I don't think there is an algebraic way to completely solve the problem.

You can start with the following. (assuming costs in cents)

w*x*y*z = 711000000
w+x+y+z = 711

Finding factors of 711
711 = 3*3*79

w*x*y*z must be divisible by 3,9,and 79

At this point you can say one of the costs should be $0.79 or a multiple of that. When you divide $7.11 by $0.79 you get a whole number (9) Dividing by $1.58 you get 4.5, and dividing by $3.16 you get 2.25. This suggests that the other 3 prices are multiples of 5 cents.

At that point it's trial and error or using some kind of solver function in a spreadsheet or calculator.