Linked below are three different sites with separate ways to construct NOUNS and わけ(だ/ではない).
わけだ Form 1 from 'japanesetest4you'
わけではない Form 1 from 'japanesetest4you'
Forms 3 from 'Learn Japanese (because of dreams)'
This has been killing me for the past few days. Please someone clarify how the heck you form わけ(だ/ではない) sentences with nouns.
わけ is an interesting little word with lots of different usages.
So it looks like the differences in the reference websites you posted are N + という + わけだ and N + である + わけだ.
There is a difference. N+という＋わけた has a nuance of "oh so that means..." or "oh from what you say I understand that..." For example:
Aさん：僕みたいな人はやっぱり夏に仕事がないからね、
People like me don't have work in the summer.
Bさん：そうか、Aさんは学校の先生というわけですね！
B: Oh, so I take it you're a school teacher?
Or
Aさん：俺の息子はアメリカで勉強｛べんきょう｝しているから、あまり帰ってこれないんですよね。
My son is studying in America, so he can't come home very often.
Bさん：息子さんは留学生｛りゅうがくせい｝というわけですか？
B: So I guess/I take it your son is doing a study abroad?
On the other hand, であるわけ implies that the speaker already knows (but perhaps needed to be reminded) of whatever is being said. There were no example sentences using this construction on the website you referenced, and only a fragment in my grammar book, which follows:
Aさん：それは、彼の責任{せきにん}であるわけだ。
As you know, that was his responsibility.
わけ is frequently used with verbs with similar meaning.
石田先生が亡くなったとき、我々は素晴｛すば｝らしい指導者｛しどうしゃ｝を失｛うしな｝ったわけですが、これからもがんばらないといけません。
As you all know (implied) When Professor Ishida died, we all lost a great leader, but we must keep moving forward.
As you can see, using わけ in this kind of context implies an "as you know" kind of feeling.
Hope that helps!
