Chapter 2 Systems of Linear Equations and Inequalities Table of Contents 2.1 Systems of Linear Equations 2.2 Solving Systems of Equations Using the Substitution

Chapter 2 Systems of Linear Equations andInequalities Table of Contents2.1 Systems of Linear Equations 2.2 Solving Systems of Equations Using the Substitution Method 2.3 Solving Systems of Equations Using the Elimination Method 2.4 Solving Linear Inequalities 2.5 Absolute Value Equations and Inequalities 2.6 Solving Systems of Linear Inequalities

Your company wants to print some flyers for advertising a new product. The printer has two options to produce the flyers. The traditional printing cost $250 for setup and $0.15 per page printed. To print the flyers digitally they charge $50 for setup and $0.20 per page printed.

a. Write equations for the cost to print n flyers using these two printing options.

2.1-12 Back to Table of Contents


b. Graph the two equations on the same calculator window.



c. Find the number of flyers that will result in the same cost for both printing methods.


Solve the following system by graphing the equations by hand

2.1-25

12

41

53

y x

y x

Back to Table of Contents

The percentage of renewable energy produced using geothermal and wind is given in the table. Source: US Statistical Abstract 2009.

a. Find a model for the percent of renewable energy

produced by using geothermal energy.

2.1-36

Year Geothermal Wind2003 5.37 1.792004 5.43 2.242005 5.28 2.82007 5.12 4.69



b. Find a model for the percent of renewable energy

produced by using wind.

2.1-37

Year Geothermal Wind2003 5.37 1.792004 5.43 2.242005 5.28 2.82007 5.12 4.69



c. Estimate the year in which the percentages of

geothermal and wind energy are the same.

2.1-38

( ) 0.82 1.03W t t

( ) 0.08 5.68G t t Year Geothermal Wind2003 5.37 1.792004 5.43 2.242005 5.28 2.82007 5.12 4.69


Use the table on the calculator to numerically find the solution to the system.

2.1-49

4 10

3 15

y x

y x


For each of the following systems of equations, determine if the system is consistent or inconsistent. If the system is consistent determine if the lines are independent or dependent. Give the solution to the system.

a.

2.1-510

6 4

6 2

y x

y x



b.

2.1-511

210

50.2 1

y x

y x



c.

2.1-512

16

22 4 24

y x

x y


Solve the system using the substitution method.

2.2-113

4 5

22

y x

y x


Using the models from Section 2.1 for geothermal and wind energy, find the year when the amount of geothermal and wind energy produced will be the same.

Where G(t) is the percentage of renewable energy produced using geothermal energy t years since 2000 and W(t) is the percentage of renewable energy produced using wind energy t years since 2000.

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( ) 0.08 5.68

( ) 0.82 1.03

G t t

W t t


Solve the following systems using the substitution method.

a.

2.2-315

3 11

5 16

h c

h c


Solve the following systems using the substitution method.

b.

2.2-316

2 3 12

5 4 39

w b

w b


As a new sales employee you are given two salary structures to choose from. The first option has a base salary of $1200 per month and 7% commission on sales made. The second option has a base salary of $800 per month and 9% commission on sales made.a. Find equations to represent the two salary options.


b. Find what sales amount will result in the same monthly salary for both options.

2.2-418

( ) 1200 0.07

( ) 800 0.09

F s s

S s s


When Raja retired she received a $760,000 lump sum retirement package. She wants to invest this amount in two accounts. The first account pays 4% simple interest and the second account pays 7% simple interest. Raja wants to earn $47,500 per year in interest to live on.a. Write a system of equations that will help Raja find the amount she should invest in each account.


b. How much should Raja invest in each account to earn the $47,500 she wants each year?

2.2-520

76,0000

0.04 0.07 47,500

A B

A B


Solve the following systems. Label each system as consistent or inconsistent. If the system is consistent, determine if the lines are independent or dependent.a.

2.2-621

2.3 4.7

5 11.5 23.5

d a

d a


Solve the following systems. Label each system as consistent or inconsistent. If the system is consistent, determine if the lines are independent or dependent.b.

2.2-622

5 10

2 10 20

m p

m p


Solve the system using the elimination method.

2.3-123

7 3 6

4 6 42

x y

x y


Yomaira need 70 ml of 45% saline solution for a science experiment. She has some 60% saline solution and some 20% saline solution. How much of each of these solutions should Yomaira combine to get the 70 ml of 45% saline solution she needs?


Solve the system of equations using the elimination method.

2.3-325

4 9 2

12 6 48

a b

a b


Solve the following systems of equations using the elimination method.a.

2.3-426

5 7 4

3 9 6

x y

x y


Solve the following systems of equations using the elimination method. b.

2.3-427

2 6 4

7 21 14

m n

m n


Solve the following inequalities.

a.

b.

2.4-128

6 10 40b

8 12 3 2g g


Using the printing flyers classroom example from section 2.1 we get the following

n = Number of flyers printed.T(n) = Cost in dollars to print flyers using the

traditional printing method.D (n)= Cost in dollars to print flyers using digital

printing.

Find for what numbers of flyers the traditional printing method would be cheaper than the digital prints.

2.4-329

( ) 250 0.15 ( ) 50 0.20T n n D n n


The percentage of United States population with high blood cholesterol is given in the table. Source: CDC 1991-2003 Behavioral Risk Factor Surveillance System.

a. Find equations for models of these data.

2.4-430

Year Percent of Women Percent of Men1991 25.4 24.91995 27.3 27.11999 27.6 28.32003 29.4 33


The percentage of United States population with high blood cholesterol is given in the table. Source: CDC 1991-2003 Behavioral Risk Factor Surveillance System.

b. Use the models to approximate when the percent of men with high blood cholesterol is higher than the percent of women.

2.4-431

Year Percent of Women Percent of Men1991 25.4 24.91995 27.3 27.11999 27.6 28.32003 29.4 33

( ) 0.33 25.1 ( ) 0.74 23.4W t t M t t


Solve the inequality numerically using the calculator table.

2.4-532

4 12 5 4x x


Solve the inequality graphically.

2.4-633

3 4 13x x


Solve the following equations.

a.

b.

2.5-134

7 4x

2 5 6 20x


Solve the following equations.

c.

2.5-135

4 12 8m


Estella is driving from Savannah, Georgia to Washington, D.C. Along the way she will pass through Richmond, Virginia. Estella’s distance from Richmond can be modeled by the function

where D(t) is Estella’s distance in miles from Richmond after driving for t hours. Find the time when Estella will be 40 miles from Richmond.

2.5-236

( ) 460 60D t t


Write the given interval using a compound inequality.

a.

b.


Solve the following inequalities. Give the solution as an inequality and graph the solution set on a number line.

a.

b.

2.5-438

4 6x

3 8 10x


A person with mild high blood pressure will have a systolic pressure (first number) that satisfies the inequality

where S is a person’s systolic pressure in millimeters of mercury (mmHg ).Find the range of systolic pressures that people with mild high blood pressure will have.

2.5-539

150 10S


Write the given interval using a compound inequality.


Solve the following inequalities. Give the solution as an inequality. Graph the solution set on a number line.

a.

b.

2.5-741

3 5x

2 4 9x


A person with high or low blood pressure will have a diastolic pressure (second number) that satisfies the inequality

where D is a person’s diastolic pressure in millimeters of mercury (mmHg).Find the diastolic blood pressures that would be considered high or low.

2.5-842

75 15D


A yard equipment manufacturer produces lawn mowers and roto-tillers in the same plant. Each mower requires 4 hours to produce while roto-tillers require 7 hours to produce. If the plant operates 70 hours per week, what combinations of mowers and roto-tillers can they produce in a week?


Graph the following inequalities by hand:

a.

2.6-244

12

3y x



b.

2.6-245

24

3y x



c.

2.6-246

2 5x y


Find the inequality for the given graph.


The yard equipment manufacturer we discussed with example 1 had a time constraint that they could not exceed 70 hours of production per week. Each mower requires 4 hours to produce while roto-tillers require 7 hours to produce. The same plant must stay in production at least 40 hours per week.a. Create a system of inequalities to model this situation.

2.6-448

4 7 70M R


The yard equipment manufacturer we discussed with example 1 had a time constraint that they could not exceed 70 hours of production per week. The same plant must stay in production at least 40 hours per week.

b. Graph the solution set for this system.

2.6-449

4 7 70

4 7 40

M R

M R


The yard equipment manufacturer we discussed with example 1 had a time constraint that they could not exceed 70 hours of production per week. The same plant must stay in production at least 40 hours per week.

c. Can the plant produce 8 mowers and 4 roto-tillers in a week?

2.6-450

4 7 70

4 7 40

M R

M R


Kian has $400,000 from a court settlement to invest. Kian plans to invest the money into two accounts, one paying 3% simple interest and the other paying 4% simple interest. Kian would like to earn at least $14,300 per year to support himself while going back to college. How much should he invest in each account to earn at least $14,300 per year?


Graph each system of inequalities by hand.

a.

2.6-652

3 4 12

2 1

x y

y x


Graph each system of inequalities by hand.

b.

2.6-653

1.2 2

65

5

y x

y x


Graph the system of inequalities using a graphing calculator.

2.6-754

3 4

2 1

y x

y x


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Chapter 2 Systems of Linear Equations and Inequalities Table of Contents 2.1 Systems of Linear Equations 2.2 Solving Systems of Equations Using the Substitution