Madem arka aktarıcı gevşek, alyan anahtarıyla vites telini ayarla.

Here madem means something like since, given that, or if that’s the case.

It is often used when the speaker treats something as already accepted and then draws a conclusion from it:

• Madem arka aktarıcı gevşek, ... = Since the rear derailleur is loose, ... = If the rear derailleur is loose, then ...

It has a slightly conversational tone. It is not just a neutral because.

Because Turkish often leaves out the verb to be in the present tense.

So:

• arka aktarıcı gevşek literally: rear derailleur loose natural English: the rear derailleur is loose

This is completely normal in Turkish.
If you wanted a more formal or emphatic version, you could say:

• arka aktarıcı gevşektir

But in everyday language, gevşek by itself is enough.

Arka aktarıcı is the bicycle part called the rear derailleur.

Breakdown:

• arka = rear, back
• aktarıcı = literally something like transferer / shifter, but in cycling it means derailleur

So this is a technical term rather than a word-for-word everyday phrase.

The ending -yla means with or by means of.

So:

• alyan anahtarıyla = with an Allen key / hex key

This comes from ile, the separate word meaning with. In Turkish, ile is very often attached to the noun:

• alyan anahtarı ile
• alyan anahtarıyla

These mean the same thing.

The y appears as a buffer consonant because the word before it ends in a vowel sound.

Because vites telini is the definite direct object of the verb ayarla.

Step 1: the basic noun phrase

• vites teli = gear cable / shift cable

This is a noun compound.

Step 2: make it the object

When a direct object is specific/definite in Turkish, it usually takes the accusative ending.

So:

• vites teli → vites telini

That -ni is the accusative here.

So vites telini ayarla means:

• adjust the gear cable

not just

• adjust a gear cable

Because vites teli is already a compound form, and then the accusative is added on top of it.

Breakdown:

• tel = wire, cable
• vites teli = gear cable
• vites teli + -(y)i accusative → vites telini

In other words, the object ending is being added to the whole compound.
This is very common in Turkish noun compounds.

Ayarla is the informal singular imperative form of ayarlamak.

• ayarlamak = to adjust, to set
• ayarla = adjust!

So the speaker is giving a command or instruction to one person.

Other possible forms:

• ayarlayın = adjust! (plural or polite/formal)
• ayarlayınız = very formal
• ayarlar mısın? = could you adjust it?

It is understood and does not need to be said.

In Turkish, subject pronouns are often omitted when the verb already makes the meaning clear.
Since ayarla is an imperative, the hidden subject is:

• sen = you (singular, informal)

So the sentence naturally means:

• You, adjust the gear cable with an Allen key, since the rear derailleur is loose.

But Turkish normally does not say sen unless it wants extra emphasis.

Not completely, but the given order is very normal.

The sentence is:

• Madem arka aktarıcı gevşek, alyan anahtarıyla vites telini ayarla.

A rough structure is:

• reason/background
• instrument
• object
• verb

Turkish usually puts the verb at the end, and that is what happens here.

You could also say:

• Madem arka aktarıcı gevşek, vites telini alyan anahtarıyla ayarla.

That is also natural. The difference is mostly one of emphasis and flow.

Turkish does not have words exactly like English the or a/an.

Instead, definiteness is shown by context and often by grammar.

In this sentence:

• vites telini is clearly specific/definite because of the accusative ending
• arka aktarıcı is understood from context as the rear derailleur
• alyan anahtarıyla can mean with an Allen key or with the Allen key, depending on context

So Turkish often lets the situation do part of the work that English articles do.

Not exactly.

madem

Used when something is taken as accepted, and the speaker draws a consequence:

• Madem arka aktarıcı gevşek, ...
• Since the rear derailleur is loose, then ...

çünkü

Means because, and usually gives an explanation rather than setting up a conclusion in the same way:

• Vites telini ayarla, çünkü arka aktarıcı gevşek.
• Adjust the gear cable, because the rear derailleur is loose.

conditional -se / -sa

This is a more straightforward if structure:

• Arka aktarıcı gevşekse, vites telini ayarla.
• If the rear derailleur is loose, adjust the gear cable.

So madem is close in meaning here, but it has its own tone: given that this is true, do this.