Solution Exercises Chapter 1 Part 3
Solution Exercises Chapter 1 Part 3
Solution Exercises Chapter 1 Part 3
a) Use the question: "The objective of this problem is to determine the share price paid for
each group."
The 3 groups are General Motors, Shell and Air France.
Hence,
x represents the share price of General Motors.
y represents the share price of Shell.
z represents the share price of Air France.
b) "The total investment made by client A is €150. He bought 100 shares from General
Motors, 200 from Shell and 200 from Air France.
The total investment made by client B is zero. He sold 100 shares of General Motors,
and he bought 100 from Shell and 100 from Air France.
The total investment made by client C is same as client A. He bought 300 shares from
General Motors, 200 from Shell and sold 300 of Air France."
To summarize:
Number of shares from
General Motors Shell Air France Total investment
100 shares 200 shares 200 shares €150
-100 shares 100 shares 100 shares €0
300 shares 200 shares -300 shares €150
15
1 2 2
[0 300
10
300| 150] R2 /300 R2
0 400 900 300
1 2 2 15/10
[0 1 1 | 1/2 ] R3 - 400R2 R3
0 400 900 300
1 2 2 15/10
[0 1 1 | 1/2 ]
0 0 500 100 R3 / (500) R3
1 2 2 15/10
[0 1 1| 1/2 ]
0 0 1 1/5 R2 - R3 R2
R1 - 2R2 R1
1 0 0 1/2
[0 1 0| 3/10]
0 0 1 1/5
"They wish to conduct an opinion survey using 600 telephones contacts and 400 house
contacts. Survey company A has personnel to do 30 telephone and 10 house contacts per hour;
survey company B has handle 20 telephone and 20 house contacts per hour."
To summarize:
Company A Company B Total
30 telephone 20 telephone 600 telephones contacts
contacts per hour contacts per hour
30 20 600
The associated augmented matrix is: [ | ]
10 20 400
[
30 20 600
| ] R1/30 R1
10 20 400 R1 - 3R2 R2
1 2/3 20
[ | ] R2/(-40) R2
0 −40 −600
1 2/3 20
[ | ] R1 - (2/3)R2 R1
0 1 15
1 0 10
[ | ]
0 1 15
Therefore, x = 10 and y = 15. Company A should schedule 10 hours while Company B has to
plan 15 hours.
0 0 1 75
[ 2500 50 1| 150] Interchange R1 and R3
10000 100 1 275
1 0 0 1/100
[0 1 0| 1 ]
0 0 1 75
Therefore, a = 1/100 = 0.01, b = 1 and c = 75.
The quadratic equation is f (t) = 0.01t² + t + 75.
b) Let's use this model to estimate the population in 2050 (t = 2050 - 1900 = 150).
f (150) = 0.01(150)² + 150 + 75 = 450.
The U.S. population in 2050 should be equal to 450 million.
Exercise 20. Applications (Airline company)
You are the director of purchasing for a large commercial airline company.
Several years ago, anticipating significant growth in the air travel industry, you placed orders
totaling $2100 million to purchase new aircraft that would seat a total of 4500 passengers.
Some of the aircraft were Boeing 747s that cost $200 million each and seat 400 passengers and
Boeing 777s that cost $160 million each and seat 300 passengers. Others were European Airbus
A321s that cost $60 million and seat 200 passengers.
At the time you were instructed to buy twice as many US manufactured aircraft as foreign
aircraft.
How many of each type of aircraft did you need to buy?
METHOD
1. Identify the unknowns: Read the question “How many of each type of aircraft…”
2. Identify and name the equations based on the data provided in the problem. We have
cost data, seat data, and a constraint on US/foreign aircrafts. Therefore, we have the
following equations:
To find the last equation, read the sentence: "At the time you were instructed to buy twice as
many US manufactured aircraft as foreign aircraft".
1 1 −2 0 400R1 - R2 R2
[400 300 200| 4500] 200R1 - R3 R3
200 160 60 2100
1 1 −2 0 R2 /100 R2
[0 100 −1000| −4500]
0 40 −460 −2100
1 1 −2 0
[0 1 40R2 - R3 R3
−10 −45 ]
|
0 40 −460 −2100
1 1 −2 0
[0 1 −10| −45] R3 /60 R3
0 0 60 300
1 1 −2 0 10R3 + R2 R2
[0 1 −10| −45]
0 0 1 5
1 1 −2 0
R1 - R2 + 2 R3 R1
[0 1 0 | 5]
0 0 1 5
1 0 0 5
[0 1 0| 5]
0 0 1 5
Therefore, 𝑥 = 5, 𝑦 = 5, 𝑧 = 5.
3. The number of silver medals was the same as the total number of gold and bronze medals
combined.
Questions:
Solution:
1) Identify the unknowns.
Read the question: “…find the number of medals of each type (gold, silver and bronze)…”,
therefore the unknowns are:
𝑥 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑔𝑜𝑙𝑑 𝑚𝑒𝑑𝑎𝑙𝑠
{ 𝑦 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑖𝑙𝑣𝑒𝑟 𝑚𝑒𝑑𝑎𝑙𝑠
𝑧 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑏𝑟𝑜𝑛𝑧𝑒 𝑚𝑒𝑑𝑎𝑙𝑠
2) Write the system.
Information 3 can be expressed as: #silver medals = #gold medals + #bronze medals
𝑦 =𝑥+𝑧
Converted to standard form:
“silver vs. gold+medals equation: −𝑥 + 𝑦 − 𝑧 = 0”
(also correct: 𝑥 − 𝑦 + 𝑧 = 0)
𝑥 + 𝑦 + 𝑧 = 30 (𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑚𝑒𝑑𝑎𝑙𝑠)
{5𝑥 + 3𝑦 + 𝑧 = 80 (𝑝𝑜𝑖𝑛𝑡𝑠)
−𝑥 + 𝑦 − 𝑧 = 0 (𝑠𝑖𝑙𝑣𝑒𝑟 𝑣𝑠. 𝑔𝑜𝑙𝑑 + 𝑚𝑒𝑑𝑎𝑙)
(also correct:
7
𝑥 + 𝑦 + 𝑧 = 30 (𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑚𝑒𝑑𝑎𝑙𝑠)
{5𝑥 + 3𝑦 + 𝑧 = 80 (𝑝𝑜𝑖𝑛𝑡𝑠) ).
𝑥−𝑦+𝑧 =0 (𝑠𝑖𝑙𝑣𝑒𝑟 𝑣𝑠. 𝑔𝑜𝑙𝑑 + 𝑚𝑒𝑑𝑎𝑙)
1 1 1 30
[ 5 3 1 |80] 5𝑅1 − 𝑅2 → 𝑅2
−1 1 −1 0 𝑅1 + 𝑅2 → 𝑅3
1 1 1 30
[0 2 4 |70] 𝑅2 ↔ 𝑅3 (𝐼𝑛𝑡𝑒𝑟𝑐ℎ𝑎𝑛𝑔𝑒 𝑅2 𝑎𝑛𝑑 𝑅3 )
0 2 0 30
We notice that 𝑅3 contains two zeros already. By switching it with 𝑅2 , we get closer to our
1 0 0
goal of turning the left part of the matrix into the identity matrix 𝐼3 = [0 1 0]
0 0 1
1 1 1 30
[0 2 0 |30] 𝑅2 ⁄2 → 𝑅2
0 2 4 70 𝑅2 − 𝑅3 → 𝑅3
We have an opportunity to simplify 𝑅2 . Continue triangulation (turn red numbers into 0 by
combining rows).
1
1 1 1 30 𝑅1 − 𝑅2 + 𝑅3 → 𝑅1
[0 1 0 | 15 ] 4
𝑅3 /(−4) → 𝑅3
0 0 −4 −40
Complete triangulation.
1 0 0 5
[0 1 0 15]
|
0 0 1 10
The solution is therefore:
𝑥=5
{𝑦 = 15
𝑧 = 10
Optional (useful if you have time): verification step (to check your work):
𝑥 + 𝑦 + 𝑧 = 5 + 15 + 10 = 30 𝑐𝑜𝑟𝑟𝑒𝑐𝑡
5𝑥 + 3𝑦 + 𝑧 = 5(5) + 3(15) + 10 = 25 + 45 + 10 = 80 𝑐𝑜𝑟𝑟𝑒𝑐𝑡
-𝑥 + 𝑦 − 𝑧 = −5 + 15 − 10 = 0 𝑐𝑜𝑟𝑟𝑒𝑐𝑡
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It is important to answer the question asked in the word problem, using the terms used in the
word problem. Here:
“Sunnyvale High’s athletes were awarded 5 gold medals, 15 silver medals and 10 bronze
medals.”
𝐴 𝑀𝑎
𝐴 0.3 0.25
𝑀= [ ]
𝑀𝑎 0.1 0.25
Task: Find the output for each sector that is needed to satisfy a final demand of $40 million for
agriculture and $40 million for manufacturing.
Solution:
0.30 0.25 40 𝑥1
𝑀=[ ] 𝐷=[ ] 𝑋 = [𝑥 ]
0.10 0.25 40 2
Total Internal External
Output Demand Demand
𝑋 = 𝑴𝑿 + 𝑫
𝑋 − 𝑀𝑋 = 𝐷
𝐼2 𝑋 − 𝑀𝑋 = 𝐷
(𝐼2 − 𝑀)𝑋 = 𝐷
(𝐼2 − 𝑀)𝑋 = 𝐷
0.7 −0.25 𝑥1 40
[ ] [𝑥 ] = [ ]
−0.1 0.75 2 40
[𝑰𝟐 − 𝑴|𝑫]
0.7 −0.25 40
[ | ] 𝑅 /0.7 → 𝑅1
−0.1 0.75 40 1
5 400
1 − 14
[ | 7 ] 𝑅1 + 10𝑅2 → 𝑅2
−0.1 0.75 40
5 400
1 − 14 7
7
[ 50 | 3200 ] 𝑅2 × 50 → 𝑅2
0 7
7
9
5 400
1 − 14 5
[ | 7 ] 𝑅1 + 14 𝑅2 → 𝑅1
0 1 64
1 0 80
[ | ]
0 1 64
𝑥1 80
[𝑥 ] = [ ]
2 64
The agriculture industry must produce $80 million worth of output and the manufacturing
industry must produce $64 million worth of output.
Task: Find the output for each sector that is needed to satisfy a final demand of $20 billion for
coal and $10 billion for steel.
Solution:
"Production of a dollar’s worth of coal requires an input of $0.10 from the coal sector and $0.20
from steel sector." Elements of the first column.
𝐶 𝑆
𝑀 = 𝐶 [0.10 0.20] = 𝑇𝑒𝑐ℎ𝑛𝑜𝑙𝑜𝑔𝑦 𝐿𝑒𝑜𝑛𝑡𝑖𝑒𝑓 𝑀𝑎𝑡𝑟𝑖𝑥
𝑆 0.20 0.40
"Production of a dollar’s worth of steel requires an input of $0.20 from the coal sector and $0.40
from the steel sector." Elements of the second column.
"...to satisfy a final demand of $20 billion for coal and $10 billion for steel."
20
𝐷=[ ]
10
𝑥1
Goal: Find 𝑋 = [𝑥 ].
2
At the equilibrium Supply = Demand, so
Total Internal External
Output Demand Demand
𝑋 = 𝑀𝑋 + 𝐷
𝑋 − 𝑀𝑋 = 𝐷
𝐼2 𝑋 − 𝑀𝑋 = 𝐷
(𝐼2 − 𝑀)𝑋 = 𝐷
0.90 −0.20 𝑥1 20
[ ] [𝑥 ] = [ ]
−0.20 0.60 2 10
[𝑰𝟐 − 𝑴|𝑫]
Let's reduce the augmented matrix [𝐼2 − 𝑀|𝐷] to find the value of 𝑥1 and 𝑥2 .
2 200
1 −9 2
[ | 9 ] 𝑅1 + 9 𝑅2 → 𝑅1
0 1 26
1 0 28
[ | ]
0 1 26
𝑥1 28
[𝑥 ] = [ ]
2 26
The coal industry must produce $28 billion worth of output and the steel industry must produce
$26 billion worth of output.
11
Solution:
"Production of a dollar’s worth of coal requires an input of
$0.20 from the coal sector and $0.40 from the transportation
sector."
Elements of the 1st column. 0.2 0 0.4
𝑀 = [ 0 0.1 0.2]
"Production of a dollar’s worth oil requires and input of $0.10 0.4 0.2 0.2
from the oil sector and $0.20 from the transportation sector."
Elements of the 2nd column. = Technology Leontief
Matrix
"Production of a dollar’s worth of transportation requires an
input of $0.40 from the coal sector, $0.20 from the oil sector,
and $0.20 from the transportation sector."
Elements of the 3rd column.
30
𝐷 = [10]=Final
20
"...a final demand of $30 billion for coal, $10 billion for oil, demand
and $20 billion for transportation."
𝑥1
𝑋 = [𝑥2 ] ?
𝑥3
At the equilibrium Supply = Demand, so
Total Internal External
Output Demand Demand
𝑋 = 𝑀𝑋 + 𝐷
𝑋 − 𝑀𝑋 = 𝐷
𝐼3 𝑋 − 𝑀𝑋 = 𝐷
(𝐼3 − 𝑀)𝑋 = 𝐷
(𝐼3 − 𝑀)𝑋 = 𝐷
0.8 0 −0.4 𝑥1 30
[ 0 0.9 −0.2] [𝑥2 ] = [10]
−0.4 −0.2 0.8 𝑥3 20
0.8 0 −0.4 30
[𝐼3 − 𝑀|𝐷]= [ 0 0.9 −0.2| 10 ]
−0.4 −0.2 0.8 20
0.8 0 −0.4 30
[ 0 0.9 −0.2| 10 ] 𝑅1 /0.8 → 𝑅1
−0.4 −0.2 0.8 20
12
1 75
1 0 −2
2
[ 0 0.9 −0.2| 10 ] 𝑅2 /0.9 → 𝑅2
−0.4 −0.2 0.8 20
1 75
1 0 −2
2
[ 0 1 − |
2 100 ] 0.4𝑅1 + 𝑅3 → 𝑅3
9 9
−0.4 −0.2 0.8 20
1
1 0 −2 75
2 2
0 1 − 9|| 100 0.2𝑅2 + 𝑅3 → 𝑅3
3 9
[0 −0.2 5
35 ]
1 75
1 0 −2
2
2| 100 9
0 1 − | 𝑅3 × → 𝑅3
9 9 5
5 335
[0 0 9 9 ]
1 75
1 0 −2 2
2 𝑅2 + 9 𝑅3 → 𝑅2
[0 1 − |
2 100 ] 1
9 9 𝑅1 + 2 𝑅3 → 𝑅1
0 0 1 67
1 0 0 71
[0 1 0| 26 ]
0 0 1 67
𝑥1 71
[𝑥2 ] = [26]
𝑥3 67
The coal industry needs to produce $71 billion worth of production, the oil industry $26 billion
worth of production, and the transportation industry $67 billion worth of production.
(Do not forget the conclusion!)
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Exercise 25. Leontief – Three-Industry Model (Agriculture, Manufacturing and Energy)
An economy is based on three sectors, agriculture, manufacturing, and energy. Production of a
dollar’s worth of agriculture requires inputs of $0.20 from agriculture, $0.20 from
manufacturing, and $0.20 from energy. Production of a dollar’s worth of manufacturing
requires inputs of $0.40 from agriculture, $0.10 from manufacturing, and $0.10 from energy.
Production of a dollar’s worth of energy requires inputs of $0.30 from agriculture, $0.10 from
manufacturing, and $0.10 from energy.
Find the output for each sector that is needed to satisfy a final demand of $10 billion for
agriculture, $15 billion for manufacturing, and $20 billion for energy.
Task: Find the output for each sector that is needed to satisfy a final demand of $20 billion for
coal and $10 billion for steel.
"Production of a dollar’s worth of agriculture requires inputs
of $0.20 from agriculture, $0.20 from manufacturing, and
$0.20 from energy."
Elements of the 1st column.
10
"...a final demand of $10 billion for agriculture, $15 billion 𝐷 = [15] = Final
for manufacturing, and $20 billion for energy."
20
Demand
14
𝑥1
𝑥
𝑋 = [ 2 ] =?
𝑥3
𝑋 = 𝑀𝑋 + 𝐷
𝑋 − 𝑀𝑋 = 𝐷
𝐼3 𝑋 − 𝑀𝑋 = 𝐷
(𝐼3 − 𝑀)𝑋 = 𝐷
(𝐼3 − 𝑀)𝑋 = 𝐷
15
1 3 25
1 −2 −8 2
7| 175 32
0 1 − 32| 𝑅3 × 25 → 𝑅3
8
25 215
[0 0 32 8 ]
25
1 3
1 −2 −8 2 3
175 𝑅1 + 8 𝑅3 → 𝑅1
− 32|
7
0 1 8 7
172 𝑅2 + 32 𝑅3 → 𝑅_2
[0 0 1 5 ]
127
1 5
1 −2 0 147 1
0 1 0| 5
𝑅1 + 2 𝑅2 → 𝑅1
0 0 1 172
[ 5 ]
401
1 0 0 10 1 0 0 40.1
147
0 1 0| 5
= [0 1 0| 29.4 ]
0 0 1 172 0 0 1 34.4
[ 5 ]
𝑥1 40.1
𝑥
[ 2 ] = [ 29.4 ]
𝑥3 34.4
The agricultural sector has to produce $40.1 billion, the manufacturing sector has to produce $29.4
billion, and the industrial sector has to produce $34.4 billion worth of output.
(Do not forget the conclusion!)
16