more with systems of equations

645

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645

More with Systems of Equations

12.1 Systems of EquationsUsing Linear Combinations to Solve

a Linear System..........................................................647

12.2 What’s for Lunch?Solving More Systems .................................................655

12.3 Making DecisionsUsing the Best Method to Solve

a Linear System..........................................................663

12.4 Going GreenUsing a Graphing Calculator to

Solve Linear Systems .................................................669

12.5 Beyond the Point of IntersectionUsing a Graphing Calculator to Analyze

a System ....................................................................675

In 2008, 4.7

million Americans went on a rafting

expedition. In Georgia, outfitters run

whitewater expeditions for ages 8 and up on the Chattooga

River.

646 • Chapter 12 More with Systems of Equations

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12.1 Using Linear Combinations to Solve a Linear System • 647

Key Term linear combinations

method (elimination)

Learning GoalsIn this lesson, you will:

Write a system of equations to represent a problem

context.

Solve a system of equations algebraically using linear

combinations (elimination).

Morse code is a communication system which allows people to “speak”

with sound. Words are transmitted using short sounds called “dits,” which

are represented in writing as dots, and long sounds, called “dahs,” which are

represented in writing as dashes. The letters of the alphabet and digits each have

their own unique collection of dits and dahs:

UVWXYZ

ABCDEFGHIJKLMNO

QP

RST

1234567890

When you combine these codes, you can produce sentences in Morse code.

Try it out! Communicate with your friends using Morse code.

Systems of EquationsUsing Linear Combinations to Solve a Linear System


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Problem 1 End of Year Trip

There are a total of 324 students enrolled in Armstrong Elementary School. The girls

outnumber the boys by 34.

1. Write an equation in standard form that represents the total number of students at

Armstrong Elementary School. Use x to represent the number of girls, and use y to

represent the number of boys.

2. Write an equation in standard form to represent the number of girls in relationship to

the number of boys.

3. How are these equations the same?

4. How are the equations different?

5. Complete parts (a) through (f) to write and solve a linear system of equations for

this situation.

a. Write a linear system for this problem situation.

b. Add the two equations together.

c. Solve the resulting equation.

d. Substitute the value of x that you obtained in part (c) into one

of the original equations and solve to determine

the value of y.

I see how you

can add equations. (4 + 2) = 6 (4 - 2) = 2

So, if I can add 6 and 2 and get 8, then that means I can add (4 + 2) and (4 - 2)

and get 8 also. So, (4 + 2) + (4 - 2) = 8.

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e. What is the solution of your linear system?

f. Check your solution algebraically.

6. Check your solution by creating a graph of your linear system on the coordinate plane

shown. Choose your variables, bounds, and intervals. Be sure to label your graph.

Variable Quantity Lower Bound Upper Bound Interval

x

y

Num

ber

of B

oys

at A

rmst

rong

Ele

men

tary

Number of Girls at Armstrong Elementary

7. Interpret the solution of the linear system in this problem situation.

8. What effect did adding the equations together have?

9. Describe how the coefficients of y in the original system are related.



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Problem 2 Hotel Promotion

The Splash and Stay Resort has an indoor water park, an arcade, a lounge, and

restaurants. The resort is offering two winter specials. The first special is two nights and

four meals for $270, while the second is three nights and eight meals for $435.

1. Write an equation in standard form that represents the first resort special. Let n

represent the cost for one night at the resort, and let m represent the cost for

each meal.

2. Write an equation in standard form that represents the second resort special.

3. How are these equations the same?

4. How are these equations different?

5. Multiply each side of the equation that represents

the first resort special by 22. Simplify the equation;

maintain standard form.

If I multiply both sides of an

equation by the same number, is the equation

still true?

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6. Write a linear system of equations using the transformed equation that represents the

first special and the equation that represents the second special.

a. How do the coefficients of the equations in your linear system of equations compare?

b. Add the equations in your linear system together. Then, simplify the result.

What does the result represent?

c. How will you determine the m value of the linear system?

d. Determine the value of m for the linear system.

e. What is the solution of the linear system?

f. Interpret the solution of the linear system in the problem situation.

g. Check your solution algebraically.


When you divide a negative value by -1, you make it positive.


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Problem 3 Linear Combinations

The method you used to solve the linear systems in Problems 1 and 2 is called the linear

combinations method. The linearcombinationsmethod is a process to solve a system

of equations by adding two equations to each other, resulting in an equation with one

variable. You can then determine the value of one variable and use it to find the value of

the other variable.

In many cases, one, or both, of the equations in the system must be

multiplied by a constant so that when the equations are added together,

the result is an equation in one variable. This means that the coefficients of

either the term containing x or y must be opposites.

For example, consider this system:

  4x 1 2y 5 3

5x 1 3y 5 4  

You can multiply the first equation by 3 and the second equation by 22 to

produce opposite coefficients for y that will eliminate each other.

Alternatively, you can multiply the first equation by 23 and the second

equation by 2.

1. Multiply the first equation by 3 and multiply the second equation by 22. Then, rewrite

each equation.

2. Solve the new linear system. Show your work.Did you check

your solution by substituting the

ordered pair back into the original equations?

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3. For each linear system shown, describe the first step you would take to solve the

system using the linear combination method. Identify the variable that will be solved

for when you add the equations.

a. 5x 1 2y 5 10 and 3x 1 2y 5 6

b. x 1 3y 5 15 and 5x 1 2y 5 7

c. 4x 1 3y 5 12 and 3x 1 2y 5 4



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4. Solve each system using linear combinations.

a.   2x 1 y 5 8

3x 2 y 5 7  

b.   4x 1 3y 5 24

3x 1 y 5 22  

c.   3x 1 5y 5 17

2x 1 3y 5 11  

Be prepared to share your solutions and methods.

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12.2 Solving More Systems • 655


Write a linear system of equations to represent a problem context.

Choose the best method to solve a linear system of equations.

What’s for Lunch?Solving More Systems

Suppose one cell phone company charges $0.10 per minute for phone calls.

Another company charges $60 per month for unlimited call time. Can you write a

system of equations to compare the two plans? Which one would be best for you?

Which one would be best for your parents?


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Problem 1 Fundraising

One day each month the Family and Consumer Science classes offer a deli lunch for the

faculty and staff of the school to purchase. The staff has a choice of either a chef salad for

$5.75 or a hoagie for $5. Today the Family and Consumer Science classes sold 85 lunches

for a total of $464. Determine how many chef salads and hoagies were sold.

1. Write an equation in standard form that gives the total number of lunches in terms of

the number of chef salads sold and the number of hoagies sold. Let x represent the

number of chef salads sold, and let y represent the number of hoagies sold.

2. Write an equation in standard form that represents the amount of money collected.

Use the same variables as those used in Question 1.

3. Write a system of linear equations to represent this problem situation.

4. What methods can you use to solve this system of linear

equations?Think about all

the strategies you have used in previous

lessons.

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5. Determine the solution of this linear system of equations by using linear combinations

and check your answer.

6. Interpret your solution to the linear system in terms of the problem situation.

Problem 2 More Fundraising

The Jewelry Club is making friendship bracelets with the school colors to sell in the school

store. The bracelets are black and orange and come in two lengths: 5 inches and 7 inches.

The club has enough beads to make a total of 84 bracelets. They have made 49 bracelets,

which represents 1 __ 2 the number of 5-inch bracelets plus 3 __

4 the number of 7-inch bracelets

they plan to make and sell.

1. Write an equation in standard form to represent the total number of bracelets the

Jewelry Club can make out of the beads that they have. Let x represent the number of

5-inch bracelets, and let y represent the number of 7-inch bracelets.

2. Write an equation in standard form to represent the number of bracelets the Jewelry

Club has made so far. Use the same variables as those used in Question 1.



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3. Write a system of linear equations that represents the problem situation.

4. Karyn says that the first step she would take to solve this system would be to

first multiply the second equation by the least common denominator (LCD) of the

fractions. Is she correct? Explain your reasoning.

5. Rewrite the equation containing fractions as an equivalent equation without fractions.

6. Determine the solution to the system of equations by using linear combinations and

check your answer.

7. Interpret the solution of the linear system in terms of the problem situation.

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8. Solve each linear system using linear combinations. Check all solutions.

a.   x 1 2y 5 2

5x 2 3y 5 229  

b.   1 __ 2

x 1 1 __ 3

y 5 3

3x 1 5y 5 36  



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c.   0.6x 1 0.2y 5 2.2

0.5x 2 0.2y 5 1.1  

d.   1 __ 2

x 1 3 __ 5

y 5 17

1 __ 5

x 1 3 __ 4

y 5 17

 

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Talk the Talk

You have used three different methods for solving systems of equations: graphing,

substitution, and linear combinations.

1. Describe how to use each method and the characteristics of the system that makes

this method most appropriate.

a. Graphing Method:

b. Substitution Method:

c. Linear Combinations Method:



Which method do

you like best?


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12.3 Using the Best Method to Solve a Linear System • 663

Making DecisionsUsing the Best Method to Solve a Linear System



Choose the best method to solve a linear system of equations using linear combinations.

Problem 1 Roller Skating, Here We Come!

The activities director of the Community Center is planning a skating event for all the

students at the local middle school. There are several skating rinks in the area, but the

director does not know which one to use. Skate Fest charges a fee of $200 plus $3 per

skater, while Roller Rama charges $5 per skater.

1. Write an equation that gives the total cost of renting Skate Fest for the skating event

in terms of the number of students attending. Define your variables.

2. Write an equation that gives the total cost of renting Roller Rama for the skating event

in terms of the number of students attending. Use the same variables you used in

Question 1.

3. Suppose the activities director anticipates that 50 students will attend.

a. Calculate the total cost of using Skate Fest.

b. Calculate the total cost of using Roller Rama.


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4. Suppose the activities director has $650 to spend on the skating event.

a. Determine the number of students who can attend if the event is held at

Skate Fest.

b. Determine the number of students who can attend if the event is held at

Roller Rama.

5. Write a system of equations to represent this problem situation.

6. When is the cost of each skating rink the same? Use an algebraic method to explain

your reasoning.

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7. Which algebraic method did you use in Question 6? Explain your reasoning.

8. Complete the table of values to show the cost for different numbers of students

attending the event at each rink.

Quantity Name

Number of Students

Skate Fest Roller Rama

Unit

Expression x

0

25

75

150

200

300

9. Is your solution confirmed by the table? Explain your reasoning.


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10. Check your solution by creating a graph of your system of equations. First, choose

your bounds and intervals. Be sure to label the graph.

Variable Quantity Lower Bound Upper Bound Interval

11. Is your solution confirmed by your graph?

12. Which skating rink would you recommend to the activities

director? Explain your reasoning.

A graph is sometimes not as

exact as I'd like it to be.

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Problem 2 Another Consideration

Super Skates offers the use of the rink for a flat fee of $1000 for an unlimited number

of skaters.

1. Write a linear equation to represent this situation. Then, add the graph of this equation

to the grid in Problem 1, Question 10.

2. Describe when going to Super Skates is a better option than going to Skate Fest or

Roller Rama.

3. The activities director’s final budget is $895, and she has chosen to host the event at

Skate Fest.

a. Write an inequality that represents this situation.

b. Solve the inequality.

c. What does this solution mean in terms of the problem situation?



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12.4 Using a Graphing Calculator to Solve Linear Systems • 669

Learning GoalIn this lesson, you will:


Going GreenUsing a Graphing Calculator to Solve Linear Systems

The world's first graphing calculator was introduced in October of 1985.

This graphing calculator contained 422 bytes of memory and could calculate

to a precision of 13 digits.

Some of the graphing calculators available in 2011 can show even

three-dimensional graphs and have 2.5 megabytes of data—over 2.5 million

bytes! That's over 6000 times more memory than the first graphing calculator!


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Problem 1 When HEVs Take Over the Market

In 2008, about 16,500 hybrid cars were sold. In 2009, about 20,000 hybrids were sold.

1. What is the rate of change in the number of hybrids sold per year from 2008 to 2009?

2. Write an equation that gives the total number of hybrid cars sold since 2008. Assume

that the rate of change in sales is constant. Define the variables.

3. In 2008, 13.2 million conventional autos were sold in the United States. In 2009, 11.4

million conventional autos were sold. What is the rate of change in the number of

conventional automobiles sold in the United States per year from 2008 to 2009?

4. Write an equation that gives the total number of conventional cars sold since 2008.

Assume that the rate of change in sales is constant. Define the variables.

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5. Write a system of linear equations that represents the total sale of hybrid cars and the

total sale of conventional automobiles since 2008.

6. If these trends were to continue, could the hybrid sales ever equal the conventional

auto sales? Use what you know about the equations of a linear system to explain

your answer.

Problem 2 Using a Graphing Calculator

There are many ways to use a graphing calculator to solve systems. You will complete a

table of values and then graph the system of equations.

1. Follow the steps shown to use the table features of your graphing calculator.

Step 1: Press Y5 and enter the system of equations.

y1 5 16,500 1 3500x

y2 5 13,200,000 2 1,800,000x



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Step 2: Set up the table. Press TBLSET (press

2nd and press WINDOW ).

TblStart is the start of your table. The

range of your independent values will be

1 to 20, so enter TblStart50.

Tbl is the increment (read “delta table”).

This value tells the table how to count.

Enter Tbl51. Then, the independent

values in your table will appear as 1, 2, 3, and so

on.

Step 3: Press TABLE (press 2nd GRAPH ).

Step 4: Use the down arrow keys to scroll through the table.

Notice that the dependent values are in scientific

notation.

a. Complete the table of values that shows the sales of hybrid cars

and conventional cars for different numbers of years.

Quantity Name

Years Since 2008

Hybrid Sales

Conventional Cars

Unit Year Cars Cars

Expression x

1

5

7

15

20

Be sure

the “"Auto"” option is selected for

both “"Indpnt"” and “"Depend"” in the TABLE SETUP.

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b. Determine whether the sales of hybrids will ever equal the sales of conventional

cars. Explain your reasoning by using the table of values.

2. Follow the steps shown to graph the system of equations using your graphing

calculator and to determine the point of intersection.

Step 1: Press Y5 and enter

y1 5 16,500 1 3500x

y2 5 13,200,000 2 1,800,000x

Step 2: Press WINDOW to set the bounds and intervals for the graph. You can use

the table of values you completed in Question 1 to determine how to set the

window.

Xmin 5 0, Xmax 5 20, Xscl 5 5, Ymin 5 0,

Ymax 5 600,000, and Yscl 5 100,000.

Step 3: Press GRAPH . The intersection of both lines should be visible. If it is

not, go back and adjust the WINDOW settings.

Step 4: Determine the point of intersection; you must be able to view the

intersection point. Press CALC (Press 2ND TRACE ). Press

5 or select 5 :intersect. Move the cursor toward the intersection

point and press ENTER three times. The point of intersection will be

displayed at the bottom of your screen.

a. What is the point of intersection?

b. What does this point of intersection mean in terms of the problem situation?



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c. In what year would the sale of hybrids equal the sale of conventional cars?

d. What can you predict about the sales of conventional cars from your graph?

Talk the Talk

Solve each system using a graphing calculator.

1.   y 5 1 __ 2

x 1 9

y 5 x __ 2

1 1 __ 2

 

2.   6x 1 3y 5 7

3x 1 2y 5 7  

3.   6x 2 8y 5 16

3x 2 4y 5 8  

4.   y 5 21.57x 2 12.4

y 5 3.65x 1 19.6  


What predictions

can you make about each solution by analyzing the equations first?

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12.5 Using a Graphing Calculator to Analyze a System • 675



Use a graphing calculator to solve a linear system of equations.

Beyond the Point of IntersectionUsing a Graphing Calculator to Analyze a System

Problem 1 What’s the Count?

Cinemaplex, the local movie theater, had a blockbuster weekend! On Friday, 218 people

attended the matinee and 753 people attended the evening shows, bringing in a total of

$8620. Saturday was even more profitable. There were 847 people who attended

the matinee while 1215 people attended the evening movies, bringing in sales of $16,385.

1. Write an equation in standard form that represents each night. Define x as the price of

matinee movies, and define y as the price of evening movies.

2. Without solving, interpret the solution to the linear system of equations.

3. Determine the solution of the linear system using your graphing calculator.

a. Rewrite each equation by solving for y. Do not perform the division.

Friday: y1 5

Saturday: y2 5

b. Enter the equations into your graphing calculator.


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c. What is the point of solution? Explain how you know.

d. Interpret the solution of this linear system of equations.

Problem 2 Job Offers

1. Alex is applying for positions at two different electronic stores in neighboring towns.

The first job offer is a $200 weekly salary plus 5% commission on sales. The second

job offer is a $75 weekly salary plus 10% commission.

a. Write a system of equations that represents the problem situation. Describe

the variables.

b. Without solving the system of linear equations, interpret the solution.

c. Solve the system of equations using your graphing calculator.

d. Interpret the solution of the system in terms of the problem situation.

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12.5 Using a Graphing Calculator to Analyze a System • 677

It is important to understand the point of intersection in this situation. Let’s use the

graphing calculator to analyze beyond the point of intersection.

2. Determine values for each graph by using the TRACE function on your graphing

calculator. You may have to adjust the WINDOW .

a. Determine how much money Alex would earn at the first store if he sold $3000.

First, set Xmax 5 4000 and Ymax 5 450.

Press GRAPH .

Then, press TRACE .

Move the cursor to y1.

Enter 3000 and press ENTER .

b. Determine how much money Alex would earn at the second store if he sold $3000.

Move the cursor to y2.

Enter 3000 and press ENTER .

c. What is the difference in the weekly pay between stores if Alex sells $3000?

3. What is the difference in pay if he sold $4225 weekly?

Alex’s sales targets for each job would be between $1500 and $3000 weekly. Each

manager told Alex the same thing: “Some weeks are better than others, depending on the

time of year and the new releases of technology.”

4. Which job offer would you recommend Alex take? Explain your reasoning.


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Look at the structure of each system before you choose your solution

method.

Talk the Talk

Solve each linear system using the substitution method,

linear combinations, or a graphing calculator.

1.   y 5 5x 1 12

y 5 9x 2 4

 

2.   15x 1 3y 5 30

8x 2 3y 5 16  

3.   4y 5 11 2 3x

3x 1 2y 5 25  

4.   15x 1 28y 5 417

31x 1 23y 5 548  


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Chapter 12 Summary • 679

Chapter 12 Summary

Key Term linear combinations method (elimination) (12.1)

Solving a System of Equations Using the Linear Combinations Method

Thelinear combinations method is a process used to solve a system of equations by

adding two equations to each other so that they result in an equation with one variable.

Then, you can determine the value of one variable and use it to determine the value of the

other variable. In many cases, you may need to multiply one or both of the equations in

the system by a constant so that when the equations are added together, the result is an

equation in one variable. This means that the coefficients of the term containing either x or

y must be opposites.

Example

5x 1 2y 5 16 and 2x 1 6y 5 22

23(5x 1 2y) 5 23(16) 215x 2 6y 5 248 5x 1 2y 5 16

2x 1 6y 5 22 2x 1 6y 5 22 5(2) 1 2y 5 16

213x 5 226 10 1 2y 5 16

213x _____ 213

5 226 _____ 213

2y 5 6

x 5 2 y 5 3

Check:

5(2) 1 2(3) 16 2(2) 1 6(3) 22

10 1 6 16 4 1 18 22

16 5 16 22 5 22

The solution is (2, 3).

Nice work! Keep it up!


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Example

1 __ 2

x 1 1 __ 3 y 5 7

→ 12 ( 1 __

2 x 2 1 __

3 y 5 7 ) →

6x 1 4y 5 84

1 __ 4 x 2 1 __

9 y 5 1 36 ( 1 __

4 x 2 1 __

9 y 5 1 ) 9x 2 4y 5 36

6x 1 4y 5 84

6x 1 4y 5 84 6(8) 1 4y 5 84

9x 2 4y 5 36 48 1 4y 5 84

15x 5 120 4y 5 36

15x ____ 15

5 120 ____ 15

4y

___ 4

5 36 ___ 4

x 5 8 y 5 9

Check:

1 __ 2

(8) 1 1 __ 3

(9) 7 1 __ 4

(8) 2 1 __ 9

(9) 1

4 1 3 7 2 2 1 1

7 5 7 1 5 1


Writing a Linear System of Equations to Represent a Problem Context

When two or more linear equations define a relationship between quantities, they form a

system of linear equations. Use the data in the problem to write two related equations.

Then, evaluate the equations by substituting the given value for x or y to determine the

value of the other variable. Interpret the solution in terms of the problem context.

Example

Ling needs to print flyers. Printer A will charge $30 plus $0.50 per flyer. Printer B will

charge $15 plus $1 per flyer. Let x represent the cost per flyer and let y represent the total

cost of the order.

Printer A: y 5 30 1 0.5x

Printer B: y 5 15 1 x

To determine how much each printer will charge to print 300 flyers, you can use

substitution and evaluate each equation.

Printer A: 30 1 0.5(300) 5 180

Printer B: 15 1 300 5 315

Printer A will charge $180 for 300 flyers, and Printer B will charge $315 for 300 flyers.

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Choosing the Best Method to Solve a Linear System of Equations

Use graphing or substitution to determine the solution of a linear system of equations.

Either graph the equations to determine their point of intersection, or use substitution

to set the equations equal to one another. These methods work well when y is the same

for both equations and the equations are in slope-intercept form. Interpret the solution in

terms of the problem context.

Example

y 5 2x 2 5 and y 5 x 1 1

Substitution: Graphing:

2x 2 5 5 x 1 1

x16

12

16

188

8

124

4

146 10200

y

18

10

14

6

2

x 5 6

y 5 6 1 1

y 5 7


Writing and Solving an Inequality to Represent a Problem Context

When a problem situation defines an upper or lower limit for the value of y, such as a

budget, write the linear equation as an inequality rather than an equality. Solve as usual

and interpret the solution in terms of the problem context.

Example

Kyle belongs to a DVD club. He pays $14 per month plus $2 per DVD. If he sets a budget

for himself of $30 per month, you can determine how many DVDs he can buy each month.

14 1 2 x 30

2 x 16

x 8

Kyle can buy up to eight DVDs each month.



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Calculate the slope of each equation by determining the rate of change for each situation.

Example

Two cars are depreciating at different rates. Car A’s value went from $28,000 when it was

sold new in 2007 to $20,000 in 2009. Car B’s value went from $34,000 when it was sold

new in 2007 to $30,000 in 2009.

Car A: 20,000 2 28,000 ________________ 2009 2 2007

5 28000 _______ 2

5 24000; y 5 28,000 2 4000x

Car B: 30,000 2 34,000 ________________ 2009 2 2007

5 24000 _______ 2

5 22000; y 5 34,000 2 2000x


When the coefficients in a system of equations are not convenient for using substitution or

linear combinations, you may need to use a graphing calculator to solve. Follow the steps

to graph the system of equations using your graphing calculator, and then determine the

point of intersection.

Example

  13x 1 5y 5 25.6

17x 2 19y 5 82  

Step 1: Press Y5 and enter

y1 5 2 13 ___ 5 x 1 25.6 _____ 5

y2 5 17 ___ 19

x 2 82 ___ 19

.

Step 2: Press WINDOW to set the bounds and intervals for the graph.

Step 3: Press GRAPH . The intersection of both lines should be visible. If it is not, go

back and adjust the WINDOW settings.

Step 4: Determine the point of intersection; you must be able to view the intersection

point. Press CALC . Press 5 or select 5: intersect.

Move the cursor toward the intersection point and press ENTER three times.

The point of intersection will be displayed at the bottom of your screen.

The solution is (2.7, 21.9).

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Find the slope of each equation by determining the rate of change for each situation.

Example

Carlos and Carrena work as waiters. Carlos earns $6 per hour plus an average of 20% of

his total bills in tips. Carrena earns $5 per hour plus an average of 23% of her total bills in

tips. Last Saturday, they worked the same number of hours and had the same total bills.

Carlos earned $322 and Carrena earned $357.

6x 1 0.2y 5 322

5x 1 0.23y 5 357

Using a Graphing Calculator to Solve a Linear System of Equations

When the coefficients in a system of equations are not convenient for using substitution or

linear combinations, you may need to use a graphing calculator to solve. Follow the steps

to graph the system of equations using your graphing calculator and determine the point

of intersection. Input the values greater than and less than the solution to interpret the

problem situation.

Example

A Movers: y 5 0.64x 1 250 y is the total spent on a moving van, and x is the

number of miles you need to drive.

B Movers: y 5 0.35x 1 453


Try 100 miles more or less than the solution.

0.64(600) 1 250 5 634; 0.35(600) 1 453 5 663

0.64(800) 1 250 5 762; 0.35(800) 1 453 5 733

If you are moving more than 700 miles away, B Movers is the better choice. If you are

moving less than 700 miles away, A Movers is the better choice. If you are moving exactly

700 miles, there is no difference between companies.



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Use substitution, linear combinations, or a graphing calculator to determine the solution

of a linear system of equations. Use substitution to set the equations equal to one another

when y is the same for both equations and the equations are in slope-intercept form.

Interpret the solution in terms of the problem context. Use linear combinations when

the coefficients of like terms are opposites or can be easily made into opposites by

multiplication. Use a graphing calculator when the coefficients in a system of equations

are complex numbers or are difficult to work with using other methods.

Example

a.   y 5 4x 1 5

3x 1 2y 5 43  

Use substitution because one equation is in slope-intercept form.

3x 1 2(4x 1 5) 5 43

3x 1 8x 1 10 5 43 y 5 4(3) 1 5

11x 5 33 y 5 17

x 5 3


b.   2x 1 6y 5 14

4x 2 6y 5 10  

Use linear combinations because the y-coefficients of both equations are opposites.

2x 1 6y 5 14 2(4) 1 6y 5 14

4x 2 6y 5 10 8 1 6y 5 14

6x 5 24 6y 5 6

x 5 4 y 5 1


c.   13x 1 5y 5 11

3x 2 16y 5 54  

Use a graphing calculator because other methods are not easily used.


more with systems of equations

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