Basically Georgia passed the Worcester v. Georgia law that allowed them control over the Native's land so they lived on the land east of the Mississippi. The government wanted that land because they thought that the Native Americans were getting in the way of westward expansion of the United Sates. This got to the Supreme Court where they ruled their decision that Georgia didn't have a right to control territory outside it's state line. Then the government passed the Indian Removal Act which allowed the government to trade the Natives land west of the Mississippi for their land in the east. This forced the Natives to be removed from their land and migrate to the other side of the Mississippi.
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Yes, your summary is largely accurate. The case you mentioned, Worcester v. Georgia, did occur in 1832. In this case, the Supreme Court ruled that Georgia's laws did not have authority over Cherokee lands and recognized Native American tribes as separate nations. However, President Andrew Jackson famously denounced the decision and refused to enforce it. Subsequently, the Indian Removal Act was passed in 1830, which allowed the federal government to negotiate treaties with Native American tribes to exchange their lands in the East for territory in the West. This led to the forced removal of thousands of Native Americans, primarily from the southeastern United States, in what is known as the Trail of Tears.