Mater et filia post laborem ad balneum lavatum eunt.

Because both words are in the nominative case, which is the case normally used for the subject of a sentence.

• mater = mother (nominative singular)
• filia = daughter (nominative singular)

They are joined by et = and, so together they form a compound subject: mother and daughter.

The verb also helps confirm this, because eunt is plural: they go.

Because the subject has two people: mater et filia.

In English, we say the mother and daughter go, not goes. Latin works the same way here:

• it = he/she/it goes
• eunt = they go

Since there are two subjects joined by and, Latin uses the plural verb eunt.

Because post is a preposition that takes the accusative case.

So:

• labor = nominative form
• laborem = accusative singular

After post, Latin uses the accusative, so post laborem means after work or after the work.

Yes. Post is one of the Latin prepositions that regularly takes the accusative case.

So you get forms like:

• post laborem
• post cenam
• post scholam

A learner often has to memorize which prepositions take which case, and post + accusative is a standard combination.

Because it follows ad, and ad also takes the accusative case.

Here ad means to or toward, showing motion in the direction of something:

• ad balneum = to the bath/bathhouse

So just as post requires the accusative, ad does too.

Lavatum is a supine, a verbal form made from the verb lavo, lavare (to wash).

In this sentence, it is the accusative supine, used after a verb of motion to express purpose.

So:

• eunt = they go
• lavatum = to wash / for washing

That means ad balneum lavatum eunt literally means something like:

• they go to the bath to wash

This is a construction English speakers often find unusual, because English normally just uses to + infinitive.

Because with verbs of motion, Latin can use the supine to show purpose.

So instead of saying:

• they go to wash

Latin can say:

• eunt lavatum

This is especially common with a limited set of verbs where someone goes, comes, or sends someone to do something.

For an English speaker, it helps to think of lavatum here as meaning to wash, even though grammatically it is not the infinitive.

They are accusative for different reasons.

• balneum is accusative because it is the object of the preposition ad
• lavatum is accusative because it is the supine in -um

So even though they look similar on the surface, they are doing different jobs:

• ad balneum = to the bath
• lavatum = to wash

They answer two different questions:

• ad balneum answers Where are they going?
• lavatum answers For what purpose are they going?

So:

• ad balneum = destination
• lavatum = purpose

Together they mean something like:

• They go to the bath in order to wash.

Because Latin has no articles.

English distinguishes between:

• the mother
• a mother

Latin usually just says mater, and the exact sense depends on context.

So:

• mater can mean mother, the mother, or sometimes a mother
• filia can mean daughter, the daughter, or a daughter

The reader figures out which is most natural from the situation.

Yes. Latin word order is much more flexible than English word order because the endings show the grammatical roles.

So this sentence could be rearranged in various ways without changing the basic meaning, for example:

• Post laborem mater et filia ad balneum lavatum eunt.
• Mater et filia ad balneum post laborem lavatum eunt.

Some orders may sound more natural or emphatic than others, but the case endings still tell you what each word is doing.

Because they belong to different declensions.

• filia is a first-declension noun
• mater is a third-declension noun

That is why their nominative singular forms look different:

• filia
• mater

A native English speaker often expects nouns of the same type to behave similarly, but in Latin the form depends heavily on the noun’s declension.

Here it is a place: the bath, bathing place, or bathhouse.

The action is expressed by lavatum.

So the sentence separates:

• the place they go to = balneum
• the purpose of going there = lavatum

That distinction is important, because balneum itself does not mean washing here; it means the place associated with bathing.

Yes, in principle. The supine with a verb of motion can stand on its own to express purpose:

• eunt lavatum = they go to wash

Adding ad balneum just makes the destination explicit:

• ad balneum lavatum eunt = they go to the bath to wash

So lavatum does not depend on balneum; it depends on the idea of motion in eunt.