According to Ramban, “vachai bahem” refers to the mishpatim, the “mitzvot ben adam l’chavero”, the social mitzvot between a person and their fellow person. These laws were given so that man should live. Following these laws ensures peace among men so that no one should cause harm to his fellowman or kill him.

A society that does not observe these social mitzvot will not be able to function and people will literally not be able to live in that type of a society.

We also learn from the words “vachai bahem” that we perform the mitzvot in order to live. If there is an issue of pikuach nefesh (danger to life) then one can break Shabbat laws as well as most of the commandments in order to make sure that you can continue to live your life.

We learn in the Gemara in Shabbat 151b: Rabban Shimon ben Gamliel says: To save a live day old baby we may profane Shabbat…This is because the Torah said in Shmot 31:16, “B’nai Yisrael shall observe Shabbat in order to perform Shabbat throughout the generations”. Profane one Shabbat on his account so that (the baby) will live and grow to observe many Shabbatot .

During the Holocaust, the issue of “vechai bahem” unfortunately came up a lot. A case that really stands out is the prayer for eating Hametz on Pesach.

By 1944 there was no question that Jews must eat hametz to stay alive. Rabbi Avraham Levisson from Holland dealt with this issue in Bergen-Belsen. In 1944 Rabbi Levisson and his father, along with the Chief Rabbi of Rotterdam Rav A.B. Davids and a number of Jews gathered in a barrack to quietly hold a seder. The Dutch rabbis, seeing the Jews could not survive without eating bread on Pesach, composed a prayer to recite upon eating hametz.

To be said with utmost concentration before eating hametz on Pessah:

Master of the Universe,
It is manifest and known to You we want to fulfill Your commandment that we celebrate the holiday of Passover by eating matza and abstaining from hametz.
But to our great sorrow our servitude prevents us from fulfilling these precepts.
We are not masters of our own fate and our lives are in danger.
Therefore we are ready and willing to keep the mitzva: “So that you shall live by them” and not die because of keeping the mitzvot. Therefore we are commanded to do what we must in order to remain alive; thus by eating hametz we will be keeping Your other precept, “Be ever so careful with your life.” (Devarim 4:9)
We pray that You keep us alive and sustain us so that we merit to survive to fulfill Your commandments wholeheartedly in the future. Amen.

We learn from here that we must do what we can to live by the commandments and that at times it is even a mitzvah to transgress them.