You need to multiply by the reciprocal because to get rid of an operation on one side, you have to perform the opposite operation. When you multiply a fraction by its reciprocal, you get one, essentially getting rid of the 4/3.
4/3 x 3/4 = 12/12 = 1
What you do to one side of an equation has to be done to the other, that's why the other side is multiplied by 3/4 also :)
