الجهاز البيضاوي جديد في الصالة الرياضية.

Because this is a present-tense nominal sentence in Arabic. In MSA, Arabic usually does not use a separate word for is/am/are in the present tense.

So:

• الجهاز البيضاوي جديد = The elliptical machine is new

If you wanted the past or future, then Arabic would use other forms, but in the present, the copula is normally understood rather than spoken.

Because in Arabic, adjectives normally come after the noun they describe.

So:

• الجهاز = the machine/device
• البيضاوي = elliptical / oval

Together:

• الجهاز البيضاوي = the elliptical machine

This is the normal Arabic order:

• noun first
• adjective second

Unlike English, where we say elliptical machine, Arabic says literally machine elliptical.

This is a very common and important point.

1. البيضاوي is an adjective describing الجهاز

Since الجهاز is definite (the machine), its adjective must also be definite:

• الجهاز = the machine
• البيضاوي = the elliptical

So الجهاز البيضاوي is one definite noun phrase: the elliptical machine.

2. جديد is the predicate

In this sentence, جديد means new and is the predicate: it tells us something about the subject.

Arabic predicates like this are often indefinite, so:

• الجهاز البيضاوي جديد
• literally: The elliptical machine [is] new

So the contrast is:

• البيضاوي = attributive adjective, part of the noun phrase
• جديد = predicate adjective, part of the sentence

Literally, it means something like:

• الجهاز = the device / machine / apparatus
• البيضاوي = oval / elliptical

So the phrase literally means the elliptical machine or the oval-shaped machine.

In context, this is the usual way to refer to an elliptical exercise machine.

Because it matches the thing being described: الجهاز البيضاوي.

The head noun is:

• الجهاز — a masculine singular noun

So the predicate adjective is also masculine singular:

• جديد = new

If the noun were feminine, you would expect a feminine form such as جديدة.

Because it describes الصالة, and الصالة is feminine.

Here the phrase is:

• الصالة الرياضية

That means literally the sports hall or the gym.

Agreement works like this:

• الصالة = feminine singular
• الرياضية = feminine singular adjective

So الرياضية has the feminine ending ـة because it must agree with الصالة.

No. It is not an iḍāfa.

It is a noun + adjective phrase:

• الصالة = the hall
• الرياضية = athletic / sports-related

So:

• الصالة الرياضية = the sports hall / the gym

A quick clue: in a noun + adjective phrase, both words can take الـ if the phrase is definite.

That is exactly what happens here:

• الصالة — definite
• الرياضية — also definite

In a typical iḍāfa, the first word usually does not take الـ in the same way.

في usually means in.

So:

• في الصالة الرياضية = in the gym

Depending on context, English might also say at the gym, but Arabic uses في very naturally here.

So the whole sentence is describing where this machine is:

• new in the gym
• or more naturally in English, There is a new elliptical machine in the gym / The elliptical machine is new in the gym, depending on context

Yes. Arabic word order is somewhat flexible, especially for emphasis or style.

The basic version here is:

• الجهاز البيضاوي جديد في الصالة الرياضية

This presents the elliptical machine as the topic, then says something about it.

You could rearrange parts in some contexts, especially the prepositional phrase, but the original sentence is a straightforward and natural structure for learners to study.

Because everyday Arabic writing usually leaves out short vowels and tanwīn.

So you see:

• جديد

But in a fully vocalized form, it could appear as:

• جديدٌ

That final ـٌ marks it as indefinite in careful grammatical spelling.

Likewise, many endings that matter grammatically are often not written in ordinary text unless the text is fully vocalized.

A simple pronunciation guide is:

• al-jihāz al-bayḍāwī jadīdun fī aṣ-ṣālah ar-riyāḍiyyah

A few notes:

• الجهاز = al-jihāz
• البيضاوي = al-bayḍāwī
• جديد = jadīd
• الصالة الرياضية is pronounced with assimilation:
  • aṣ-ṣālah ar-riyāḍiyyah

That happens because ص and ر are sun letters, so the l sound of الـ is not pronounced separately.

Grammatically, it is an adjective modifying الجهاز.

But in real usage, الجهاز البيضاوي also works like a set expression for a specific kind of exercise machine: the elliptical.

So both things are true:

• grammar: noun + adjective
• meaning in context: a specific gym machine name

That kind of combination is very common in Arabic.