For full context:
<http://www3.nhk.or.jp/news/easy/k10011221251000/k10011221251000.html>

The sentence in question:

> ＪＲ東日本は、 **新幹線で** 無線のＬＡＮのサービスを始めて、 **無料で** インターネットが利用 **できるようにすることにしました**

My problems are the parts in bold. First, my attempt at translation:

> JR-East-Japan has decided that, establishing wireless LAN service in
> shinkansen, they make it so that you can use the internet for free.

So, first there 新幹線で and 無料で. In the way I translated these parts, it feels
like I'm wrong. Whenever で is a particle that marks a location, it usually was
in context of an action happening in this place. In this case, "establishing
WLAN" can be interpreted as an action happening there, however, the cases I
used to encounter with で expressing the semantics I mentioned above were more
of this sort: キャシーがパーティーでなに何をき着ていたかおぼ覚えていますか。 And I think that this is
something else, although I could be wrong as well of course.

Second, 無料で. It's a similar problem here. I interpreted 無料で as 無料 +
conjunctive form of だ. In this case, the literal translation would be:

> ...being for free, they can use the internet.

However, this already expresses something else than what I translated above, I
think. I can't think of another way to translate this meaningfully, and it
fits into the context very well.

Third, ように in インターネットが利用できる **ように** することにしました. I translated it in a final
sense like in "They do X, **so** that they achieve Y." However, combined with
することにしました it sounds rather overcomplicated to me: "They decided that they do,
so that X/in order to X...". That's why I feel the need to ask for
confirmation...^^ I also know a different meaning for verb+ように, which would be
something along these lines: 学生に図書館で物を食べないように注意しました。 Here, my textbook says
that "this structure is used to quote the contents of an order or request.".
Since sometimes the definitions turned out be defined too narrow, I wanted to
bring this up in context of this question.
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grammar
What do で and ように express here?
> 新幹線で

This is just representing the location of 始まる and explains where the action
will occur. Your confusion may have been caused by the often ill-explained
difference between に and で when marking location. JSL schools tend to teach
that で marks the location of action and に marks the location of existence but
it seems that a better interpretation is that に marks the location when it is
critical to the verb and で when it is not. See [に and で
revisited](https://japanese.stackexchange.com/questions/2197/%e3%81%ab-
and-%e3%81%a7-revisited).

> 無料で

One of the uses of で is to indicate cost. I'm not certain and I hope someone
else will address the question of whether a native Japanese would consider
this で a case particle or the て形 of だ／です which can also be interpreted as an
adverb and in this case describe the extent of the verb.

> 利用できるようにすることにしました

I'm going to try explaining this in pieces without specifying any nouns.

Aにしました  
Someone decided A (as you've identified)

Bすること  
Doing B

利用できるように  
So that someone is able to use something (indicating purpose as you guessed)

So all in all, someone decided to do something so that someone is able to use
something

> ＪＲ東日本は、新幹線で無線のＬＡＮのサービスを始めて、無料でインターネットが利用できるようにすることにしました
>
> JR East-Japan is beginning wireless LAN service on the shinkansen and
> decided to make it so that passengers are able to use the internet for free.
> (This could obviously be worded better in English but I'm trying to make a
> point).

As usual, I welcome comments about any errors I may have made. I am merely a
student after all.
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answer