わけ

わけ is a noun that means “conclusion”, “reason”, or even “meaning”. 

• 不合格したわけを聞く
  Ask the reason he failed.
  
  
• 言いわけは要らない
  I don’t need a (verbal) excuse. 
  
  
• わけがあるんだ。。
  There’s a reason.
  
  

わけ is very useful and common; and is used in a lot of different expressions such as:

• わけだ – Conclusion, So that’s why
• わけではない・わけじゃない – That’s doesn’t mean, It’s not
• わけがない – There’s no way
• わけにはいかない – It can’t be allowed
• わけがわからない – Completely un-understandable

わけだ

Reason or Conclusion
Plain form verb/い Adjective + わけだ
Noun/な Adjectives + なわけだ

The most basic usage of わけ is with だ。It expresses a reason or conclusion.
Its similar to だから, but explains something you were wondering about.
It’s sort of like “that’s why” instead of “because“.

わけ can also be used to explain things to other people.

わけ does not require から, since it fills that role.

Examples

• 結婚したことになったから、嬉しいわけだ
  He’s going to get married, so no wonder he’s happy. 
  
  
• 彼は今年は最後らしい。なるほど寂しいわけだ
  I heard this is the last year. Oh, that’s why everyone’s sad.
  
  
• 明日は試験だよ。ああ、道理で誰もいないわけだ.
  There’s a test tomorrow. Oh no wonder no one’s here.
  
  
  

わけではない

Its not that/It doesn’t mean that
Plain form verb/い Adjective + わけだ
Noun/な Adjectives + なわけだ

Despite わけではない being the negative of わけだ, it doesn’t (entirely) mean the opposite.
It has 2 usages, to express “It doesn’t mean” and “Its not that”.

The first usage of わけではない  is to deny an inference that might have been drawn from your statement.

The second use is to partially deny a statement or indirectly deny a statement.
It sounds confusing, but its basically denial with some wiggle room for possibility.
So if I say:
納豆が食べられないわけではないが。。- Its not that I can’t eat natto.
It means that I don’t really like or want to eat natto, but if a situation came up I might be able to.

Although わけではない is more common, わけじゃない may also be used.

わけではない is frequently used with という, particularly with nouns. 

Examples

Its not that…

• 魚が完全に嫌いなわけではない
  Its not that I completely hate fish. (I don’t hate fish a lot).
  
  
• 何もしないわけではないが。。
  Its not that I don’t do anything (I just don’t do much/I can’t do more).
  
  
• 彼は頭がよくないわけではないが。。
  It’s not that he isn’t smart but..(He does stupid stuff).
  
  

It doesn’t mean that..

• 彼といつも話しているが、友達だというわけじゃない.
  I talk to him a lot, but it doesn’t mean we’re friends.
  
  
• ウエブサイトを作っているが、Javaがかけるわけではない
  I’m making a website, but it doesn’t mean I can write Java.
  
  
• そろそろ帰る。もう帰る？
  I’m going to go home now. You’re going home already?
  
  帰りたいというわけではないが、家事があるんだ
  Its not like I want to go home, but I have chores to do.
  
  

わけがない

There’s no way
Plain form verb/い Adjective + わけだ
Noun/な Adjectives + なわけだ

わけがない means “There’s no way”.
Literally it probably means something like “There’s no reason that could cause this to happen”
が may be abbreviated in informal speech.

Another way to express this is はずがない

Examples

• 殴るわけがない。
  There’s no way I’d punch (something).
  
  
• 学校をさぼるわけがない
  There’s no way I’d skip school. 
  
  
• あれは犬なわけがない
  There’s no way that’s a dog.
  
  
• 彼が優しいわけがない
  There’s no way he’s (actually) nice. 
  
  
• うんこがかわいいわけがないだろう.
  There’s no way poop is cute, right?

わけにはいかない

It can’t be allowed
Plain form verb/い Adjective + わけだ
Noun/な Adjectives + なわけだ

This one’s a little tricky.
Its わけ (reason)+にはいかない (Won’t go/pass).
It’s used to describe a situation where something won’t be allowed by something other than the speaker, usually society.
This phrase may indicate that it is contrary to the speakers wishes. “I want to, but I can’t”
It’s kind of like てはならない

Examples

• 簡単に許すわけにはいかない
  I can’t just forgive you easily.
  
  
• 金をとらないわけにはいかない
  I can’t just not take money. (Not fair, rules…)
  
  
• 何もせず通すわけにはいかない
   I can’t just let you pass without doing anything.
  
  
• 何も飲まずに帰るわけにはいかない
  You can’t go home without drinking anything. (looks bad)
  
  

わけがわからない

Un-understandable
訳が分からない

わけがわからない is a phrase that means “un-understandable”. (Don’t’ understand the meaning at all).
It comes from the “meaning” meaning of わけ。It is best thought of as a phrase since it functions mostly like an adjective (Verb modification). 

Examples

• 訳の分からないことを言うな
  Don’t say un-understandable things.* 　This is a が replacing の (will add a link when the page is made)
• 言葉の訳が分からない
   I don’t know the meaning of what he’s saying. 
  
  
• 手紙をもらったけど、訳が分からない内容だ
  I got a letter, but I have no idea what it means.
  
  
• なまりが強くて何を言っているのかわけがわからない 
  He’s got a strong accent, and I don’t understand what he’s saying at all.