I am not completely sure about what you mean when you say “enforcing the economic system.” Zinn does not use these words in Chapter 2 and the idea of “enforcing” an economic system is not a common one. I will assume that you are asking why slavery was necessary for maintaining the economic system that was in place in the Southern colonies of what is now the US.
Slavery was necessary for the economic system because the economic system was based on a few planters owning large plantations. This was not like the North where the economic system was based largely on small farmers who owned their own land and worked for themselves. Instead, the South’s system was based on large plantations that needed a great deal of labor.
In the colonies at that time, there was plenty of “free” or relatively cheap land. This meant that it was practically impossible to get people to voluntarily work on the plantations for someone else. Free people preferred to work for themselves. The planters, therefore, needed an unfree source of labor. They first tried indentured servants but later came to find that slavery worked better for them.
Slavery was necessary for the economic system, then, because it was not possible to get free people to agree to do that kind of work for other people rather than doing it as their own bosses.