That would depend on when and where.

In the 1843, Life of a Slave by Moses Grandy, he discusses the severe discipline that existed on the MacPherson plantation. Any day where a slave didn't complete their assigned task they were tied up and flogged, with pickle brine rubbed into the wounds, and then left to hang for a day. According to Grandy these floggings were severe enough that sometimes the entrails would be left visible. He relates multiple instances of slaves dying because they had become sick or otherwise feeble. Under such circumstances few slaves lived long enough to become elderly. So there was little need for elder care.

In Henry Clay Bruce's 1895 autobiography, Twenty-Nine Years a Slave, he highlights how the older slaves (especially the women) were given the charge of raising the slave children. In his accounts these "uncles" and "aunts" were particularly kindly treated and given especially good food in comparison to the regular field hands.

Clearly Grandy and Bruce had very different experiences.

Another common solution was to emancipate elderly slaves. By doing so masters discharged all obligation to care for the elderly slave. It became a sufficient problem that some states started legislating against the practice. In Mrs. Dred Scott: A Life on Slavery's Frontier, Lea VanderVelde highlights Missouri's laws. To free a slave over 45 years old, an owner had to post a $1,000 bond to cover the care of the freed person, if they weren't able to care for themselves. They also were liable for all unpaid taxes a former slave might accrue. At least three members of Dred Scott's family were freed in Missouri under these laws. That made just turning out a former slave a very expensive proposition. But such laws were far from universal, and indigent freed elderly slaves were quite common.