Thank you for the salvation question, “How did Jesus save everyone by dying on the cross?”
The atoning death of Jesus Christ DOES NOT SAVE EVERYONE, but MAKES AVAILABLE God’s salvation to ANYONE who believes in his substitutionary death for him or her. This is an important distinction for the question being pose would inferred that everyone is saved and none will be condemned. Salvation is available to everyone or anyone, but only applicable to the ones who believe that Jesus Christ died for one’s sin. For God so loved the world that he gave his One and Only Son that WHOEVER BELIEVES SHALL NOT PERISH BUT HAVE EVERLASTING OR ETERNAL LIFE (John 3:16).
To have a greater appreciation and depth of insight into the atonement, the Old Testament pictures for us the covering and removal of sins against God or other human beings. The Day of Atonement in Leviticus 16 depicts for us how sin is atone for in the Old Testament and their foreshadowing or parallelism in the New Testament.
Leviticus 16 says this, “15 “He shall then slaughter the goat for the sin offering for the people and take its blood behind the curtain and do with it as he did with the bull’s blood: He shall sprinkle it on the atonement cover and in front of it. 16 In this way he will make atonement for the Most Holy Place because of the uncleanness and rebellion of the Israelites, whatever their sins have been. He is to do the same for the tent of meeting, which is among them in the midst of their uncleanness. 17 No one is to be in the tent of meeting from the time Aaron goes in to make atonement in the Most Holy Place until he comes out, having made atonement for himself, his household and the whole community of Israel.
18 “Then he shall come out to the altar that is before the LORD and make atonement for it. He shall take some of the bull’s blood and some of the goat’s blood and put it on all the horns of the altar. 19 He shall sprinkle some of the blood on it with his finger seven times to cleanse it and to consecrate it from the uncleanness of the Israelites. 20 “When Aaron has finished making atonement for the Most Holy Place, the tent of meeting and the altar, he shall bring forward the live goat. 21 He is to lay both hands on the head of the live goat and confess over it all the wickedness and rebellion of the Israelites—all their sins—and put them on the goat’s head. He shall send the goat away into the wilderness in the care of someone appointed for the task. 22 The goat will carry on itself all their sins to a remote place; and the man shall release it in the wilderness” (A searchable online Bible in over 150 versions and 50 languages). Citations are in the New International Version.
One goat is sacrificed to atone, to cover the sins against God while the other live goat bears all the confessed sins of Israel and is taken away into the wilderness. God’s righteousness is satisfied by the shed blood of the goat and God doesn’t look upon one’s sins anyone as the lived goat carries away one’s sins. What a beautiful picture for the Israelites.
The author of Hebrews states that the blood of animals can never remove sin nor can the priest permanently cease making those sacrifices for the Israelites would sin daily and on the Day of Atonement the national sins would be covered over for another year. In Hebrews 9, the writer explains the foreshadowing of a permanent solution to sin.
Hebrews 9 says this, “23 It was necessary, then, for the copies of the heavenly things to be purified with these sacrifices, but the heavenly things themselves with better sacrifices than these. 24 For Christ did not enter a sanctuary made with human hands that was only a copy of the true one; he entered heaven itself, now to appear for us in God’s presence. 25 Nor did he enter heaven to offer himself again and again, the way the high priest enters the Most Holy Place every year with blood that is not his own. 26 Otherwise Christ would have had to suffer many times since the creation of the world. But he has appeared once for all at the culmination of the ages to do away with sin by the sacrifice of himself. 27 Just as people are destined to die once, and after that to face judgment, 28 so Christ was sacrificed once to take away the sins of many; and he will appear a second time, not to bear sin, but to bring salvation to those who are waiting for him” (NIV).
In other words, Jesus being the God became the God-Man on earth. He lived a sinless life and offered his own life to satisfied the righteousness demand of God against sins by dying on the cross. Because of his infinite value as the God-Man, his shed blood became the infinite price to pay the penalty of finite sins by humans. For the blood of bulls and goats can’t take away sins but only cover them (Heb. 10:3–4). There had to come someone or something that can satisfy the holiness of God. Jesus did that by paying the price of sin by dying in our behalf. God is now able to forgive the repentant sinners for all their sins because Jesus’ blood is more than sufficient to pay the debt of those sins. Jesus took the punishment and wrath of God in our behalf. He bore our sins on his own body so that we can be set free from sins and the bondage to sins.
The Apostle Paul said this in II Corinthians 5, “We are therefore Christ’s ambassadors, as though God were making his appeal through us. We implore you on Christ’s behalf: Be reconciled to God. 21 God made him who had no sin to be sin for us, so that in him we might become the righteousness of God” (NIV).
SUMMARY: The infinite worth of Jesus in his infinite sacrifice was able to fully atone all the sins against God. The animal sacrifices in the Old Testament depicted/foreshadow someone who would pay the ultimate price to remove sins before God’s presence. Jesus’ shed blood on the cross satisfied the holiness and righteousness of God so that God can forgive the person who believes that Jesus died for him or her.
For more perspectives:
https://thechristiancorner.quora.com/How-did-Jesus-save-everyone-by-dying-on-the-cross