ずに expresses doing one action without doing another action. It is formed from a verb’s 未然形, the form used before certain negative-related endings, followed by ずに. The ず itself can express the meaning “without doing,” as in 食べず、学校へ行った, while に gives the common connective form ずに. This construction is more formal and often more common in written or formal contexts than ないで, although it is also used in careful speech. The action before ずに normally has the same subject as the main action.