I cannot understand the meaning of jiayi. The dictionary simply tells how to deal with something that is mentioned before. Please hive me as many examples of sentences with jiayi that are short sentences with easy characters.
Also please tell me, if these sentences are correct.

这个农村加以进展。
我不知道怎么样加以回答

谢谢

Surprisingly enough, someone left a pretty good explanation on Yahoo! Answers:
http://answers.yahoo.com/question/index?qid=20140205085000AA0ZSe3

(from the thread, edited slightly for style)
"加以" is an extremely formal phrase/structure that would only be used in newspapers or formal speeches. It can be compared with the English phrase "enter into.[verb+ing]..of", "engage in the..[verb+ing] of" or "endeavor to [verb]".

"加以" is a very similar, more formal version of "进行" (enter into [an action]; engage in.