well, there is a "hidden assumption" in the problem, which is:

that the bar charged accurately for the drinks. to amplify, suppose the cost of drink w is odd (in cents). buying 0.5 of a drink would result in a cost of a fractional amount of cents. now, the error introduced by "rounding up" to the nearest penny, could be 0.5 cents. this makes little difference to the total a customer pays, but it drastically affects the solution space of the problem.

by playing around with the values of the fourth column of the augmented matrix, i discovered (largely by trial-and-error, i am sure there is a more precise numerical method) the following solution:

w = $1.23

x = $1.44

y = $1.39

z = $1.55

assuming the bar rounds up for 0.5 a drink, this makes the "true value" of the fourth column of the matrix:

8.975 (drinker 1 paid 1.5 cents extra for his 3 0.5 drinks)

10.45 (drinker 2 paid 1 cent extra for his 3 0.5 drinks, one of which is actually "fair price", since the cost per drink is an even number of cents)

11.78 (drinker 3 didn't pay extra because his one 0.5 drink was at a "fair price")

7.985 (drinker 4 paid 0.5 extra for his one 0.5 drink).