's laws, the ten commandements, are written on the hearts of believers:
Jer 31:33 But this shall be the covenant that I will make with the house of Israel; After those days, saith the LORD, I will put my law in their inward parts, and write it in their hearts; and will be their God, and they shall be my people.
Heb 8:10 For this is the covenant that I will make with the house of Israel after those days, saith the Lord; I will put my laws into their mind, and write them in their hearts: and I will be to them a God, and they shall be to me a people:
Heb 10:16 This is the covenant that I will make with them after those days, saith the Lord, I will put my laws into their hearts, and in their minds will I write them;
2Co 3:3 Forasmuch as ye are manifestly declared to be the epistle of Christ ministered by us, written not with ink, but with the Spirit of the living God; not in tables of stone, but in fleshy tables of the heart.
This process of God writing his laws upon our hearts is only possible through the sacrifice of Christ. We call it "having the spirit of God", or the indwelling of the holy spirit.
The Ark of the Covenant was a type of the human heart. In the ark, God specified that the ten commandments be placed in it:
Deu 10:2 And I will write on the tables the words that were in the first tables which thou didst break, and thou shalt put them in the ark.
Deu 10:3 And I made an ark of shittim wood, and hewed two tables of stone like unto the first, and went up into the mount, having the two tables in mine hand.
Deu 10:4 And he wrote on the tables, according to the first writing, the ten commandments, which the LORD spoke unto you in the mount out of the midst of the fire in the day of the assembly: and the LORD gave them unto me.
Deu 10:5 And I turned myself and came down from the mount, and put the tables in the ark which I had made; and there they be, as the LORD commanded me.
The ten commandments were the heart of the old covenant:
Exo 34:28 And he was there with the LORD forty days and forty nights; he did neither eat bread, nor drink water. And he wrote upon the tables the words of the covenant, the ten commandments.
Those who enter in the new covenant with God pledge to follow these same laws, with a distinct advantage. All we have to do is allow the spirit of God to lead us.
This "writing on our hearts" manifests itself as Godly love. ...love toward God and love toward others.
Mat 22:37 Jesus said unto him, Thou shalt love the Lord thy God with all thy heart, and with all thy soul, and with all thy mind.
Mat 22:38 This is the first and great commandment.
Mat 22:39 And the second is like unto it, Thou shalt love thy neighbor as thyself.
Mat 22:40 On these two commandments hang all the law and the prophets.
The ten commandments ARE the written expression of Godly love toward God and others.
What precisely does it mean to have the laws written upon our hearts?