Why did African slavery immediately become the dominant labor system in South Carolina, while it was slow to dominate in the other Southern colonies
Keep in mind this is a pretty specific question, and your instructor may be looking for a specific response from class notes or your textbook. In my opinion, there were two key reasons slavery so quickly became the dominant aspect of the South Carolinian economy:
1) Charleston, South Carolina was the largest entry port in the colonies for slaves from Africa and the West Indies, therefore slaves were somewhat cheaper and more readily available in that particular colony.
2) South Carolina farmed different cash crops than North Carolina and Virginia, specifically indigo and rice. Slaves were imported to South Carolina from tribes that already farmed rice in Africa, giving them a skilled labor force that needed little training. This simply accelerated the economy's dependence on slave labor in that location.
In other colonies north of there, plantation owners had experimented with indentured servants as a labor force for a time as the tobacco economy developed there. By the 1670s when the Carolinas were settled, large scale indentured servitude had been abandoned in favor of the more economical slavery.
South Carolina also engaged in the export of Indians as slaves for quite some time, and tens of thousands were exported for other labor markets in the late 1600s and early 1700s, so in addition to slaves on plantations, the slave trade itself was central to its early economy.