I know that yields the correct answer but how do I know I should divide the expression by two from the problem statement?

You could do a couple of steps forward in solving the equations but the intuitive explanation is something along the lines of: Assuming the ball costs nothing the total comes to $1. If the total is more than that then it means that both the ball and the bat have an increased price by an equal amount. So the difference from $0 of the ball is going to be half the difference between the total and $1.