How were people saved in the old covenant?

Jeremiah 31:31-33
31“Behold, the days are coming, declares the Lord, when I will make a new covenant with the house of Israel and the house of Judah, 32not like the covenant that I made with their fathers on the day when I took them by the hand to bring them out of the land of Egypt, my covenant that they broke, though I was their husband, declares the Lord. 33But this is the covenant that I will make with the house of Israel after those days, declares the Lord: I will put my law within them, and I will write it on their hearts. And I will be their God, and they shall be my people.

God makes promises to us as long as we keep up our end of the agreement. This is a covenant, a promise in other words or an agreement. Much like marriage is an agreement, we make vows to love through the bad times and the tough times and to stick it out and make it work.

The ten commandments are the terms of this covenant which were written on the tablets by the hand of God and then given to Moses and the people of Israel.  

There has to be a leader. Now lets get to know our leader.
What does God promise to those who keep the terms of this relationship?

1. Mercy
Promise: To mercifully forgive repentant sinners.

Exodus 34:6-7
The Lord, the Lord, a God merciful and gracious, slow to anger, and abounding in steadfast love and faithfulness, keeping steadfast love for thousands, forgiving iniquity and transgression and sin, but who will by no means clear the guilty, visiting the iniquity of the fathers on the children and the children’s children, to the third and the fourth generation.”

There are two kinds of people.
1- Those who are presumptuous and unrepentant God will not forgive them.
2- Those who’s sin have crushed them and brought them to their knees, broken them and led them to God’s mercy for forgiveness.

God is mercifully willing to forgive those who seek out forgiveness.

A covenant relationship based on mercy from the beginning

There are many bible teachers who say that there are two covenants, mainly because of Jeremiah 31. But this new covenant is not really different to the first only in that God supernaturally now writes the commandments not a stone tablets but on our hearts.

When God says in verse 10, "Behold, I make a covenant!" right after declaring himself to be a God who is merciful and who forgives iniquity, transgression, and sin, can we believe that this covenant is not based on mercy? Can we believe that the covenant has no merciful provision for forgiveness in it? And if it is based on mercy and does provide forgiveness, how can it be a covenant of works? If a person sins under this covenant and flees to God for mercy and finds forgiveness, how can we say the covenant is based on merit? Is it merit we offer to God when we plead his mercy and ask for forgiveness?  

maybe this is a covenant of works based on merit even though God shows some mercy and forgives some sins." like and employer - employee relationship.

The main point of Exodus 32–34 is to show that if merit were basis of God's dealings with this people, they would have been destroyed long ago.

2- The basic requirement is worship (and not really following the letter of the law)

The reason is that the basic requirement of the covenant is worship not work. But you can't worship an employer, whose needs you are meeting, in order to earn his wages.

The only kind of being you can truly worship is one whose fullness meets your needs. If God wanted to be pictured in this covenant as an employer who pays wages of blessing to employees who supply him service valuable enough to earn these wages, then he would not have written a so-called job description requiring worship above all else. You can't worship an employer who depends on you to meet his needs.

Exodus 34:13 Break down their altars, smash their sacred stones and cut down their Asherah poles.

The reason for tearing down pagan altars is to guard their hearts for Yahweh alone. The reason for not making covenants with pagan peoples is to escape the snare of divided loyalties. The commands of the covenant don't describe the services God needs as an employer.

They describe the faithfulness he wants from his wife. It's as though he said, 'Don't make dates with other men; don't keep the pictures of your old boyfriends on the dresser, lest they become a snare for you and draw your heart away from me."

God is a jealous husband
"For you shall worship no other god, for the LORD, whose name is Jealous, is a jealous God." The demand of the covenant is for single-minded worship of God alone.

This picture is confirmed by verses 15 and 16 which warn Israel against playing the harlot with other gods. The demand of the covenant is: don't be a harlot. Don't commit adultery against God. Don't let your heart turn from him and go after other things. For your God, your husband, whose name is Jealous, is a jealous God.

Two Reasons to Stress This Covenant's Merciful Basis

There are two reasons I have stressed that the covenant of Exodus 34 is not a covenant of works but is based on mercy.

1. To Appreciate the Unity of the Bible
One is to help us appreciate and benefit from the unity of the Bible. The covenant made with God's people at Mount Sinai is the same kind of covenant made with Abraham and made with us in the death of Christ. It is based on mercy, it provides forgiveness, it has divine promises and warnings and commandments, and its basic requirement is single-minded devotion to God alone.

2. To Understand the Context of God's Jealousy
But there is another reason why I have stressed that the covenant of Exodus 34 is not a covenant of works but is based on mercy. I wanted to make sure that we saw the jealousy of God in its true context.
God is not jealous like an insecure employer who fears that his employees might get lured away by a better salary elsewhere. God's jealousy is not the reflex of weakness or fear.
Instead God is jealous like a powerful and merciful king who takes a peasant girl from a life of shame, forgives her, marries her, and gives her not the chores of a slave, but the privileges of a wife—a queen. His jealousy does not rise from fear or weakness but from a holy indignation at having his honor and power and mercy scorned by the faithlessness of a fickle spouse.

The ten commandments are not a job description for God's employees. They are the wedding vows that the peasant girl takes to forsake all others and to cleave to the king alone and to live in a way that brings no dishonor to his great name

The Threat and the Comfort of God's Jealousy

God is infinitely jealous for the honor of his name, and responds with terrible wrath against those whose hearts should belong to him but go after other things. For example, in Ezekiel 16:38–40 he says to faithless Israel, "I will judge you as women who break wedlock and shed blood are judged, and bring upon you the blood of wrath and jealousy. And I will give you into the hand of your lovers and they shall throw down your vaulted chamber . . . they shall strip you of your clothes and take your fair jewels, and leave you naked and bare. They shall bring up a host against you and cut you to pieces with swords."

I urge you to listen to this warning. The jealousy of God for your undivided love and devotion will always have the last say. Whatever lures your affections away from God with deceptive attraction will come back to strip you bare and cut you in pieces. It is a horrifying thing to use your God-given life to commit adultery against the Almighty.

But for those of you who have been truly united to Christ and who keep your vows to forsake all others and cleave only to him and live for his honor—for you the jealousy of God is a great comfort and a great hope. Since God is infinitely jealous for the honor of his name, anything and anybody who threatens the good of his faithful wife will be opposed with divine omnipotence.

(much of this writing comes from desiringgod.org, John Piper)