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A furniture company makes dining room furniture.
A buffet requires 40 hours for construction 10 hours for finishing.
A chair requires 20 hours for construction and 20 hours for finishing.
A table requires 40 hours for construction and 70 hours for finishing.
The construction department has 500 hours of labor
and the finishing department has 200 hours of labor available each week.
How many pieces of each type of furniture should be produced each week if the factory is to run at full capacity?
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X buffets; Y chairs; Z tables.
Write equations for full capacity as you read the problem
40X + 20Y + 40Z = 500 hours (1) (the construction department time)
10X + 20Y + 70Z = 200 hours (2) (the finishing department time)
Simplify equations by dividing each term by 10
4X + 2Y + 4Z = 50 (1')
X + 2Y + 7Z = 20 (2')
From equation (2'), Z may have only three possible values: Z = 0, or 1, or 2.
If Z= 0, then equations (1'), (2') take the form
4X + 2Y = 50
X + 2Y = 20
and have a UNIQUE solution X= 10, Y= 5. (You can use the Elimination method to get this solution).
If Z= 1, then equations (1'), (2') take the form
4X + 2Y = 46
X + 2Y = 13
and have a UNIQUE solution X= 11, Y= 1. (You can use the Elimination method to get this solution).
If Z= 2, then equations (1'), (2') take the form
4X + 2Y = 42
X + 2Y = 6
and have NO a solution in integer non-negative numbers.
Thus the problem has two possible solutions: (X,Y,Z) = (10,5,0) and/or (X,Y,Z) = (11,1,1).
There are no other solutions.
Solved.
B, C, and T must be non-negative integers:
Subtract the 2nd eq. from the 1st.
Eliminate the constant, multiply 1st eq. by 2, and 2nd eq. by 5
Answer: 10 buffets, 5 chairs, and no tables OR
11 buffets, 1 chair, and 1 table.
Edwin