Exodus 21:3

If he came in alone, he shall go out alone: but if (he be) the husband of a wife, a daughter of Israel, his wife shall go out with him.

THEN HIS WIFE SHALL GO OUT WITH HIM. Rashi commented: “But who brought her in, that Scripture need say that she shall go out with him? But [by saying this], Scripture tells us that he who acquires a Hebrew servant is obliged to provide for the food of his wife and children.” This is a Midrash of the Sages. Now [even though the children are not mentioned in this verse, but only his wife,] the Rabbis have included the children together with the wife in this duty of the master, on the basis of what is written further on, Then he shall go out from thee, he and his children with him.
I am not clear on this law as to whether the earnings of the woman and children belong to the master during the time he is supporting them. It appears to me that the master takes the place of the husband [in this respect]. For the Torah had compassion on the wife and children, whose lives are hanging in suspense, and who expect [to be supported from] the husband’s earnings, since now that he is sold as a servant, they are in danger of being lost in their misery. Therefore the Torah commanded the master who is now entitled to the servant’s labor, to act towards them as he [the servant] would. If so, the master only has to assume the responsibilities of the husband, [and no more]; thus he is entitled to their labor as is the husband, and in return he must feed and support them. This is the meaning of the expression, then his wife shall go out with him, since the servant’s wife was together with him as a handmaid to his master, for the labor of both of them belongs to him, in return for which he is obliged to give them food. Thus the only difference between husband and wife is that the wife has a right to go away as she pleases, [and is not bound to work for her husband’s master if she does not want to be supported by him, whilst the husband, who is the servant, is bound to the master]. Similarly, the master’s obligation to support the children is limited to the time that the father is responsible for them, namely when they are minors, or as long as is customary to feed them, as Rashi explained in Tractate Kiddushin. All this is out of G-d’s compassion for them [the wife and children], and for the servant as well so that he should not die in his anguish, in the knowledge that whilst he is toiling in a strange house, his children and wife are neglected. Now even though he is not obliged by law of the Torah to support them, as has been explained in the Talmud, Tractate …

אם בגפו יבא בגפו יצא. The Torah first mentions the subject in general terms, then proceeding to details. For instance, if the servant was unmarried at the time the court sold him, or even if his master has given him a wife while he was in his employ, he leaves without his wife seeing that the woman given to him as a wife remains in the house of the master, her employer. Having said this, the Torah continues to explain how this works אם בעל אשה הוא, if he was already married, his wife (who will also be a charge on his master) will leave with him when he is released from this involuntary contract after 6 years. However, if his master had provided him with a wife during the period of his forced employ, the wife in question remains with her master. The “wife” the master had assigned to his Jewish servant was a gentile servant maid whose body belongs to her master.

ויצאה אשתו עמו, “his wife will leave that master’s employment with him (at the same time). Rashi asks correctly, granted that we have been told that his wife leaves with him, but where did we hear that this applies to his children? As an answer he cites the Talmud, tractate Kiddushin folio 22, that we learn from this verse that the master of the servant of which the Torah speaks here is obligated to look after the servant’s wife’s and family’s needs, if he had such when he was sold to him. Whereas it is clear that the master has to look after such a servants’ wife, it does not appear clear from the text that he is also obligated to look after the children of such a servant. We need to find a nuance in the text that would support this contention. Our author therefore suggests that this can be found in Leviticus 25,41, where the subject is someone who was forced to sell his labour as he had fallen on hard times, debts, etc; the Torah there states that the “master” must release both him and his family in the Jubilee year, showing that there is an unnecessary word. The word שכיר appears both in Leviticus and that same word appears again in Deut. 15,18. The scholars that do not accept this g’zeyrah shaveh as something that is traditional, but simply consider (Talmud Kiddushin there) the phrasing of the verse as sufficient.