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I am currently learning Russian using Glossika and it gives the following as the translation of the sentence "I'm going out with some friends tonight. Why don't you come too?":
Мы идём гуля́ть с друзья́ми сего́дня ве́чером. Почему́ бы тебе́ не пойти́ с на́ми?
I cannot understand two things about this sentence: (1) Why is the subject of the sentence in the dative case? and (2) Why is the verb in the infinitive rather than the past tense which I was taught to use with бы constructions?
Почему бы тебе не пойти с нами?
For practical reasons, you can look at it this way: the sentence is elliptical, that is, some words are missing, but everybody can reconstruct them and understand the sentence. Look at it like this, the missing words are in square brackets:
Почему бы тебе не пойти с нами?
Why would — Почему бы
[it be good] for you — тебе
not to go with us? — не пойти с нами?
That is a why? question, the possible answers can be 'because I'm busy', 'because I don't like you', or 'great, I'm going!', etc., but still you will always answer the question 'Why isn't it good for you to go or not to go with us?'
"Почему бы [кому-либо] не [сделать] что-либо?" is a template, where [кому-либо] is a noun or a pronoun in dative case, and [сделать] is a verb in perfect infinitive. It means "Why don't [someone] [do something]?". It's a polite form of speaking.
It's possible to use "Ты бы пошёл с нами?" instead. No dative and past tense, I believe this is the one you've been looking for. Though this is less polite. Friends could use use "Пойдешь с нами?" and even shorter "А ты?".
Почему бы тебе не пойти с нами?
It's a special form of a polite suggestion. We don't use nominative case and past form of the verb after why in such sentences. It's very close in form to Would you like to go? where an infinitive is used. You can make a suggestion without why with the verb in the past (Ты бы пошел с нами?Would you go with us?) or in the present (Ты пойдешь с нами?Пойдешь с нами?Will you join us?) But to say Почему бы ты не пошел is incorrect. Perhaps the sentence Why not joining us? corresponds better.
Why is the verb in the infinitive rather than the past tense which I was taught to use with бы constructions?
Well, there are quite a few different constructions with "бы". Do not expect a politeness to be a simple thing.
А ты бы пошёл с нами --> this is also to motivate / slightly push rather than simply ask; need right stress not to confuse with a question (i.e. do not stress the verb here);
А ты хотел бы пойти с нами? --> again could be confused with a plain question; put the stress over "хотел" to show it's an invitation too;
Пойти бы с вами (or Хорошо бы пойти с вами) --> infinitive + "бы" shows an inclination to do something; missing any "who" points to the speaker himself, i.e. it's the same thing as I'd like;
Тебе пойти бы с нами --> here I say that such an inclination to go should belong to you; so it means you should better;
Почему бы не пойти? --> this is sort of "is there any reason not to go?", which in fact stands for soft variant of "let's go": why don't we go;
Почему бы тебе не пойти с нами? --> now it's like a previous one, but points to you, so it's a polite form for asking to go with us: why don't you.
