Here we'll talk about a Kena'ani (Canaanite) slave who was owned equally by two partners, 50-50, and then one of them freed him. So he's now half a slave and half a free man, a ben chorin (free man). The Mishnah says: Someone who's half a slave and half a free man works one day for his master and then one day for himself. The money that he gets for his own day of work, he keeps, and the money he generates when he's working that one day for his master, he gives to his master. That's the opinion of Beit Hillel.
With this setup that you just made, you handled things for his master. You fixed his master's problem of how to get productivity, how to get money out of the slave. But what about him? You didn't solve anything for the slave himself. He's stuck in a very sticky, tricky situation. He can't marry a regular Kena'ani slave woman because half of him is free, and therefore half of him has the laws of a regular Jew, and a regular Jew is not allowed to marry a Shifcha Kena'anit, a non-Jewish slave woman. He's not allowed to marry a regular free Jewish, a non-Jewish slave woman as well, because half of him is a slave, and a regular Jewish woman is not allowed to marry a slave. Should he give up on marriage altogether and not have any children? The whole world was created only for the purpose that people have children and that we would populate the world. As the Pasuk (verse) in Yeshayahu says, “Hashem didn't create the world to be deserted. He created it in order for it to be inhabited.” So Hashem's plan: He created it in order for people to have kids. Therefore, we have to find a way to let the half-slave marry and have children.
So Beit Shammai gives a solution. For the benefit of society, we want to make the slave have the option to have children and get married, so we force his master to free him, to let him go, to make him a full free man, so that he can marry now a free Jewish woman. And to compensate the master who owns fifty percent of his wealth, we make the slave write an IOU that obligates him to pay his master for half of his value. When the freed slave eventually makes that money, he would then have to pay the master who let him go for free. When that happened, Beit Hillel retracted their original ruling and they held like Beit Shammai that a person to find the child has a right to marry. So then He asked to have the slave. When that happened, Beit Hillel retracted their original ruling and they held like a Mother depois de la Shem Ik, that a person has a right to marry and to have children with the children, to free that slave and let him go fully free and allow him to get married.