Yes there is a relationship between the Old Testament and the law , as the book of Moses Leviticus is full of laws the Jews had to follow.
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The New Testament can be described as The New Law as The Old Testament is to The Old Law. Christ brought a new law with Him which was to replace the old law as taught in the Old Testament. He taught that He was the Christ, the son on a living God. He also gave new commandments which the people were to live by doing away with the old, such as An eye for an eye. He taught that we were to love those who did harm to us and to love our neighbour as ourselves. He gave us the ten commandments and taught us how to pray. To go the extra mile when the need arose. He taught that everyone would be resurrected.
No, the Law had not yet been given. Abraham lived by faith and obeyed God.
The first five books of the Bible are the Law of Pentateuch.
The Old Testament is the Mosaic covenant of works The New Testament is the Abrahamic covenant of grace Law of Moses: the ministration of death Gospel of Christ: the ministration of life
Up until Christ was born the children would have been taught from the old law, which was called the Law of Moses. When Christ came He did away with the old law and gave the people a New Law to live by. The New Testament contains that law and the Ten Commandments are a part of that law. As a point of interest the Jewish religious leaders even corrupted the Old Law by adding to and taking away many true and precious things.