You Are My Sister

(בראשית יב, יג) אִמְרִ י־נָא אֲחֹתִי אָתְ לְמַעַן יִיטַב־לִי בַעֲבוּרֵךְ וְחָיְתָה נַפְשִי בִגְלָלֵךְ:

The Medrash states that this Pasuk teaches that one does shecht an animal for a sick person on Shabbos. How do we understand this Medrash?

1. We find that Sarah was allowed to be exposed to an aveira in order to save Avrohom. Even though this concept is derived from other places, since these two great people of Israel did such a public action regarded in the Torah, this teaches that such an action can be taught publicly, plus, in event of need we do not look for a non-Jew, but a Jew should take the action of saving a life.
2. Avrohom was in danger of his life if he stated that Sarah was his wife. The Egyptians thought it was less of a problem to kill the husband than to transgress with a married woman (even though both are forbidden to non-Jews). Similarly, one does a one-time stringent transgression of schechting on Shabbos (punishment of death by stoning) rather that providing a non-kosher piece of meat to be eaten which is not as stringent (lav) but involves a repeated transgression with each bit.
3. Avrohom could have prevented Sarah from possible transgression by divorcing her. However, divorce was not an option since they had a status of Bnai Noach. Therefore, this incident shows that one can perform a transgression (lying) in order for another to gain. Similarly one can perform a transgression on Shabbos in order to feed another (sick person that requires the meat).
4. The Medrash is offering that if one schechts for a sick person on Shabbos then a healthy person may also eat from that meat. Similarly the main actions of Sarah and Avrohom was to save a life, nevertheless they also accepted and benefitted from the gifts that were given to them
5. By means of a remez (allusion) the seven letters of words אחותי את reference the reasons of the seven Tana’im in Yoma 85 that offer opinions for the source of the Halacha h that saving the life of a Jew takes precedence over the laws of Shabbos.

TannaSourceLetterR. YishmaelDerived from the Pasuk "Im ba'Machteres Yimatzei ha'Ganav ... " (If one is permitted to kill [with all its implications] in place of Safek Piku'ach Nefesh, then one should certainly be permitted to break the Shabbos for that). א first letter of the pasukאR. ElazarDerived from the Mitzvah of Milah - because if one may break Shabbos to rectify one limb then one may certainly do so to rectify the entire body. ח for the eighth day of the MilahחShmuelIt is best learned from the Pasuk "ve'Chai Bahem ("Live with the Mitzvos", 'and do not die with them'). ו first letter of the pauskוR. Yonasan b. YosefLearns it from "Kodesh Hi Lachem" (Shabbos is handed over to us [to observe], and not us to it [to die]). ת part of the name יונתןתR. AkivaInferred it from the Pasuk "veki yazud ish al reaihu ... me'Im Mizbechi Tikachenu Lamus", (If we will even interrupt the Avodah (which over-rides Shabbos) to call a witness (whose testimony may not be of any value), to save someone's life then we should certainly break the Shabbos in order to do so. י from the word יזדיR. Yosi b. R. YehudahLearns it from the excluding word "Ach" (to teach us that there are cases where one is permitted to break the Shabbos). א from אךאR. Shimon b. MenasyahDerives it from "ve'Shameru B'nei Yisrael Es ha'Shabbos" (Torah says one should break one Shabbos to enable the person who will otherwise die, to keep many Shabbasos). ת from the word תורהת

Further, the Gematria of the phrase את למען ייטב לי בעבורך counted with the numbers of words equals the Gematria of the phrase מבאו נלמד ששוחטים לחולה בשבת