Really the answer can be fairly simple, but you could also make it a lot more complex.
Simply put,
1. God basically says, "you can get to heaven if you don't sin" or "If you sin I will crush you". Also the Old covenant - "keep all of these rules, do these sacrifices,etc... and I won't crush you, and I will bless you"
2. We sin. We do not keep all of the rules, and have a tough time with all of the sacrifices.
3. According to God's law, the punishment for our sin is death. Not keeping the rules and sacrifices is punishable by death. W
4. We have no recourse. Because we've sinned and are guilty, we can't just go to God and plead our case, we cannot just say "hey look, we tried, but can't do it, please don't crush us"
5. Jesus is born. He is fully God and fully man. He is also perfect. Never sins, never fails to do as God asks.
6. Because Jesus has not sinned, he can go to the "gates" of Heaven. The impassable gates that previously kept you and I out. He throws open the gates. For the first time in history, the gates can be opened by a Man, because that man was perfect.
7. He then "bargains" with God, saying "Father, place their guilt, and the punishment for all of their sin on ME" and they will then no longer be guilty.(he is able to take all of this guilt, and stand in our place, because he is fully God) and able to bargain with God, because he hasn't done anything that God could find him guilty for.
8. In that act, Jesus essentially opens the door to heaven for us, and nullifies the requirements of the old covenant. At the last supper, He then issues us a new covenant, where all that we have to do for our end is to partake in Him, to fulfill our end.
9. So the old covenant was fulfilled because Jesus was able to bargain with God on our behalf, and change the rules.
With all of that said, here is a great passage from about 350 years ago from John Flavel's "The Covenant of Redemption between the Father and the Redeemer"
Father: My son, here is a company of poor miserable souls, that have utterly undone themselves, and now lie open to my justice! Justice demands satisfaction for them, or will satisfy itself in the eternal ruin of them: What shall be done for these souls And thus Christ returns.
Son: O my Father, such is my love to, and pity for them, that rather than they shall perish eternally, I will be responsible for them as their Surety; bring in all thy bills, that I may see what they owe thee; Lord, bring them all in, that there may be no after-reckonings with them; at my hand shalt thou require it. I will rather choose to suffer thy wrath than they should suffer it: upon me, my Father, upon me be all their debt.
Father: But, my Son, if thou undertake for them, thou must reckon to pay the last mite, expect no abatements; if I spare them, I will not spare thee.
Son: Content, Father, let it be so; charge it all upon me, I am able to discharge it: and though it prove a kind of undoing to me, though it impoverish all my riches, empty all my treasures, (for so indeed it did, 2 Cor. 8: 9. “Though he was rich, yet for our sakes he became poor”) yet I am content to undertake it.
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