Kate sold 12 of her paintings and purchased 7 more. if kate has 21 paintings now, how many paintings did she have to begin with?

First, let x or any variable represent the number of paintings Kate initially has. After selling 12 of it, her paintings now becomes only x - 12. Then, she purchased 7 more making the number of her paintings become x - 12 + 7 = x - 5. This number tantamount to 21.

x - 5 = 21; x = 26

Thus, Kate initially had 26 paintings.