First; preliminary — the initial stage or round (e.g., first exam, preliminary selection).
一次試験を通過して、次は面接です。
I passed the first (preliminary) exam; next is the interview.
大会は一次リーグと決勝トーナメントに分かれている。
The tournament is divided into a preliminary (first) league and a final knockout stage.
一次対応は済ませましたが、詳しい調査が必要です。
We've handled the initial response, but a detailed investigation is needed.
Primary/original — referring to original or primary sources, industries, or sources (as opposed to secondary).
研究では一次資料を直接確認することが重要だ。
In research it's important to check primary sources directly.
この町は漁業などの一次産業が盛んだ。
This town's primary industries like fishing are thriving.
一次ソースから情報を得ると、信頼性が高い。
Obtaining information from a primary source increases reliability.
Linear / first-order (mathematics) — indicating linearity, as in linear functions or first-order equations.
y = 2x + 1 は一次関数の例だ。
y = 2x + 1 is an example of a linear (first-order) function.
一次方程式は未知数が1次の項しか含まない方程式だ。
A first-order equation is one that contains only first-degree terms in the unknowns.