Final answer:
Around the time of the Judiciary Act of 1789, which established the federal judiciary, the first U.S. Congress passed the 1789 Tariff Act, and the process for establishing the Bill of Rights began.
Step-by-step explanation:
The major U.S. event that took place around the same time as the establishment of the federal judiciary system under the Judiciary Act of 1789 was the first operation of the U.S. Congress under the new Constitution. This Congress passed its first major piece of legislation, the 1789 Tariff Act, designed to raise revenue and strengthen the national government's economic position. Furthermore, another significant development was the addition of the Bill of Rights to the Constitution, a process which began shortly thereafter and led to the ratification of the first ten amendments. These events were pivotal in shaping the United States government and legal system.