Question

A ceramics workshop makes wreaths, trees, and sleighs for sale at Christmas. A wreath takes 3 hours to prepare, 2 hours to paint, and 10 hours to fire. A tree takes 17 hours to prepare, 3 hours to paint, and 4 hours to fire. A sleigh takes 4 hours to prepare, 15 hours to paint, and 7 hours to fire. If the workshop has 119 hours for prep time, 87 hours for painting, and 108 hours for firing, how many of each can be made?

Answer 1

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

Then multiply the prep equation by 2 and the painting equation by 3 and subtract the painting equation from the prep equation:

6w + 30t + 8s = 258
6w + 9t + 39s = 225
——————————
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
—————————
1193s = 3579

Divide each side by 1193: s = 3.

Use whichever of the last two equations you prefer (I show "8t + 45s = 183" below) and substitute s = 3 into it to find "t":

8t + 45s = 183
8t + 45*3 = 183
8t + 135 = 183
8t = 183 - 135
8t = 48

t = 6

And finally, pick one of the original equations and substitute s = 3 and t = 6 to find "w":

3w + 15t + 4s = 129
3w + 15*6 + 4*3 = 129
3w + 90 + 12 = 129
3w + 102 = 129
3w = 129 - 102
3w = 27

w = 9 and you're done.

People often prefer Cramer's rule because, as complicated as it looks, the arithmetic is pretty simple while the eliminate and substitute method can lead to all kinds of errors (like seeing 8s and 39s, forgetting you are subtracting a positive number from a smaller one, and getting 31s, not -31s or actually adding to get 47s; it's especially error-prone when you have something like 8s and –39s and get -31s, not 47s when subtracting the negative 31s) and takes a ton of work. But you learn the eliminate and substitute method first and everyone hates matrices...

Answer 2

these are the answers i have for u
4 wreaths, 3 trees, 8 sleighs

3 wreaths, 8 trees, 4 sleighs

8 wreaths, 4 trees, 3 sleighs

9 wreaths, 5 trees, 4 sleighs
just only one is correct