130

3-51. Plan

130 Chapter 3 Real Numbers and the Coordinate Plane

3-5

New Vocabulary solution, linear equation

To use tables, equations, and graphs to solve problems

What You’ll Learn

Equations, Tables, and Graphs

For help making a table of data, go to Lesson 1-1, Example 4.

1. Vocabulary Review What do you call a symbol that stands for one or more numbers?

Evaluate for

2.

3.

4.

Lesson 1-1

a � 4.

6 21a �

13 2� a

5 8a �

variable

3

21

28

See back of book.

Why Learn This?You can use equations, tables, and graphs

to represent the same data. For example, you

can use a table of values for plant growth

to write an equation or make a graph.

Given a word problem, you can sometimes

make a table of data. Then you can write

an equation to model the situation.

Making Tables and Writing Equations

Suppose you save $3 each week. Make a table and write an equation to

represent your total savings after a given number of weeks.

The equation models your total savings.

1. You buy CDs from a music store. Each CD costs $15. Make a table

and write an equation to represent the total cost of buying a given

number of CDs.

Total Savings(dollars) Expression

0

3

6

9

t

3(0)

3(1)

3(2)

3(3)

3(w)

Numberof Weeks

0

1

2

3

w

Look for a pattern in thetable. Your total savings for a given week is 3 times the number of weeks you havebeen saving.

Let t representyour total savings.

Let w represent thenumber of weeks.

t w� 3

Objective

To use tables, equations, and graphs to solve problems

Examples

1

Making Tables and Writing Equations

2

Graphing Linear Equations

Math Understandings:

p. 104C

Math Background

The solutions of an equation in two variables, such as can be graphed on a coordinate plane as a line formed by an infinite set of ordered pairs. Tables can be used to represent a set of

x

- and

y

-coordinates that satisfy the equation. The set of

x

-values is known as the domain and the set of corresponding

y

-values is known as the range. In real-world linear situations, the domain and range must be reasonable values.

More Math Background:

p. 104C

Lesson Planning and

Resources

See p. 104E for a list of the resources that support this lesson.

y x� � 5,

Bell Ringer Practice

Check Skills You’ll Need

Use student page, transparency, or PowerPoint. For intervention, direct students to:

Algebraic Expressions and Order of Operations

Lesson 1-1Extra Skills and Word Problems

Practice, Ch. 1

Special Needs

Students work in pairs. Match those who have difficulty graphing equations on the grids with those who can do so more easily. The first student can make the data tables while the partner can graph the equations. Then partners can check each other’s work.

L1

learning style: visual

Below Level

To help students remember the order of a coordinate pair, have them write

x

over the first value and

y

over the second value.

L2

learning style: visual

phm07c3_te_0305.fm Page 130 Thursday, May 25, 2006 3:56 PM

131

3-5 Equations, Tables, and Graphs 131

Any ordered pair that makes an equation true is a solution of the

equation. For example, (2, 6) is a solution of because

An equation with two variables can have many solutions. You can show

these solutions on a graph. An equation is a linear equation if all of its

solutions lie on a line.

Graphing Linear Equations

Jerrod keeps track of how much dry food is in his cat’s feeder. Graph the

linear equation where y represents the cups of food left

and x represents the number of days since he filled the twelve-cup feeder.

Step 1 Make a table. Step 2 Graph the ordered pairs and draw

a line through the points.

y x� 3 6 3 2� ( ).

y x� � �12 12,

x

0

y � � x � 1212

� (0) � 12 � 1212

� (4) � 12 � 1012

� (9) � 12 � 712

12

� (16) � 12 � 412

4

9

164

4

0 8Days Passed

Cu

ps

of

Fo

od

8

12

12

16

y

x

Each point (x, y) on the graph represents a solution of the equation. For

example, the point (4, 10) means that after 4 days, 10 cups are left.

2. Graph the linear equation where y represents the

temperature in of a chemical solution after x minutes.

y x� �5 50,

�F

A plant is 4 cm tall and grows 2 cm per day. Predict how tall the plant

will be after 8 days.

Roberto’s Method

I can make a table of data.

5

14

6

16

8

20

7

18

0

4

Days Passed

Height

1

6

2

8

3

10

4

12

Height of Plant

After 8 days, the plant will be 20 cm tall.

VVocabulary Tipocabulary TipThe term linear means “relating to a line.”

See back of book.

Activity Lab

Use before the lesson.

Student Edition

Activity Lab, Data Analysis 3-5a, Tables and Graphs, p. 129

Teaching Resources

Activity Lab 3-5:

Hidden Equation

Additional Examples

Suppose you buy a bag of food for your pet dog every week. Dog food costs $4 per bag. Make a table and write an equation to represent the total cost of buying dog food for any number of weeks.

Graph the linear equation where

y

represents the pressure inside a deflating balloon after

x

seconds.

c w� 4

Numberof Weeks

Cost ofDog Food

1

4

2

8

3

12

4

16

w

c

y x� � � 3,

Seconds

Pressure

0

3

1

2

2

1

3

0

2

Pre

ssur

e 3

1

0 321Seconds

Advanced Learners

The equation represents the height of water in a bottle pouring out over time. Do all of its solutions apply to the graph of the equation?

no

Which of the solutions to the equation should be shown on the graphed line?

Sample: the positive values, including zero

L4y x� �6 3

learning style: verbal

English Language Learners

Students may be comfortable mentally calculating the answer for the

Choose a Method

example since the numbers are small integers. Stress that the tables and graphs also allow them to make predictions, whether the numbers are easy or hard to mentally calculate.

learning style: verbal

2. Teach

phm07c3_te_0305.fm Page 131 Thursday, May 25, 2006 3:56 PM

132

132 Chapter 3 Real Numbers and the Coordinate Plane

Jasmine’s Method

I can make a graph. Let x represent the number of days that have

passed. Let y represent the height of the plant.

I can make a table of solutions. Three

points on the graph are (1, 6), (2, 8),

and (3, 10).Days Passed (x)

Height (y)

1

6

2

8

3

10

Height of Plant

I can draw a line through the points.

Then I can use the graph to find the

height y when

After 8 days, the plant will be

20 cm tall.

0

8

16

24

2 4 6 8Days Passed

He

igh

t (c

m)

y

x

x � 8.

Choose a MethodA bag of rice weighs 80 oz. If a serving of rice is 2 oz, how much rice

will be left after you prepare 10 servings? Explain why you chose the

method you used.

Check Your UnderstandingCheck Your Understanding

1. Vocabulary Which statement about linear equations is not true?

The graph of a linear equation is a line.

Every point on the graph of a linear equation is a solution.

A point that does not lie on the graph of a linear equation may

still be a solution of the equation.

You can write solutions of a linear equation as ordered pairs.

2. A leaky pipe loses 0.75 gallons of water every minute. Complete the

data table below.

3. Use the table from Exercise 2. Write a linear equation to represent

the amount of water lost from the leaky pipe.

4. Suppose you wanted to graph the equation in Exercise 3. Use the

table from Exercise 2 to name four points that lie on the graph.

Water Loss

1 2 3 4Number of Minutes (t)

Gallons of Water Lost (g) ■ ■ ■ ■

60 oz; check students’ methods.

C

See margin.

g t� 0 75.

4. (1, 0.75), (2, 1.5), (3, 2.25), (4, 3)

Guided Instruction

Example 1

Ask:•

How do the

x

-values change as you move down the table?

increase by 1

•

How do the

y

-values change?

increase by 3

•

If this pattern continues, what would be the next ordered pair in the table?

(4, 12)

Error Prevention!

In a real-world situation, students need to consider not only coordinates that are solutions to the equation, but values that are reasonable for the situation. Ask:

Why would you

not

choose negative values for x in Example 2?

There cannot be a negative number of days.

Teaching Resources

• Daily Notetaking Guide 3-5• Adapted Notetaking 3-5

L3L1

Closure

•

How do you graph a linear equation in two variables?

Sample: Make a table of values by choosing several

x

-values and substituting them into the equation and simplifying. Graph the resulting ordered pairs and connect them in a straight line.

•

In many cases when a linear equation represents a real-world situation, such as height vs. age, which values cannot apply to the situation? In which quadrant will the graph be shown?

negative nonzero

x

- and

y

-values; first quadrant

2.

Number of Minutes (t) 1 2 3 4

Gallons of Water Lost (g) 0.75 1.5 2.25 3

Water Loss

phm07c3_te_0305.fm Page 132 Thursday, May 25, 2006 3:56 PM

133

3-5 Equations, Tables, and Graphs 133

Homework ExercisesHomework ExercisesFor more exercises, see Extra Skills and Word Problems.

5. In 2000, about four babies were born world-wide every second.

Make a table and write an equation to represent the total number of

babies born over time.

6. The temperature drops every hour. Make a table and write an

equation to represent the total temperature drop over time.

7. For a certain repair, an auto shop charges a $20 fee for materials

plus $40 per hour for labor. Graph the linear equation

where y represents the total cost and x represents

the hours of labor.

8. On a 100-point test, each question is worth 5 points. Partial answers

receive partial credit. Graph the linear equation

where y represents your score and x represents the number of

incorrect answers.

9. Guided Problem Solving In the design at

the right, 12 squares surround a row of

3 circles. Predict the number of squares

needed to surround a row of 10 circles.

• Make a Plan Make a table of values.

Graph the ordered pairs from the table and draw a line through

the points. Then use the graph to find the answer.

• Carry Out the Plan Complete the table below.

Graph each linear equation.

10. 11. 12.

13. Writing in Math

Four of the five points below are solutions of the

same linear equation. Which one is not? Explain.

A(2, 1) D(4, 4) E(3, 2)

14. The table below shows the cost of buying class mugs online. Write

an equation to model the data.

15. Engraving a key chain costs $10 plus $1.50 for each engraved letter.

You can only spend $20. What is the maximum number of letters

you can engrave? Solve by making a table and writing an equation.

2�F

y x� �40 20,

y x� �100 5 ,

1 2 3 4 5 6Number of Circles (x)

Number of Squares (y) ■ ■ ■ ■ ■■

y x� � �23

3 y x� � �35

2 y x� �1 5 4.

B( , )0 4� C( , )1 2�

3

29

4

37

5

45

Number of Mugs

Total Cost

1

13

2

21

For Exercises See Examples

5–6 1

7–8 2

nline

Visit: PHSchool.comWeb Code: ase-0305

lesson quiz, PHSchool.com, Web Code: asa-0305

See margin.

See margin.

7–8. See back of book.

26 squares

10–12. See back of book.13. A; A is not on the line passing through the other points.

14. Let m the number of mugs. Let t the total cost.

��

t m� �8 5

See left.

See left.

See back of book.

Adapted Practice 3-5 L1

Use the equation y � �2x � 1. Complete each solution.

1. (0, 9) 2. (�5, 9) 3. (20, 9) 4. (�68, 9)

5. Determine whether each ordered pair is a solution of y � 3x � 8.

a. (0, �8) b. (6, �10) c. (�2, �2) d. (4, 4)

6. Determine whether each ordered pair is a solution of y � �5x + 19.

a. (�3, 4) b. (0, 19) c. (2, 9) d. (�4, 39)

Graph each linear equation.

7. y � �4x + 6 8. 9.

10. 11. y � �2x + 7 12. y � �3x � 1

13. Jan wants to buy maps for her trip. The maps cost $2 each andshe has $25. Make a table and write an equation to represent theamount she will have left if she buys m maps.

y 5 12 x 2 1

2

y 5 212 x 1 3y 5 5

2 x 2 5

Practice 3-5 Equations, Tables, and Graphs

x

y

2 4 6-6 -4 -2

2

O

4

6

-4

-2

-6y � �4x � 6

x

y

2 4 6-6 -4 -2

2

O

4

6

-4

-2

-6

y � x � 552

x

y

2 4 6-6 -4 -2

2

O

4

6

-4

-2

-6

y � � x � 312

x

y

2 4 6-6 -4 -2

2

O

4

6

-4

-2

-6

y � x �12

12

x

y

2 4 6-6 -4 -2

2

O

4

6

-4

-2

-6

y � �2x � 7

x

y

2 4 6-6 -4 -2

2

4

6

-4

-2

-6

O

y � �3x � 1

5 maps

1 11 –39 137

yes no no yes

no yes yes yes

L3

3-5 • Guided Problem Solving

Student Page 133, Exercise 15:

Engraving a key chain costs $10 plus $1.50 for each engraved letter.You can only spend $20. What is the maximum number of letters youcan engrave? Solve by making a table and writing an equation.

Understand

1. What are you being asked to do?

Plan and Carry Out

2. Write an expression for the cost l per letters in a key chain.

3. What is the flat fee charged per key chain?

4. Write an equation for the cost c of engraving a key chain.Be sure to include the flat fee and the cost per letter.

5. Use the equation in Step 4 to complete the table.

6. What is the maximum number of letters you can engrave for $20

or less?

Check

7. Compare the cost of your answer with the cost of having onemore letter engraved. How should these two amounts compare

to $20?

Solve Another Problem

8. Suzy is going bowling at EZ Lanes. The cost to rent shoes is $4,and each bowling game costs $3.50. If Suzy needs to rent shoesand has $16, how many games can she play? Solve by drawing atable and writing an equation.

GPS

Number Expression Total Costof Letters (dollars)

1 10 � 1.5(1) $11.50

2 10 � 1.5(2) $13.00

3 10 � 1.5(3) $14.50

4 10 � 1.5(4) $16.00

5 10 � 1.5(5) $17.50

6 10 � 1.5(6) $19.00

7 10 � 1.5(7) $20.50

find the maximum number of letters

that can be engraved for $20

1.5l

10

c � 10 � 1.5l

6 letters

the cost for 6 letters should be less than or equal

to $20, and the cost for 7 letters should be more

Suzy can bowl 3 games.Number Expression Total Cost

of Games (dollars)

1 4 � 3.50(1) $7.50

2 4 � 3.50(2) $11.00

3 4 � 3.50(3) $14.50

4 4 � 3.50(4) $18.00

5 4 � 3.50(5) $21.50

c � 4 � 35(g)

L3

Assignment Guide

Check Your Understanding

Go over Exercises 1–4 in class before assigning the Homework Exercises.

Homework Exercises

A

Practice by Example 5–8

B

Apply Your Skills 9–17

C

Challenge 18Test Prep and

Mixed Review 19–24

Homework Quick Check

To check students’ understanding of key skills and concepts, go over Exercises 6, 7, 13, 15, and 16.

3. Practice

5–6. See back of book.

phm07c3_te_0305.fm Page 133 Thursday, May 25, 2006 3:56 PM

134

134 Chapter 3 Real Numbers and the Coordinate Plane

16. Choose a Method You start an exercise routine by lifting 3 lb and

increase the weight by 2 lb per month. Predict how much weight you

will lift after 5 months. Explain why you chose the method you used.

17. Error Analysis Gina owes her father $200. During each week, she

pays $20. They both draw graphs to represent the money she owes.

Who is correct? Explain.

18. Challenge A club sells calendars for $4 each. It spends $2 to make

each calendar and $20 on film. Write and graph two equations to

represent income and expenses. Where do the graphs intersect?

Test Prep and Mixed ReviewTest Prep and Mixed Review Practice

19. The graph of is shown on

the coordinate grid at the right. Which

table of ordered pairs contains only

points on this line?

0

2001601208040

2 4 6Week

Gina

Mon

ey I O

we

y

x 0

2001601208040

2 4 6Week

Gina’s Dad

Mon

ey G

ina O

wes

Me

y

x

y x� �12 1 y

xO�2 2

2

4

x

�4

2

3

1

2

2.5

y x

�2

1

4

0

1.5

3

y x

0

1

2

0

2

�2

y x

�3

0

5

2

3.5

�1.5

y

20. Audrey bought a box of cereal and some bananas for $4.69. If the

cereal cost $3.99 and the bananas were on sale for $0.28 per pound,

how many pounds of bananas did Audrey buy?

0.42 lb 2.2 lb 2.5 lb 4.2 lb

21. A ferry travels at 20 knots, which is about 23 miles per hour. How

should Sam find the number of miles per hour that equals 1 knot?

Divide 20 by 23. Divide 3 by 20.

Divide 23 by 20. Divide 3 by 23.

Solve each equation.

22. 23. 24.b � �6 10 k � �1 24 � � �4 40n

Multiple Choice

13 lb; check students’ methods.

See left.

See back of book.

B

H

B

44254

17. Gina; the graph drawn by Gina’s father shows the amount owed after Gina gives him $40 each week, not $20.

For Exercises See Lesson

22–24 1-6

Enrichment 3-5 L4

You can use equations, tables, and graphs to represent the same data.

Suppose you spend $8 each day on food. Make a table and write an equation to represent the totalamount of money spent after a given number of days. Then graph your equation.

Let x represent the number of days.

Let y represent the total money spent.

Find at least three solutions.

The equation y � 8x models your totalspending.

Show your solutions on a graph. Graph the ordered pairs and draw a line throughthe points.

y

O x

10

20

30

40

2 4 6 8

Tot

al S

pend

ing

($)

Days Passed

1. A tree is 2 feet tall and grows 2 feet per year.Complete the table and graph each (x, y)solution. How tall will the tree be in 5 years?

2. Mary buys bottled water for her family.Graph the linear equation y � �x � 10,where x represents the number of days sinceshe bought 10 bottles and y represents thenumber of bottles left.

Reteaching 3-5 Equations, Tables, and Graphs

1 4

52

3

Days Passed Total Spending Expression(dollars)

0 0 8(0)

1 8 8(1)

2 16 8(2)

3 24 8(3)

x y 8(x)

Years Passed (x) Height in Feet (y)

0 2

1 4

2 63 84 105 12

y

O x

5

10

15

20

2 4 6 8

Hei

ght (

ft)

Years

y

O x

2

4

6

8

2 4 6 8

Bot

tles o

f Wat

er

Days Passed

L2

Test Prep

Resources

For additional practice with a variety of test item formats:• Test-Taking Strategies, p. 151• Test Prep, p. 155• Test-Taking Strategies with Transparencies

Lesson Quiz

1.

Suppose you make $8 per hour at an after-school job. Make a table and write an equation to represent your total pay after 6 hours of work.

2.

Membership at a video store costs $5 per month, plus $1.50 to rent each movie. Graph the linear equation where

y

represents the total cost in a month and

x

represents the number of movies rented each month.

3.

Suppose you rent 6 movies in a month, in the situation above. Make a table that represents your total costs.

4.

The air pressure in a tire is 32 pounds per square inch. Every hour, air is leaking out at the rate of 3 pounds per square inch. Write an equation that describes this situation.

Sample:

y x� �5 1 50. ,

p h� �32 3

Alternative Assessment

Students write a paragraph explaining how to graph the following situation in the coordinate plane: A scientist is releasing 20 drops of liquid from a tube every 10 seconds. Students write an equation that describes the situation and draw the graph.

4. Assess & Reteach

1–3. See back of book.

phm07c3_te_0305.fm Page 134 Thursday, May 25, 2006 3:56 PM