A ceramics workshop makes wreaths, trees and sleighs for sale at Christmas. A wreaths takes 3 hours, 2 hours to paint, and 9 hours to fire. A tree takes 15 hours to prepare, 3 hours to paint , and 4 hours to fire. A sleigh takes 4 hours to prepare, 14 hours to paint, and 7 hours to fire. If the workshop has 114 hours for preparation time, 15 hours for painting, and 122 hours for firing, how many of each can be made ?

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cyaspain11 cyaspain11 · Answer 1 · 2016-10-05T21:28:00-04:00

w = wreaths produced
t = trees produced
s = sleighs produced

Prep: 3w + 15t + 4s = 129
Painting: 2w + 3t + 13s = 75
Firing: 8w + 4t + 7s = 117

8w + 12t + 52s = 300
8w + 4t + 7s = 117
——————————
8t + 45s = 183

6w + 30t + 8s = 258
6w + 9t + 39s = 225
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21t - 31s = 33

Now you have "w" gone and just the other two variables. Multiply "8t + 45s = 183" by 21, then "21t - 31s = 33" by 8, then subtract it from the first:

168t + 945s = 3843
168t - 248s = 264
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1193s = 3579

Divide each side by 1193: s = 3.

(I show "8t + 45s = 183" below) and substitute s = 3 into it to find "t":

8t + 45s = 183
8t + 453 = 183
8t + 135 = 183
8t = 183 - 135
8t = 4

substitute s = 3 and t = 6 to find "w":

3w + 15t + 4s = 129
3w + 156 + 4*3 = 129
3w + 90 + 12 = 129
3w + 102 = 129
3w = 129 - 102
3w = 27

w = 9 and you're done.