The solution, born in God's grace, was the notion of vicarious sacrifice. By identifying with the sacrifice [see also Leviticus 4], the penitent could acknowledge and embrace the just retribution against his or her sin, while living on to experience reconciliation with God. God and the sinner would be seeing sin the same way, and God and vicariously punished sinner could be at one again. Atonement is about being on track with God, after the just dust of punishment has settled.
Sin is untenable if you think about it--withstanding an omniscient, omnipresent, omnipotent, omnitemporal God just doesn't have a future! For God to allow sin to continue, sin would have to be part of who he was. Since he cannot cease being what he is, sin must be eradicated. That is, really, what death is: the quashing of sin (the Lake of Fire is the logical and ultimate conclusion to the problem of sin among spiritually eternal beings). From God's perspective, then, sin has already been dealt with, for all sinners die and then face the Lake of Fire.
From our perspective, that's not pleasant news at all. There is in the human soul a bit of Adam and Eve cowering in the bushes with the foreboding knowledge that our dying and death is the result of being rebels at odds with God. We're in a quandary: we don't want to die, but at the same time, we want to do what we want to do, even if God doesn't like it. Even if God waved a magic wand over us and returned us to life, without a change of heart we'd be dead again in a heartbeat!
So, to take care of sin, we first must repudiate it. Then its effects must be reversed and our consciences cleansed of the memory. Only then could we be emboldened enough to come out from behind the bushes and meet our Maker face to face. Then we'd have to go on with him, if he'd have us, walking trustingly, always in agreement with him.
If such an atonement is to take place, God must take the initiative, for we have no power or standing to do so. His heart must make a way to reach past the wall of sin and death and bring home the prodigals who would want to come home. In doing so, he'd have to retain the perfection of his justness (Godhood) while simultaneously justifying sinners who are not just. Substitution is the only avenue.
That which isn't guilty, must willingly adopt our sin as his own and take the heat for it, literally. Thereby, the cathartic that frees the penitent to stand unashamed in the presence of the holy God would be provided. This was accomplished in Christ Jesus. The sinless Son of God, became a man, offered himself in our place, to be our sin and to suffer our death.
The punishment of God against us was fully poured out on Jesus Christ. He died in our sin, forsaken by the Holy God. However, since he was faithful to God's will unto death, he rose from the dead on the third day, victor over our sin and death. By such, he became the means by which we can stand assured before and reconciled to God. Those that trust in this atonement made by Christ are, by the means of it, made one with God.