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Applications – leontief models

Solve the following homogeneous system.

x + y + z = 0 size 12{x+y+z=0} {}
3x + 2y + z = 0 size 12{3x+2y+z=0} {}
4x + 3y + 2z = 0 size 12{4x+3y+2z=0} {}

( t size 12{t} {} , 2t size 12{ - 2t} {} , t size 12{t} {} )

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Solve the following homogeneous system.

x y z = 0 size 12{x - y - z=0} {}
x 3y + 2z = 0 size 12{x - 3y+2z=0} {}
2x 4y + z = 0 size 12{2x - 4y+z=0} {}
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Chris and Ed decide to help each other by doing repairs on each others houses. Chris is a carpenter, and Ed is an electrician. Chris does carpentry work on his house as well as on Ed's house. Similarly, Ed does electrical repairs on his house and on Chris' house. When they are all finished they realize that Chris spent 60% of his time on his own house, and 40% of his time on Ed's house. On the other hand Ed spent half of his time on his house and half on Chris's house. If they originally agreed that each should get about a $1000 for their work, how much money should each get for their work?

Chris = $ 1250 size 12{"Chris"=$"1250"} {} , Ed = $ 1, 000 size 12{"Ed"=$1,"000"} {}

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Chris, Ed, and Paul decide to help each other by doing repairs on each others houses. Chris is a carpenter, Ed is an electrician, and Paul is a plumber. Each does work on his own house as well as on the others houses. When they are all finished they realize that Chris spent 30% of his time on his own house, 40% of his time on Ed's house, and 30% on Paul's house. Ed spent half of his time on his own house, 30% on Chris' house, and remaining on Paul's house. Paul spent 40% of the time on his own house, 40% on Chris' house, and 20% on Ed's house. If they originally agreed that each should get about a $1000 for their work, how much money should each get for their work?

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Given the internal consumption matrix A size 12{A} {} , and the external demand matrix D size 12{D} {} as follows.

A = . 30 . 20 . 10 . 20 . 10 . 30 . 10 . 20 . 30 size 12{A= left [ matrix { "." "30" {} # "." "20" {} # "." "10" {} ##"." "20" {} # "." "10" {} # "." "30" {} ## "." "10" {} # "." "20" {} # "." "30"{}} right ]} {} D = 100 150 200 size 12{D= left [ matrix { "100" {} ##"150" {} ## "200"} right ]} {}

Solve the system using the open model: X = AX + D size 12{X= ital "AX"+D} {} or X = I A 1 D size 12{X= left (I - A right ) rSup { size 8{ - 1} } D} {}

315 . 34 383 . 52 440 . 34 size 12{ left [ matrix { "315" "." "34" {} ##"383" "." "52" {} ## "440" "." "34"} right ]} {}
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Given the internal consumption matrix A size 12{A} {} , and the external demand matrix D size 12{D} {} as follows.

A = . 05 . 10 . 10 . 10 . 15 . 05 . 05 . 20 . 20 size 12{A= left [ matrix { "." "05" {} # "." "10" {} # "." "10" {} ##"." "10" {} # "." "15" {} # "." "05" {} ## "." "05" {} # "." "20" {} # "." "20"{}} right ]} {} D = 50 100 80 size 12{D= left [ matrix { "50" {} ##"100" {} ## "80"} right ]} {}

Solve the system using the open model: X = AX + D size 12{X= ital "AX"+D} {} or X = I A 1 D size 12{X= left (I - A right ) rSup { size 8{ - 1} } D} {}

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An economy has two industries, farming and building. For every $1 of food produced, the farmer uses $.20 and the builder uses $.15. For every $1 worth of building, the builder uses $.25 and the farmer uses $.20. If the external demand for food is $100,000, and for building $200,000, what should be the total production for each industry in dollars?

Farming = $ 201 , 754 . 38 size 12{"Farming"=$"201","754" "." "38"} {} , Building = $ 307 , 017 . 54 size 12{"Building"=$"307","017" "." "54"} {}
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An economy has three industries, farming, building, and clothing. For every $1 of food produced, the farmer uses $.20, the builder uses $.15, and the tailor $.05. For every $1 worth of building, the builder uses $.25, the farmer uses $.20, and the tailor $.10. For every $1 worth of clothing, the tailor uses $.10, the builder uses $.20, the farmer uses $.15. If the external demand for food is $100 million, for building $200 million, and for clothing $300 million, what should be the total production for each in dollars?

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Suppose an economy consists of three industries F size 12{F} {} , C size 12{C} {} , and T size 12{T} {} . The following table gives information about the internal use of each industry's production and external demand in dollars.

F size 12{F} {} C size 12{C} {} T size 12{T} {} Demand Total
F size 12{F} {} 30 10 20 40 100
C size 12{C} {} 20 30 20 50 120
T size 12{T} {} 10 10 30 60 110

Find the proportion of the amounts consumed by each of the industries; that is, find the matrix A size 12{A} {} .

30 / 100 10 / 120 20 / 110 20 / 100 30 / 120 20 / 110 10 / 100 10 / 120 30 / 110 size 12{ left [ matrix { "30"/"100" {} # "10"/"120" {} # "20"/"110" {} ##"20"/"100" {} # "30"/"120" {} # "20"/"110" {} ## "10"/"100" {} # "10"/"120" {} # "30"/"110"{}} right ]} {}
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Source:  OpenStax, Applied finite mathematics. OpenStax CNX. Jul 16, 2011 Download for free at http://cnx.org/content/col10613/1.5
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