済む may conjugate: 済んだ (past), 済めば (conditional), etc.
The grammar pattern ずに済む expresses that something was resolved or turned out okay without having to do a particular action. It conveys a sense of relief or fortune that an undesirable or burdensome action was avoided. It translates naturally as "to get by without doing ~" or "to manage without having to ~."
Attach ずにすむ済む to the negative stem of a verb (the same stem used for ない). For most verbs, replace ない with ず:
The irregular verb する becomes せずにすむ済む (not ×しずに済む).
This pattern typically implies that the speaker is glad or relieved that the action was unnecessary. The avoided action is usually something unpleasant, costly, or troublesome.
The pattern frequently appears in the past tense (ずにすんだ済んだ / ずにすみました済みました), reporting that something has already been avoided. However, it can also appear in non-past forms to express hope or possibility: