Таким образом, мы решили проблему.

Breakdown of Таким образом, мы решили проблему.

мы
we
проблема
problem
решить
to solve
таким образом
thus, in this way

Questions & Answers about Таким образом, мы решили проблему.

What literally is Таким образом, and why is it in the instrumental case?
It literally translates to "by such a manner" or "in such a way." The noun образ can mean form, shape, or manner. Russian often uses the instrumental case to describe how an action is performed, which is why Таким образом is a set phrase meaning "thus" or "therefore."
Could I just use Так instead of Таким образом?
Yes, you could say Так, мы решили... which means "So, we solved..." However, Таким образом is a slightly more formal and conclusive transition phrase. It feels more like "Thus," "Therefore," or "In this way" in English.
Why use the perfective verb решили instead of the imperfective решали?
We use the perfective решили because the action is fully completed and produced a result: the problem is successfully solved. If you used the imperfective решали, it would mean "we were solving" or "we were trying to solve" the problem, but maybe did not actually finish.
Why does the word проблема change to проблему here?
It changes to проблему because it is the direct object of the verb решили (it receives the action of being solved), which means it must be in the accusative case. Feminine nouns ending in -а change their ending to -у in this case.

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