How did blood atone for sin?

This is the atonement question.

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Sin causes death (Ro 6:23). Without the shedding of blood, there was no forgiveness of sins (He 9:22). You could shed your blood and die for your sins. However, the merciful God offered an alternative: You could shed an animal's blood to cover your sins so that you could continue to live. Blood sacrifices were central to the Israelite sacrificial system.

Le 17:

10 “If any one of the house of Israel or of the strangers who sojourn among them eats any blood, I will set my face against that person who eats blood and will cut him off from among his people.

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There is something sacred about blood.

11 For the life [H5315] of the flesh is in the blood, and I have given it for you on the altar to make atonement for your souls [H5315], for it is the blood that makes atonement by the life [H5315].

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God reserved blood to be used exclusively to atone for people's sins. Don't eat it.

How did blood atone for sin?

Blood contains the H5315-life of the flesh.
Shedding animal blood atones for you H5315-souls.

This is the concept of substitutionary atonement. In this concept, the life of the animal (represented by its blood) is given in place of the sinner's life.

How often did the Israelites have to offer animal sacrifices?

Daily and regularly.

Animal sacrifice was only a shadow type of substitutionary atonement. The true type is found in the sacrifice of Jesus, the Son of God.

1 John 1:

7 The blood of Jesus His Son cleanses us from all sin.

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Jesus' death on the cross fulfilled the animal sacrifices perfectly, once and for all. We no longer need to shed animal blood to atone for our sins. We can live forever. We have eternal life in Jesus' blood.