Learn Icelandic: Ég geri gatið með borvélinni til að setja myndina á vegginn. - I make the hole with the drill to put the picture on the wall.

The preposition með is used here to show the tool used to do something. When með introduces an instrument or tool, it always forces the following noun into the dative case.

How is the word borvélinni formed?

It comes from borvél (a drill), which is a feminine noun. Because it follows með as a tool, it must be dative. Since we are talking about a specific drill, we add the feminine dative definite article -inni to the end, giving us borvélinni.

Why does á vegginn use the accusative case instead of the dative?

Prepositions like á (on) take the accusative case when they describe motion towards a destination. Because you are actively moving the picture onto the wall, the masculine noun veggur (wall) becomes the accusative vegginn. If the picture were already hanging on the wall without moving, you would use the dative case instead.

What is the role of til að in this sentence?

Til að translates to 'in order to' when expressing the purpose of an action. Here, you are making a hole in order to put the picture on the wall. It is always followed by a verb in its dictionary (infinitive) form, like setja (to put).