Final answer:
Medicare is the federal government program that subsidizes health care costs for the elderly, covering those over age 65.
Step-by-step explanation:
The federal government program that subsidizes health care costs for the elderly is called Medicare. Medicare provides health insurance to all individuals over the age of 65, and it was established as a result of the Social Security Act of 1965. In addition to Medicare, the U.S. government also funds other health-care programs such as Medicaid for those with low incomes, the Veterans Health Administration for military veterans, and the Children's Health Insurance Program for children in families with relatively low incomes.