We eat ______ we may live.

A clause which is used to tell you about the purpose of the verb in the main clause is called adverb clause of purpose.
Adverb clauses of purpose are introduced by the subordinating conjunction that, so that, in order that, in order to and lest. Moreover, these expressions are usually followed by modal auxiliary verbs such as will, can, may, could or might.

• We play well so that we may win.
• Play carefully lest you should lose.
• He gripped his brother’s arm lest he be trampled by the mob.
• We left early in order that guests might reach home before us.
• I learn English in order to increase the standard of living.