Final answer:
The federal government supported railroads and western settlement primarily through the passage of the Homestead Act and the Pacific Railway Act, along with providing important infrastructure like the transcontinental railroad and the construction of forts for protection and trade.
Step-by-step explanation:
The central way the federal government supported railroads and western settlement was through legislative actions, particularly by passing the Homestead Act and the Pacific Railway Act in 1862. These laws aimed to encourage the expansion of a "free soil" mentality across the West and to populate the region with settlers who opposed the expansion of slavery. Moreover, the government facilitated the creation of the transcontinental railroad by providing land grants and resources, which in turn connected the East Coast to the West Coast, dramatically increasing the value of government lands and spurring further development and settlement.
Along with this infrastructure, the government constructed numerous forts such as Fort Laramie and Fort Apache, which served to facilitate trade and provide protection, thus aiding settlers during their migration. Financially, land grants given to railroads by the government served as an indirect way to finance their construction, as these lands would become more valuable once connected by rail, leading to higher revenue from land sales.