Az úgynevezett megoldás egyszerű.

Here az is the definite article, so it means the.

Hungarian has two forms of the definite article:

• a before a consonant
• az before a vowel

Since the next word, úgynevezett, begins with a vowel sound, Hungarian uses az.

So:

• a megoldás = the solution
• az úgynevezett megoldás = the so-called solution

In this sentence, no: it is understood as the, not that.

Hungarian az can also be a demonstrative (that), but when az stands directly before a noun phrase like this, it is normally just the article.

If you wanted that so-called solution, Hungarian would usually make the demonstrative clearer, for example with a fuller demonstrative structure in context.

So in Az úgynevezett megoldás egyszerű, the natural reading is:

• The so-called solution is simple.

Úgynevezett means so-called.

It is built from:

• úgy = so / thus / in that way
• nevezett = called / named (historically from the verb nevez, to call/name)

So the literal idea is something like thus-called.

In modern Hungarian, learners should usually just treat úgynevezett as a fixed adjective meaning so-called.

Because in Hungarian it functions as a fixed lexical item, essentially a set adjective.

English writes so-called with a hyphen, but Hungarian standard spelling writes úgynevezett as one word.

So:

• úgynevezett megoldás = so-called solution

Not necessarily.

This is an important nuance. In English, so-called often suggests doubt, criticism, or irony:

• their so-called expert can sound dismissive.

Hungarian úgynevezett can sometimes have that tone too, depending on context and intonation, but it is often much more neutral than English so-called. It can simply mean:

• what is called
• what is known as
• the term used for

So this sentence could be neutral or slightly skeptical depending on context. The word itself does not force sarcasm.

Megoldás is a noun meaning solution.

It comes from the verb:

• megold = to solve

and the noun-forming ending:

• -ás / -és

So:

• megold = solve
• megoldás = solution / solving

In this sentence, megoldás is the head noun of the subject:

• Az úgynevezett megoldás = the so-called solution

Because Hungarian usually leaves out the verb to be in the present tense, third person when the sentence is of the form X is Y.

So:

• A megoldás egyszerű. = The solution is simple.

There is no overt word for is.

This is completely normal Hungarian grammar.

But the verb does appear in other persons:

• Egyszerű vagyok. = I am simple.
• Egyszerű vagy. = You are simple.

And it also appears in other tenses:

• A megoldás egyszerű volt. = The solution was simple.

Because egyszerű is the predicate adjective: it tells you what the subject is like.

The sentence structure is:

• Az úgynevezett megoldás = subject
• egyszerű = predicate adjective (simple)

A neutral English translation is:

• The so-called solution is simple.

Putting egyszerű at the end is the normal neutral order here.

Hungarian word order can change for emphasis. For example, if you front egyszerű, it can sound contrastive or emphatic:

• Egyszerű az úgynevezett megoldás.

That is closer to:

• The so-called solution is SIMPLE.
• It is simple, that so-called solution.

So the original sentence has a calm, neutral order.

Everything here is in the nominative (basic dictionary form). There are no extra case endings in this sentence.

• az = the
• úgynevezett = so-called
• megoldás = solution
• egyszerű = simple

This makes sense because the sentence is simply stating something about the subject.

Hungarian is much simpler than many European languages in this respect.

There is:

• no grammatical gender
• usually no adjective agreement before the noun

So úgynevezett does not change for masculine/feminine/neuter, because Hungarian has no such system.

Also, the attributive adjective before the noun stays the same:

• az úgynevezett megoldás = the so-called solution
• az úgynevezett megoldások = the so-called solutions

Notice that úgynevezett does not change.

However, predicate adjectives do often show plural when the subject is plural:

• A megoldás egyszerű. = The solution is simple.
• A megoldások egyszerűek. = The solutions are simple.

So in your sentence, egyszerű is singular because megoldás is singular.

A few useful points:

• egy is roughly like edgy without the final vowel, but this is only an approximation.
• sz is pronounced like English s.
• ű is a long vowel that English does not really have. It is like a long ü sound, similar to German ü but held longer.

So egyszerű may be the hardest word in the sentence to pronounce naturally.

Also remember that Hungarian stress normally falls on the first syllable of each word:

• AZ
• ÚGYnevezett
• MEGoldás
• EGYszerű

That first-syllable stress is very consistent in Hungarian.