Kyle works at a donut​ factory, where a​ 10-oz cup of coffee costs 95cents​, a​ 14-oz cup costs​ $1.15, and a​ 20-oz cup costs​ $1.50. During one busy​ period, Kyle served 30 cups of​ coffee, using 464 ounces of​ coffee, while collecting a total of ​$37.20. How many cups of each size did Kyle​ fill?

Answer:4, 16 and 10Step-by-step explanation:Here we would like to work with a system of 3 equations, we need ti find them. Lets call s the 10oz coffees, m the 14oz and l the 20oz.As Kyle sold $37.2 selling 3 different types of coffee at their respective prices we can say:revenue = s*0.95 + m*1.15 + l*1.5 = 37.2 [equation 1]And as he sold 30 cups:s + m + l = 30 [equation 2]And as the total volume of coffee is 464 oz:s*10 + m*14 + l*20 = 464 [equation 3]So, we have a system of 3 equations with three unknowns s, m and l.Take the eq 2 and subtract m and l in both sides:s = 30 - m - lNow, take it and replace in the other 2:(30 - m - l)*10 + 14m + 20l = 464 [eq 3]0.95(30 - m - l) + 1.15m + 1.5l = 37.2 [eq 1]Working with equation 3:300 -10m -10l + 14m + 20l = 464300 + 4m + 10l = 4644m + 10l = 164 [equation 3']Now with eq 1:0.95(30 - m - l) + 1.15m + 1.5l = 37.228.5 - 0.95m - 0.95l + 1.15m + 1.5 l = 37.228.5 + 0.2m + 0.55l = 37.20.2m + 0.55l = 8.7 [equaiton 1']Now we passed to a 2-equation system with eq 1' and eq 3'.Take eq 3' and subtract 4m in both sides:10l = 164 - 4mNow divide by 10 in both sidesl = [164 - 4m]/10Now, with this value for l replace in eq 1':0.2m + 0.55 ( [164 - 4m]/10) = 8.7 0.2m + 0.055 (164 - 4m) = 8.70.2m + 9.02 - 0.22m = 8.7- 0.02m = -0.32m = -0.32/-0.02m = 16 ----> 16 14oz coffeesNow replace m=16 in eq 1' or 3' to find l:4m +10l = 16464 + 10l = 16410l = 100l = 10 ----> 10 20oz coffeesSo, finally we replace l=10 and m=16 in equaiton 2:s + m + l = 30s + 16 +10 = 30s = 30 - 26s = 4 So, Kyle filled 4 10oz coffes, 16 14oz coffees and 10 20oz coffees.