Question

To manufacture an automobile requires​ painting, drying, and polishing. Epsilon Motor Company produces three types of​ cars, the​ Delta, the​ Beta, and the Sigma. Each Delta requires 11 hours for​ painting, 5 hours for​ drying, and 3 hours for polishing. A Beta requires 17 hours for​ painting, 8 hours for​ drying, and 4 hours for​ polishing, and a Sigma requires 4 hours for​ painting, 5 hours for​ drying, and 1 hour for polishing. If the company has 253 hours for​ painting, 149 hours for​ drying, and 63 hours for polishing per​ month, how many of each type of car are​ produced?

Answer 1

Explanation:

d = number of Delta cars

b = number of Beta cars

s = number of Sigma cars

253 = 11d + 17b + 4s

149 = 5d + 8b + 5s

63 = 3d + 4b + 1s

s = 63 - 3d - 4b

149 = 5d + 8b + 5(63 - 3d - 4b) =

= 5d + 8b + 315 - 15d - 20b = -10d - 12b + 315

-166 = -10d - 12b

-83 = -5d - 6b

83 = 5d + 6b

5d = 83 - 6b

d = (83 - 6b)/5

with these 2 "identities" we are going into the 3rd equation and solve for the third variable (b), followed by solving the first 2 identities :

253 = 11((83 - 6b)/5) + 17b + 4(63 - 3((83 - 6b)/5) - 4b)

1265 = 11(83 - 6b) + 85b + 20×63 - 12(83 - 6b) - 80b

1265 = -(83 - 6b) + 5b + 1260

5 = -83 + 6b + 5b

88 = 11b

b = 8

d = (83 - 6b)/5 = (83 - 48)/5 = 35/5 = 7

s = 63 - 3d - 4b = 63 - 3×7 - 4×8 = 63 - 21 - 32 = 10

7 Delta cars

8 Beta cars

10 Sigma cars

were produced.