System using augmented matrices question

cptsimose

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A small craft business makes woven blankets, wooden carvings, and baskets. Each blanket requires 4 hours to prepare the materials, 24 hours for crafting, and 15 hours for finishing. Carvings require 5 hours for preparation, 30 hours for crafting, and 18 hours for finishing. Baskets require 3 hours for preparation, 12 hours for crafting, and 9 hours for finishing. The labor hours available are 59 for preparation, 306 for crafting, and 201 for finishing. How many of each item can be made?

I just need a little help setting this problem up. I don't want the answer. Just the equation to help me solve the problem. This question was given to me without being taught how to do it.

Thanks for the help.
 
cptsimose said:
A small craft business makes woven blankets, wooden carvings, and baskets. Each blanket requires 4 hours to prepare the materials, 24 hours for crafting, and 15 hours for finishing. Carvings require 5 hours for preparation, 30 hours for crafting, and 18 hours for finishing. Baskets require 3 hours for preparation, 12 hours for crafting, and 9 hours for finishing. The labor hours available are 59 for preparation, 306 for crafting, and 201 for finishing. How many of each item can be made?


Use prep time for the top linear equation, crafting for the 2nd, and finishing for the 3rd.

Your prep times are 4, 5, and 3

Craft times are 24, 30, and 12

Finishing times are 15, 18, and 9

Set up a system of equations and set them to the given times allowed.

\(\displaystyle 4x+5y+3z=59\\24x+30y+12z=306\\15x+18y+9z=201\)

Solve the system.
 
Hello, cptsimose!

A small craft business makes woven blankets, wooden carvings, and baskets.
Each blanket requires 4 hours to prep, 24 hours for crafting, and 15 hours for finishing.
Carvings require 5 hours for prep, 30 hours for crafting, and 18 hours for finishing.
Baskets require 3 hours for prep, 12 hours for crafting, and 9 hours for finishing.
The labor hours available are 59 for preparation, 306 for crafting, and 201 for finishing.
How many of each item can be made?

A chart helps to organize the information.
Code:
          |  Prep | Craft | Finsh |
- - - - - + - - - + - - - + - - - +
 Blankets |   4   |   24  |   15  |
- - - - - + - - - + - - - + - - - +
 Carvings |   5   |   30  |   18  |
- - - - - + - - - + - - - + - - - +
 Baskets  |   3   |   12  |   9   |
- - - - - + - - - + - - - + - - - +
  Total   |  59   |  306  |  201  |
- - - - - + - - - + - - - + - - - +

Let \(\displaystyle \,x\) = number of blankets
. . . \(\displaystyle y\) = number of carvings
. . . \(\displaystyle z\) = number of baskets

Reading down the columns, we have these equations:

. . \(\displaystyle \L\begin{array}{ccc}4x\,+\,5y\,+\,3z & = & 59 \\ 24x\,+\,30y\,+\,12x & = & 306 \\ 15x\,+\,18y\,+\,9z & = & 201\end{array}\)

Edit: Too fast for me, Cody!
 
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