5. Based on the phrase, "and you will honor it by not...pursuing YOUR affairs" (Isaiah 58:13), our Sages inferred that only involvement with one's personal affairs is forbidden on Shabbos, but involvement with "heavenly matters" ("cheftzei shomayim" - that is, mitzvos) is permitted. Accordingly, one may walk, on Shabbos, to the end, or just before the end of the "techum" (1), and wait there until nightfall in order to depart immediately after Shabbos on a journey associated with a mitzvah (2).
Similarly, one may show concern for matters of communal necessity; for example, one may go to speak to a ruler or to ruling body on behalf of the community, because communal affairs are equivalent to matters involving a mitzvah.
Similarly, one may talk to a teacher about agreeing to teach one's child Torah, or even a trade, because teaching a trade is also a mitzvah, since if a person does not have a trade with which he can earn his livelihood, he will end up stealing form others.
It is, however, forbidden to actually hire a teacher on Shabbos. Hiring an employee is an unqualified rabbinical prohibition ("Shvus Gemura"), and is not permitted even for matters associated with a mitzvah. Only those activities forbidden solely because of the verse quoted above: "honoring it...by refraining from pursuing your affairs and speaking [of mundane] things" are permitted when a mitzvah is involved.
One may announce the discovery of a lost article on Shabbos, since the return of a lost article is a mitzvah (3)
(1) The "techum" is the area within which one is permitted to walk on Shabbos, that is, 2000 "amos" (an "ama," or cubit is 48-60cm depending on which authority one follows) from one's place of residence; this concept will be explained in detail in chapter 95.
(2) If the journey was not for the sake of a mitzvah, it would be prohibited to wait at the end of the "techum" in order to shorten the journey after Shabbos. (See HY 90:3).
(3) Even if the object is mutkzeh and cannot be returned until after Shabbos.