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The Old Testament covenant had some laws that apply only to Jews, and some laws that only apply when a temple is in place.
After Jesus came, and the Temple was destroyed, and the curtain of the temple ripped open, then there was a New Covenant.
The old temple laws and festival laws do not apply anymore, and neither do the parts specific only to Jews.
However the moral laws still stand.
The Old Testament Law was the law of Severity, and it teaches how to avoid going to hell, or getting punished, but it doesn't focus on how to be redeemed through the Law.
The New Law is the law of love, mercy, forgiveness, charity. This is the law that redeems one from the Old Law. The New Law is telling you to go above and beyond in righteousness.
For example, while a Jew under the Old Law has the right to fight back when someone attacks him, the Christian is urged to turn the other cheek, to forgive, to show endless mercy.
Many of the sayings of Jesus was already in the Old Testament, but it's easy for them to get overlooked amongst all the other civil and moral laws.
Following Old Testament Law is like getting 50% on the post-death life-review exam. Following New Testament Law is like getting 100% on the post-death life-review exam.