遊びます　vs　遊べます - 'be' meaning "you can"?


I'm going through a Nintendo DS instruction manual and trying to do a bit of
translation. I'd like to ask about the difference between 遊びます and 遊べます。

[code]

    本ゲームは、　ボタン操作とタッチスクリーン操作の　どちらでも遊べます。
    
[/code]

I'm sure this is "you can play using both the buttons and the touch screen".
Is this a simple rule? I can change a verb -masu form from 'bi' to 'be' (for
example) to become "can"?

I am certain I've seen this elsewhere, the changing of 'i' to 'e'

[code]

    本が　読みます - The book is read
    先生が本を読めます - The teacher reads the book
    
[/code]

I'm a bit confused as to how/when this changes.




**I - CONSTRUCTION**

_1 - Godan verbs._

The potential form of the Godan verbs is constructed by changing the /u/ sound
to /e/ and adding る :

> 読む → 読める ; 帰る → 帰れる ; 買う → 買える

It's actually a contraction of 読まれる, 帰られる and 買われる which are now only used as
the passive form (I think).

_2 - Ichidan verbs._

The potential form of the Ichidan verbs is constructed like the passive form
by adding られる to the stem (verb at it's simple form minus る) :

> 食べる → 食べられる ; 忘れる → 忘れられる

The contracted forms 食べれる and 忘れれる, though widely used, is not considered
gramaticaly correct.

_3 - Irregular verbs._

Watchout for ある which have a different kind of potential form.

> 来る → 来られる ; する → 出来る{できる}

**II - Meaning**

_1 - Potential._

> 忘れられない人。 Someone unforgettable.
>
> 世界大戦が起これる。 A world war can happen.

_2 - Capacity_

> 犬は泳げる。 Dogs know how to swim.
>
> ついに私は読める! I'm finally able to read.

**III - Expressions with similar meaning.**

_1 - ことができる._

Verb (simple form) + ことができる has the same meaning as the potential form
(potential + capacity).

> 飛ぶことができると思います。 I believe I can fly.

_2 - 得る{える} and 得る{うる}._

Verb stem + える (literary) or うる (more literary) has the meaning of potential
but not capacity. It's also the only potential form of ある.

> そんな馬鹿な事はありえないよ。
>
> Something that stupid can't exist!
>
> ガラスは圧力で割れ得る。
>
> Glass can break under pressure.

**IV - Regarding the particles and your examples.**

The basic sentence is (neutral and polite) :

> 私は本を読む。 私は本を読みます。 I read a book.

The passive form isn't 本が読みます* but :

> 私に(は)本が読まれる。 私に(は)本が読まれます。 The book is read by me. (unnatural)

The gramatical subject of the potential verb in japanese isn't the one being
able to do but the thing that can be done :

> 私(に)は本が読める。私(に)は本が読めます。 I can read a book. (Litteraly) The book can be read
> by me.

Due to the influence of the occidental languages, and mainly english, you can
also find が (insisting on the object "the book") replaced by を (insisting on
the action "to read").

[The difference between が and を with the potential form of a
verb.](https://japanese.stackexchange.com/questions/609/the-difference-
between-%E3%81%8C-and-%E3%82%92-with-the-potential-form-of-a-verb)

