であるために is same meaning as なので and だから. であるために is a literary style of them. and であるように can use many situations. perhaps you saw であるように in something like this; 父が医者であるように、私も医者である。 it means I'm a doctor as my father. in this case, であるように can say in this words; "と同じように". 　and it's used in these other cases.. 1. seem, like, as if 　　彼はまるで先生であるように話す。( 彼はまるで先生であるかのように話す is more natural ) -----he talks as if he were a teacher. 　　彼は病気であるように見える。　----- he seems to be rich. 2. such that......in this case, it same as であるような or ような anyway, we don't use both two (especially latter) phrases in our usual conversations. it might be used in any books or newspapers.

である = to be ...and is the newspaper version of だ。 ために　= for the sake of; for the purpose of ように has quite a few uses. You can find a lot of examples in a dictionary.