Get away without: 〜ないで済む・〜ずに済む

済む means “to be settled, to be over”. 〜ないで済む means you've managed to avoid something unpleasant: “to get away without having to…, to end up not needing to…”. Noun + で済む says things were limited to something not very serious.

• 友達が車で送ってくれたので、タクシーを使わないで済んだ。ともだちがくるまでおくってくれたので、たくしーをつかわないですんだ。A friend drove me home, so I didn't need a taxi.
• 早めに予約したので、並ばずに済みました。はやめによやくしたので、ならばずにすみました。I'd booked early, so I didn't have to queue.
• 事故に遭ったが、軽い怪我で済んだ。じこにあったが、かるいけがですんだ。I had an accident, but I got away with a minor injury.

済む (すむ) is “to be settled, to work out”. So 〜ないで済む reads “it got settled without…”: you feared you'd have to do something unpleasant, and in the end you escaped it — English “get away without”, “not have to after all”. The sentence almost always carries relief.

It's often in the past (済んだ, 済みました), after a cause explaining why you were lucky: 〜ので、〜ないで済んだ.

• 近くにコンビニがあったので、遠くまで行かないで済んだ。ちかくにこんびにがあったので、とおくまでいかないですんだ。There was a convenience store nearby, so I didn't have to go far.

Three variants for the verb: 〜ないで済む (everyday), 〜なくて済む (same meaning, very common in speech) and 〜ずに済む (more written, more formal). For ずに, take the ない-base; the only exception: する → せずに (not “しずに”).

With a noun, 〜で済む means “to get away with only…”: the consequence was limited to that. 〜だけで済む stresses it further.

• 地図アプリのおかげで、道に迷わずに済んだ。ちずあぷりのおかげで、みちにまよわずにすんだ。Thanks to the map app, I didn't get lost.
• 修理は千円だけで済みました。しゅうりはせんえんだけですみました。The repair only cost a thousand yen.

〜なくてもいい (N5) grants permission, often for the future: “you don't have to”. 〜ないで済む notes that a constraint was avoided thanks to circumstances, with relief. The first is about rules, the second about luck.

In the present, 〜ないで済む often takes the form of a wish or a calculation: 〜ば、〜ないで済む, “if…, we won't have to…”.

• 明日は来なくてもいいですよ。あしたはこなくてもいいですよ。You don't have to come tomorrow. (permission)
• 今やっておけば、明日は来ないで済む。いまやっておけば、あしたはこないですむ。If I do it now, I won't have to come in tomorrow.