Beginning of year:assets=27,000, liabilities=19,000, owners equity=x
during the year:assets increase by $2,000, liabilities decrease by $4,000

This is a very simple and straightforward problem.
First you have to find beginning owners equity.

The formula you need to know is the basic accounting equation, which is:
Assets = Liabilities + Owners Equity

This can be rewritten as:
Owners Equity = Assets - Liabilities

First you have to solve for beginning owners equity, so:
Beginning Owners Equity = Beginning Assets - Beginning Liabilities

Next you have to find ending assets and ending liabilities, so:
Take your beginning assets and add the assets increase by amount given to get ending assets. Now take beginning liabilities and subtract the liabilities decrease amount given to get ending liabilities.

Now since you have ending assets and ending liabilities you can solve for ending owners equity.
Simply take your ending assets minus ending liabilities to get ending owners equity.