to fasten / to determine
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to fasten
To securely attach something (e.g., a boat, a tent, a load) to a fixed point with a rope or another fastening so that it cannot move.
Der Kapitän befahl der Mannschaft, das Schiff im Hafen festzumachen.
The captain ordered the crew to fasten the ship in the harbor.
'Festmachen' (to fasten / to determine) is a separable verb. The prefix 'fest-' comes at the end in a main clause, stays connected to the verb in subordinate clauses, and in the infinitive 'zu' comes between 'fest' and 'machen'.
Die Matrosen machen das Boot am Kai fest.
The sailors fasten the boat to the quay.
In the literal meaning, it is often stated where something is fastened. For this, the preposition 'an' is used, followed by the dative.
Wir müssen das Zelt mit Heringen im Boden festmachen.
We have to fasten the tent to the ground with pegs.
In the figurative meaning 'stützen auf' (to base on), the structure is always 'etwas (Akkusativ) an etwas (Dativ) festmachen'. The thing one is basing something on is in the dative.
Die Polizei konnte den Verdacht an keinen Beweisen festmachen.
The police could not base the suspicion on any evidence.
When it is about agreeing on something in a binding way, 'festmachen' (to set / to agree on) is often used with abstract nouns such as 'Termin' (appointment/date), 'Preis' (price), or 'Details' (details).
Lass uns die Details für den Vertrag nächste Woche festmachen.
Let’s settle the details for the contract next week.
to solve / loosen
'lösen' and 'festmachen' are clear opposites here. 'festmachen' means that you fasten something securely, for example with a rope or cord. It is then not supposed to move. In this context, 'lösen' means that you loosen or open this fastening. So you first fasten something, and later you can loosen or untie it again.
Am Steg macht der Fischer sein Boot mit einem dicken Seil fest.
At the dock, the fisherman fastens his boat with a thick rope.
Vor der Abfahrt löst Tom das Seil vom Fahrradträger.
Before departure, Tom unties the rope from the bike rack.