Реакция была быстрой.

Because реакция is a feminine singular noun.

In the past tense, Russian verbs agree with the subject in gender and number:

• был = masculine singular
• была = feminine singular
• было = neuter singular
• были = plural

So:

• реакция была = correct
• ответ был = masculine example
• письмо было = neuter example

Since реакция is feminine, была is the right form.

Here быстрой is the instrumental singular feminine form of быстрый.

After past-tense быть (was / were), Russian often uses the predicate adjective in the instrumental case, especially in more neutral or standard phrasing:

• Реакция была быстрой.

You may also hear Реакция была быстрая, especially in more conversational speech, but быстрой is very natural and standard here.

So the key idea is:

• быстрая = nominative feminine singular
• быстрой = instrumental feminine singular

Because the adjective is being used as part of the predicate after была, the instrumental form is very common.

The dictionary form is быстрый.

That is the masculine singular nominative form, which is how adjectives are usually listed in dictionaries.

Some related forms:

• быстрый = masculine
• быстрая = feminine
• быстрое = neuter
• быстрые = plural
• быстрой = feminine instrumental singular

So быстрой is just one grammatical form of the adjective быстрый.

Реакция is in the nominative singular.

It is the subject of the sentence, so nominative is what you expect here.

You can tell from the ending -я, which fits a feminine noun in nominative singular:

• реакция = nominative singular
• реакции = could be genitive singular or nominative plural, depending on context
• реакцию = accusative singular
• реакцией = instrumental singular

In this sentence, реакция is the thing that was quick, so it stays nominative.

Yes. The sentence is in the past tense because of была.

Russian past tense is formed from the verb stem plus endings that show gender/number:

• был = was (masculine)
• была = was (feminine)
• было = was (neuter)
• были = were

So Реакция была быстрой is clearly a past-time statement.

In the present tense, Russian usually omits the verb быть.

So instead of saying a literal equivalent of The reaction is quick, Russian normally says:

• Реакция быстрая.

Not:

• Реакция есть быстрая. ← unnatural in normal Russian

This is a very important pattern:

• past: Реакция была быстрой.
• present: Реакция быстрая.
• future: Реакция будет быстрой.

So in the present tense, there is often no word for “is”.

Yes. Russian word order is more flexible than English word order.

The neutral order here is:

• Реакция была быстрой.

But other orders are possible for emphasis or style, for example:

• Быстрой была реакция.
• Была быстрой реакция.
  This is less neutral and more marked.

Even when the order changes, the endings still show the grammatical relationships. That is why Russian can move words around more freely.

Still, for a learner, Реакция была быстрой is the safest and most natural basic version.

Russian does not have articles like English a / an / the.

So реакция can mean:

• a reaction
• the reaction

The exact meaning depends on context.

That means Russian learners of English have to learn articles, and English speakers learning Russian have to get used to their absence.

In this sentence, whether it means a reaction or the reaction would be understood from the wider situation.

The stress is:

• реа́кция была́ бы́строй

A rough pronunciation guide:

• реа́кция ≈ ree-AK-tsi-ya
• была́ ≈ by-LA
• бы́строй ≈ BIS-tray or BYES-tray depending on how close you want to get in English approximation

A few useful notes:

• Unstressed vowels are often reduced in Russian.
• The ы sound in была and быстрой is not exactly like any common English vowel.
• The й at the end of -ой gives a glide, so -ой is not pronounced like English oy in exactly the same way, though it can sound somewhat similar.

In theory, yes: быстра is the short feminine form of быстрый.

So you might encounter:

• Реакция была быстра.

But this sounds more literary, elevated, or less common in everyday speech. For most learners, the normal and useful version is:

• Реакция была быстрой.

So it is good to recognize быстра, but not necessary to use it early on.