This pattern is the literary equivalent of ないで, meaning 'without doing ~'. It is formed by taking the negative stem of a verb and adding ずに. For example, 行かずに means 'without going' and 食べずに means 'without eating'. For the verb する, the irregular negative stem is せ, so it becomes せずに. This pattern is used in both spoken and written Japanese, but it has a slightly formal or stiff nuance compared to ないで. It is common in formal speech, written instructions, and news reports. In casual conversation, people often prefer ないで, but using ずに is not wrong and can sound a bit dramatic or emphatic. The action that is not done is the one in the ずに clause, and the main action follows. For example, 朝ごはんを食べずに学校へ行った means 'I went to school without eating breakfast.' Note that ずに is generally used with verbs, not with nouns or adjectives.