The Compromise of 1850 included several key provisions, and the statements can be evaluated as follows:
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All slavery was eliminated in the District of Columbia: This is inaccurate. The Compromise of 1850 did not eliminate slavery in the District of Columbia; rather, it abolished the slave trade there but allowed slavery to continue.
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Texas was allowed to annex New Mexico, extending slavery into that territory: This is misleading. Texas did not actually annex New Mexico as part of the Compromise, but the boundaries of Texas were reduced, and the issue of slavery in the New Mexico territory was to be decided by popular sovereignty.
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Popular sovereignty was established, allowing states to vote on whether or not to have slavery: This is accurate. The Compromise of 1850 allowed the territories of New Mexico and Utah to decide for themselves through popular sovereignty whether to allow slavery.
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California was allowed to enter the Union as a free state with no slavery: This is also accurate. One of the key provisions of the Compromise was the admission of California as a free state.
In summary, the most accurate statements are that popular sovereignty was established and California was allowed to enter the Union as a free state.