Lesson Notes Solving Equations - Ms. Astetemsastete.com/yahoo_site_admin1/assets/docs/Chpt1-3Answers... · 18 Chapter 1 Equations and Inequalities Lesson Notes Scaffolding Questions

18 Chapter 1 Equations and Inequalities

LessonNotes

Scaffolding Questions Have students read the Why? section of the lesson.Ask:• What does the variable m represent?

mile• Is the length of a kilometer greater

than or less than the length of a mile? less than

• About how many kilometers are in 12 miles? about 19.3 kilometers

1 FOCUS1 FOCUS

2 TEACH2 TEACH

Vertical Alignment

Before Lesson 1-3Use properties of real numbers to evaluate expressions.

Lesson 1-3Translate verbal expressions into algebraic expressions and equations, and vice versa. Solve equations using the properties of equality.

After Lesson 1-3Solve systems of equations.

Resource Approaching-Level On-Level Beyond-Level English LearnersTeacher Edition • Differentiated Instruction, p. 20 • Differentiated Instruction, p. 21 • Differentiated Instruction, p. 21 • Differentiated Instruction, p. 20

Chapter ResourceMasters

• Study Guide and Intervention, pp. 17–18

• Skills Practice, p. 19• Practice, p. 20• Word Problem Practice, p. 21

• Study Guide and Intervention, pp. 17–18• Skills Practice, p. 19• Practice, p. 20• Word Problem Practice, p. 21• Enrichment, p. 22• Graphing Calculator Activity, p. 23

• Practice, p. 20• Word Problem Practice, p. 21• Enrichment, p. 22

• Study Guide and Intervention, pp. 17–18

• Skills Practice, p. 19• Practice, p. 20• Word Problem Practice, p. 21

Transparencies • 5-Minute Check Transparency 1-3 • 5-Minute Check Transparency 1-3 • 5-Minute Check Transparency 1-3 • 5-Minute Check Transparency 1-3

Other• Study Notebook • Teaching Algebra with Manipulatives

• Study Notebook • Teaching Algebra with Manipulatives

• Study Notebook • Study Notebook • Teaching Algebra with Manipulatives

Lesson 1-3 Resources


Solving EquationsWhy?The United States is one of the few countries in the world that measures distances in miles. When traveling by car in different countries, it is often useful to convert miles to kilometers. To find the approximate number of kilometers k in miles m, divide the number of miles by 0.62137.

m miles × 1 kilometer __

0.62137 mile ≈ k kilometers

m _ 0.62137

≈ k kilometers

Verbal Expressions and Algebraic Expressions Verbal expressions can be translated into algebraic expressions by using the language of algebra.

EXAMPLE 1 Verbal to Algebraic Expression

Write an algebraic expression to represent each verbal expression.

a. 2 more than 4 times the cube of a number 4x3 + 2

b. the quotient of 5 less than a number and 12 n - 5 _

12

Guided Practice✓ 1A. the cube of a number increased by 4 times the same number p3 + 4p

1B. three times the difference of a number and 8 3(x - 8)

Personal Tutor glencoe.com

A mathematical sentence containing one or more variables is called an open sentence. A mathematical sentence stating that two mathematical expressions are equal is called an equation.

EXAMPLE 2 Algebraic to Verbal Sentence

Write a verbal sentence to represent each equation.

a. 6x = 72 The product of 6 and a number is 72.

b. n + 15 = 91 The sum of a number and 15 is ninety-one.

Guided Practice✓ 2A, 2B. See margin.

2A. g - 5 = -2 2B. 2c = c2 - 4


Open sentences are neither true nor false until the variables have been replaced by numbers. Each replacement that results in a true sentence is called a solution of the open sentence.

ThenThenYou used properties of real numbers to evaluate expressions. (Lesson 1-2)

NowNow Translate verbal

expressions into algebraic expressions and equations, and vice versa.

Solve equations using the properties of equality.

NGSSS

Reinforcement of MA.912.A.3.1 Solve linear equations in one variable that include simplifying algebraic expressions.

New VocabularyNew Vocabularyopen sentenceequationsolution

FL Math Online

glencoe.com

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All of the Lesson ResourcesLesson Resources are

leveled for students who are

below grade levelbelow grade level, on grade level on grade level,

and above grade level,above grade level, and for

students who are English English

language learners.language learners.

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Lesson 1-3 Solving Equations 19

Verbal Expressions and Algebraic ExpressionsExample 1 shows how to translate verbal expressions into algebraic expressions. Example 2 shows how to translate algebraic expressions into verbal expressions.

Formative AssessmentUse the Guided Practice exercises after each example to determine students’ understanding of concepts.

Additional Examples also inInteractive Classroom PowerPoint® Presentations

Additional Examples


a. 7 less than a number n - 7

b. the square of a number decreased by the product of 5 and the number x2 - 5x


a. 6 = –5 + x Six is equal to –5 plus a number.

b. 7y - 2 = 19 Seven times a number minus 2 is 19.

11

22

Properties of EqualityExample 3 shows how to identify properties of equality.

✓ ✓

Additional Example

Name the property illustrated by each statement.

a. a - 2.03 = a - 2.03 Reflexive Property of Equality

b. If 9 = x, then x = 9. Symmetric Property of Equality

33Additional Answers (Guided Practice)

2A. The difference of a number and 5 is –2.

2B. Two times a number is equal to the difference of that number squared and 4.

INTERACTIVEWHITEBOARDREADY

IWBIWB


Diophantus of Alexandria (c. 200–284) Diophantus was famous for his work in algebra. His main work was titled Arithmetica and introduced symbolism to Greek algebra as well as propositions in number theory and polygonal numbers.

Properties of Equality To solve equations, we can use properties of equality. Some of these properties are listed below.

Key Concept

Property Symbols Examples

Reflexive For any real number a, a = a. b + 12 = b + 12

Symmetric For all real numbers a and b,if a = b, then b = a.

If 18 = -2n + 4, then -2n + 4 = 18.

Transitive For all real numbers a, b, and c,if a = b and b = c, then a = c.

If 5p + 3 = 48 and 48 = 7p - 15, then 5p + 3 = 7p - 15.

Substitution If a = b, then a may be replaced by b and b may be replaced by a.

If (6 + 1)x = 21, then 7x = 21.

Properties of Equality

EXAMPLE 3 Identify Properties of Equality


a. If 3a - 4 = b, and b = a + 17, then 3a - 4 = a + 17. Transitive Property of Equality

b. If 2g - h = 62, and h = 24, then 2g - 24 = 62. Substitution Property of Equality

Guided Practice✓ 3. If -11a + 2 = -3a, then -3a = -11a + 2. Symmetric


To solve most equations, you will need to perform the same operation on each side of the equals sign. The properties of equality allow for the equation to be solved in this way.

Key Concept

Addition and Subtraction Properties of Equality

Symbols For any real numbers, a, b, and c, if a = b, then a + c = b + c and a - c = b - c.

Examples If x - 6 = 14, then x - 6 + 6 = 14 + 6.If n + 5 = -32, then n + 5 - 5 = -32 - 5.

Multiplication and Division Properties of Equality

Symbols For any real numbers, a, b, and c, c ≠ 0, if a = b,

then a · c = b · c and a_c =b_c .

Examples If m_8 = -7, then 8 · m_8 = 8 · (-7).

If -2y = 12, then -2y_-2 = 12_

-2.

StudyTip

Checking Answers When solving for a variable, you can use substitution to check your answer by replacing the variable in the original equation with your answer.

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Examples 4 and 5 show how to solve one-step and multi-step equations.

Additional Examples

Solve each equation. Check your solution.

a. m - 5.48 = 0.02 5.5

b. 18 = 1 _ 2 t 36

Solve 53 = 3(y - 2) - 2(3y - 1). –19

44

55

Tips for New Teachers

Checking Solutions Explain that checking solutions to discover possible errors is a vital procedure when you use math on the job.

Tips for New Teachers

Sense-Making Help students to remember the name of the Reflexive Property by relating a = a to seeing your reflection in a mirror.

Focus on Mathematical Content

Rules for Solving Equations The rules used to solve equations are based on the Properties of Equality. When a number is added to or subtracted from each side of an equation, the result is an equivalent equation. This equivalent equation will have the same solution as the original.

Differentiated Instruction

If students have difficulty transitioning from verbal expressions to algebraic expressions and vice versa,

Then pair these students with students who are not having trouble. Let them act as a mentor to help the students having difficulties.

ELL AL


EXAMPLE 4 Solve One-Step Equations


a. n - 3.24 = 42.1

n - 3.24 = 42.1 Original equation

n - 3.24 + 3.24 = 42.1 + 3.24 Add 3.24 to each side.

n = 45.34 Simplify.

The solution is 45.34.

CHECK n - 3.24 = 42.1 Original equation

45.34 - 3.24 � 42.1 Substitute 45.34 for n.

42.1 = 42.1 ✔ Simplify.

b. - 5 _ 8 x = 20

- 5 _ 8 x = 20 Original equation

- 8 _ 5 (- 5 _

8 ) x = - 8 _

5 (20) Multiply each side by - 8_

5 .

x = -32 Simplify.

The solution is -32.

CHECK - 5 _ 8 x = 20 Original equation

- 5 _ 8 (-32) � 20 Replace x with -32.

20 = 20 ✔ Simplify.

Guided Practice✓ 4A. x - 14.29 = 25 39.29 4B. 2

_ 3 y = -18 -27


To solve an equation with more than one operation, undo operations by working backward.

EXAMPLE 5 Solve a Multi-Step Equation

Solve 5(x + 3) + 2(1 - x) = 14.

5(x + 3) + 2(1 - x) = 14 Original equation

5x + 15 + 2 - 2x = 14 Apply the Distributive Property.

3x + 17 = 14 Simplify the left side.

3x = -3 Subtract 17 from each side.

x = -1 Divide each side by 3.

Guided Practice✓Solve each equation.

5A. -10x + 3(4x - 2) = 6 6 5B. 2(2x - 1) - 4(3x + 1) = 2 -1


StudyTip

Multiplication and Division Properties of Equality Example 4b could also have been solved using the Division Property of Equality. Note that dividing each side of

the equation by - 5_8

is the same as multiplying each side by - 8_

5 .

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Study TipsStudy Tips offer

students helpful

information about the

topics they are

studying.

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Example 6 shows how to use properties to solve a formula for a specified variable. Example 7 shows how to solve a standardized test question using the Addition Property of Equality.

Additional Examples

GEOMETRY The formula for the surface area S of a cone is S = πr� + πr2, where � is the slant height of the cone and r is the radius of the base. Solve the

formula for � . � = S - πr2 _ πr

STANDARDIZED TEST PRACTICE If 4g + 5 = 4 _ 9 ,

what is the value of 4g - 2? C

A – 41 _ 36

C – 59 _ 9

B – 41 _ 9

D – 67 _ 7

66

77

Differentiated Instruction

Extension The formula for the perimeter of a rectangle is P = 2� + 2w. Find the area of a rectangle that has a perimeter P of 22 inches and a width w of 3 inches. (Hint: Begin by solving the perimeter formula for �.) 24 i n 2

The Differentiated Homework

Options provide leveled

assignments. Many of the

homework exercises are paired,

so that the students can do the

odds one day and the evens on

the next day.

Every chapter includes one or

more worked-out Standardized Standardized

Test ExamplesTest Examples that are similar

to problems found on state

assessments.

OL BL

BLOG Have students write a blog entry to summarize how to solve one-step equations. Make sure that students use the concept of inverse operations in their explanations.Lesson 1-3 Solving Equations 21

6. h = S - 2πr 2 _

2πr

Test-TakingTip

Using Properties There are often many ways to solve a problem. Using the properties of equality can help you find a simpler way.

PRACTICE EXAMPLE 7 912.A.3.5NGSSS

If 6x - 12 = 18, what is the value of 6x + 5?

A. 5 B. 11 C. 35 D. 41

You can use properties to solve an equation for a variable.

EXAMPLE 6 Solve for a Variable

GEOMETRY The formula for the area A of a trapezoid is b1

h

b2

A = 1 _ 2 h(b1 + b2), where h represents the height, and b1

and b2 represent the measures of the bases. Solve the formula for b2.

A = 1 _ 2 h(b1 + b2) Area formula

2A = 2[ 1 _ 2 h(b1 + b2)] Multiply each side by 2.

2A = h(b1 + b2) Simplify.

2A _ h

= h(b1 + b2)

_ h

Divide each side by h.

2A _ h = b1 + b2 Simplify.

2A _ h - b1 = b1 + b2 - b1 Subtract b1 from each side.

2A _ h - b1 = b2 Simplify.

Guided Practice✓ 6. The formula for the surface area S of a cylinder is S = 2πr2 + 2πrh, where r is the

radius of the base and h is the height of the cylinder. Solve the formula for h.


Read the Test ItemYou are asked to find the value of 6x + 5. Note that you do not have to find the value of x. Instead, you can use the Addition Property of Equality to make the left side of the equation 6x + 5.

Solve the Test Item 6x - 12 = 18 Original equation

6x - 12 + 17 = 18 + 17 Add 17 to each side because -12 + 17 = 5.

6x + 5 = 35 Simplify.

The answer is C.

Guided Practice✓ 7. If 5y + 2 = 8 _

3 , what is the value of 5y - 6? G

F. -20 _

3 G. -16

_ 3 H. 16

_ 3 I. 32

_ 3


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Formative AssessmentUse Exercises 1–21 to check for understanding.

Use the chart at the bottom of this page to customize assignments for your students.

Additional Answers

3. The sum of five times a number and 7 equals 18.

4. The difference between the square of a number and 9 is 27.

5. The difference between five times a number and the cube of that number is 12.

6. Eight more than the quotient of a number and four is -16.

3 PRACTICE3 PRACTICE

✓ ✓

Differentiated Homework OptionsLevel Assignment Two-Day Option

AL Basic 22–50, 62, 64–82 23–49 odd, 67–70 22–50 even, 62, 64–66, 71–82

OL Core 23–51 odd, 52, 53–57 odd, 59–62, 64–82

22–50, 67–70 51–62, 64–66, 71–82

BL Advanced 51–74, (optional: 75–82)

Check Your UnderstandingCheck Your Understanding

exercises are intended to be

completed in class. Example Example

referencesreferences show students where

to look back for review.


✓ Check Your Understanding

Write an algebraic expression to represent each verbal expression. Example 1p. 18

1. the product of 12 and the sum of a number and negative 3 12 [x + (- 3)]

2. the difference between the product of 4 and a number and the square of the number 4x - x2

Write a verbal sentence to represent each equation. 3–6. See margin.Example 2p. 18

3. 5x + 7 = 18 4. x2 - 9 = 27

5. 5y - y3 = 12 6. x _ 4 + 8 = -16

Name the property illustrated by each statement. Example 3p. 19

7. (8x - 3) + 12 = (8x - 3) + 12 8. If a = -3 and -3 = d, then a = d.

Reflexive Property Transitive PropertySolve each equation. Check your solution.Examples 4 and 5

p. 20 9. z - 19 = 34 53 10. x + 13 = 7 -6 11. -y = 8 -8

12. -6x = 42 -7 13. 5x - 3 = -33 -6 14. -6y - 8 = 16 -4

15. 3(2a + 3) - 4(3a - 6) = 15 3 16. 5(c - 8) - 3(2c + 12) = -84 8

17. -3(-2x + 20) + 8(x + 12) = 92 4 18. -4(3m - 10) - 6(-7m - 6) = -74 -5

Solve each equation or formula for the specified variable.Example 6p. 21

19. 8r - 5q = 3, for q q = 8r - 3 _ 5 20. Pv = nrt, for n Pv _

rt = n

21. MULTIPLE CHOICE If y _

5 + 8 = 7, what is the value of

y _

5 - 2? BExample 7

p. 21A -10 B -3 C 1 D 5

Practice and Problem Solving= Step-by-Step Solutions begin on page R20.

Extra Practice begins on page 947.

Write an algebraic expression to represent each verbal expression. Example 1p. 18

22. the difference between the product of four and a number and 6 4n - 6

23. the product of the square of a number and 8 8x2

24. fifteen less than the cube of a number x 3 - 15

25. five more than the quotient of a number and 4 x _ 4 + 5

26. Four less than 8 times a number is 16.Write a verbal sentence to represent each equation.Example 2

p. 18 26. 8x - 4 = 16 27. x + 3

_ 4 = 5 28. 4y2 - 3 = 13

2929 BASEBALL During a recent season, Miguel Cabrera and Mike Jacobs of the Florida Marlins hit a combined total of 46 home runs. Cabrera hit 6 more home runs than Jacobs. How many home runs did each player hit? Define a variable, write an equation, and solve the problem.

Name the property illustrated by each statement. 30. Subtr. (=)Example 3p. 19

30. If x + 9 = 2, then x + 9 - 9 = 2 - 9 31. If y = -3, then 7y = 7(-3) Subst.

32. If g = 3h and 3h = 16, then g = 16 33. If -y = 13, then -(-y) = -13 Mult. (=)Transitive Property

27. The quotient of the sum of 3 and a number and 4 is 5.28. Three less than four times the square of a number is 13.

29. n = number of home runs Jacobs hit; n + 6 = number of home runs Cabrera hit; 2n + 6 = 46; Jacobs: 20 home runs, Cabrera: 26 home runs.

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Chapter 1 21 Glencoe Algebra 2

1-3 Word Problem PracticeSolving Equations

1. AGES Robert’s father is 5 years older than 3 times Robert’s age. Let Robert’s age be denoted by R and let Robert’s father’s age be denoted by F. Write an equation that relates Robert’s age and his father’s age.

2. AIRPLANES The Citation Sovereign is a small jet that can carry up to 2,650 pounds. The number of passengers p and the number of suitcases s that the airplane can carry are estimated by the equation 180p + 60s = 2,650. If 10 people board the aircraft, how many suitcases can the airplane carry?

3. GEOMETRY The length of a rectangle is 10 units longer than its width. If the total perimeter of the rectangle is 44 units, what is the width?

w

w + 10

4. SAVINGS Jason started with d dollars in his piggy bank. One week later, Jason doubled the amount in his piggy bank. Another week later, Jason was able to add $20 to his piggy bank. At this point, the piggy bank had $50 in it. What is d?

5. DOMINOES Nancy is setting up a train of dominos from the front entrance straight down the hall to the kitchen entrance. The thickness of each domino is t. Nancy places the dominoes so that the space separating consecutive dominoes is 3t. The total distance that N dominoes takes up is given by d = t(4N + 1).

3t

t

a. Nancy measures her dominoes and finds that t = 1 centimeter. She measures the distance of her hallway and finds that d = 321 centimeters. Rewrite the equation that relates d, t, and N with the given values substituted for t and d.

b. How many dominoes did Nancy have in her hallway?

F = 3R + 5.

20 suitcases

w = 6 units

15

321 = 4N + 1.

80 dominoes

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Word Problem Practice p. 21 AL OL BL ELL

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1-3 PracticeSolving Equations


1. 2 more than the quotient of a number and 5 2. the sum of two consecutive integers

3. 5 times the sum of a number and 1 4. 1 less than twice the square of a number


5. 5 - 2x = 4 6. 3y = 4y3

7. 3c = 2(c - 1) 8. m − 5 = 3(2m + 1)


9. If t - 13 = 52, then 52 = t - 13. 10. If 8(2q + 1) = 4, then 2(2q + 1) = 1.

11. If h + 12 = 22, then h = 10. 12. If 4m = -15, then -12m = 45.


13. 14 = 8 - 6r 14. 9 + 4n = -59

15. 3 − 4 - 1 −

2 n = 5 −

8 16. 5 −

6 s + 3 −

4 = 11 −

12

17. -1.6r + 5 = -7.8 18. 6x - 5 = 7 - 9x

19. 5(6 - 4v) = v + 21 20. 6y - 5 = -3(2y + 1)

Solve each equation or formula for the specified variable.

21. E = mc2, for m 22. c = 2d + 1 − 3 , for d

23. h = vt - gt2, for v 24. E = 1 − 2 Iw2 + U, for I

25. GEOMETRY The length of a rectangle is twice the width. Find the width if the perimeter is 60 centimeters. Define a variable, write an equation, and solve the problem.

26. GOLF Luis and three friends went golfing. Two of the friends rented clubs for $6 each. The total cost of the rented clubs and the green fees for each person was $76. What was the cost of the green fees for each person? Define a variable, write an equation, and solve the problem.

y −

5 + 2 n + (n + 1)

5(m + 1) 2y 2 - 1

5–8. Sample answers are given.

The difference of 5 and twice a Three times a number is 4 times number is 4. the cube of the number.

The quotient Three times a number is twice the of a number and 5 is 3 times the difference of the number and 1. sum of twice the number and 1.

Symmetric (=) Division (=)

Subtraction (=) Multiplication (=)

-1 -17

1 − 4 1 −

5

8 4 − 5

3 −

7 1 −

6

m = E − c 2

d = 3c - 1

− 2

v = h + g t 2

− t

I = 2 (E - U) −

w2

w = width; 2(2w) + 2w = 60; 10 cm

g = green fees per person; 6(2) + 4g = 76; $16

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Practice p. 20 AL OL BL ELL

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1-3 Study Guide and InterventionSolving Equations

Verbal Expressions and Algebraic Expressions The chart suggests some ways to help you translate word expressions into algebraic expressions. Any letter can be used to represent a number that is not known.

Word Expression Operation

and, plus, sum, increased by, more than addition

minus, difference, decreased by, less than subtraction

times, product, of (as in 1 − 2 of a number) multiplication

divided by, quotient division

Write an algebraic expression to represent 18 less than the quotient of a number and 3.n − 3 - 18

Write a verbal sentence to represent 6(n - 2) = 14.

Six times the difference of a number and two is equal to 14.

Exercises


1. the sum of six times a number and 25

2. four times the sum of a number and 3

3. 7 less than fifteen times a number

4. the difference of nine times a number and the quotient of 6 and the same number

5. the sum of 100 and four times a number

6. the product of 3 and the sum of 11 and a number

7. four times the square of a number increased by five times the same number

8. 23 more than the product of 7 and a number


9. 3n - 35 = 79

10. 2(n3 + 3n2) = 4n

11. 5n − n + 3

= n - 8

Example 1 Example 2

6n + 25

4(n + 3)

15n - 7

100 + 4n

3(11 + n)

4n2 + 5n

7n + 23

Sample answers are given.

The difference of three times a number and 35 is equal to 79.

Twice the sum of the cube of a number and three times the square of the number is equal to four times the number.

The quotient of five times a number and the sum of the number and 3 is equal to the difference of the number and 8.

9n - 6 − n

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Study Guide and Intervention pp. 17–18 AL OL ELL

C

1-3 EnrichmentUnited States’ Gross National Product

The Gross National Product, GNP, is an important indicator of the U.S. economy. The GNP contains information about the inflation rate, the Bond market, and the Stock market. It is composed of consumer goods, investments, government expenditures, exports, and imports.

Calculated from GNP = C + I + G + X - M, where

C is consumer goods (e.g. TVs, cars, food, furniture, clothes, doctors’ fees, and dining)

I is investments (e.g. factories, computers, airlines, and housing) G is government spending and investments (e.g. ships, roads, schools, NASA,

and bombs) X is exports (e.g. corn, wheat, cars, and computers) M is for imports, (e.g. cars, computer chips, clothes, and oil) X - M is exports minus imports and equals trade surplus or deficit.

1. The most important sector of the U.S. economy is consumption. It makes up about 60% of the entire GNP. In 2000, the U.S.’s GNP was 10.5 trillion dollars. In the same year, there were 1 trillion dollars in investments, but a 1 trillion dollar trade deficit. Assuming that consumption made up 60% of the GNP, how much did the government budget for spending?

2. In 2001, the U.S. trade deficit remain at 1 trillion dollars, investments also remain

10.5 = 6.3 + 1 + G - 1. Therefore G = 4.2 trillion dollars.

005_042_A2CRMC01_890526.indd 22 4/11/08 12:50:05 AM

Enrichment p. 22 OL BL


34. MONEY Aiko and Kendra arrive at the state fair with $32.50. What is the total number of rides they can go on if they each pay the entrance fee? n = number of rides; 2(7.50) + n(2.50) = 32.50; 7

Solve each equation. Check your solution. Examples 4 and 5p. 20

35. 3y + 4 = 19 5 36. -9x - 8 = 55 -7

37. 7y - 2y + 4 + 3y = -20 -3 38. 5g + 18 - 7g + 4g = 8 -5

3939 5(-2x - 4) - 3(4x + 5) = 97 -6 40. -2(3y - 6) + 4(5y - 8) = 92 8

41. 2 _

3 (6c - 18) + 3 _

4 (8c + 32) = -18 -3 42. 3

_ 5 (15d + 20) - 1 _

6 (18d - 12) = 38 4

43. GEOMETRY The perimeter of a regular pentagon is 100 inches. Find the length of each side. s = length of a side; 5s = 100; 20 in.

44. MEDICINE For Nina’s illness her doctor gives her a prescription for 28 pills. The doctor says that she should take 4 pills the first day and then 2 pills each day until her prescription runs out. For how many days does she take 2 pills?

Solve each equation or formula for the specified variable. Example 6p. 21

45. E = mc2, for m m = E _ c2

46. c(a + b) - d = f, for a a = f + d _ c - b

47. z = πq3h, for h h = z _ πq3

48. x + y

_ z - a = b, for y y = z(a + b) - x

49. y = ax2 + bx + c, for a a = y - bx - c

_ x2

50. wx + yz = bc, for z z = bc - wx _ y

51. GEOMETRY The formula for the volume of a cylinder with

h

rradius r and height h is π times the radius times the radius times the height.

a. Write this as an algebraic expression. V = π × r × r × h

b. Solve the expression in part a for h. h = V _ πr 2

52. AWARDS BANQUET A banquet room can seat a maximum of 69 people. The coach, principal, and vice principal have invited the award-winning girls’ tennis team to the banquet. If the tennis team consists of 22 girls, how many guests can each student bring? n = number of guests that each student can bring; 22n + 25 = 69; 2 guests


53. 5x - 9 = 11x + 3 -2 54. 1 _ x + 1 _

4 = 7 _

12 3

55. 5.4(3k - 12) + 3.2(2k + 6) = -136 -4 56. 8.2p - 33.4 = 1.7 - 3.5p 3

57. 4 _

9 y + 5 = -

7 _

9 y - 8 - 117 _

11 58.

3 _

4 z -

1 _

3 = 2 _

3 z + 1 _

5 32 _

5

59. FINANCIAL LITERACY Benjamin spent $10,734 on his Expense Annual Cost

Electric $622

Gas $428

Water $240

Renter’s Insurance $144

living expenses last year. Most of these expenses are listed at the right. Benjamin’s only other expense last year was rent. If he paid rent 12 times last year, how much is Benjamin’s rent each month? x = the cost of rent each month; 622 + 428 + 240 + 144 + 12x = 10,734; $775 per month

$7.50Entrance Fee:

$2.50 eachRides:

44. x = the number of days she takes 2 pills; 4 + 2x = 28; 12 days

B

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Multiple Representations In Exercise 61, students use a number line and a table of values to illustrate the workings of absolute values.

Additional Answers

60c. Yes; it seems reasonable that two crews working 4 miles apart would be able to complete the same amount of miles in the same amount of time.

61b. Integer Distance

from Zero–5 5–4 4–3 3–2 2–1 10 01 12 23 34 45 5

Watch Out!

Error Analysis In Exercise 62, encourage students to give a complete explanation of the error. For example, “Steven should have subtracted b from the entire left side, 2A _

h , in the last step

of the equation.”

61c. y

x4 8

8

4

-4

-8

-4-8

61d. For positive integers, the distance from zero is the same as the integer. For negative integers, the distance is the integer with the opposite sign because distance is always positive.

Find the ErrorFind the Error exercises help

students identify and address

common errors before they

occur. Prompts in the margin of

the Teacher Edition help you

guide students toward

understanding.


60. BRIDGES The Sunshine Skyway Bridge spans Tampa Bay, Florida. Suppose one crew began building south from St. Petersburg, and another crew began building north from Bradenton. The two crews met 10,560 feet south of St. Petersburg approximately 5 years after construction began.

a. Suppose the St. Petersburg crew built an average of 176 feet per month. Together the two crews built 21,120 feet of bridge. Determine the average number of feet built per month by the Bradenton crew. 176 ft

b. About how many miles of bridge did each crew build? 2 mi

c. Is this answer reasonable? Explain. See margin.

6161 MULTIPLE REPRESENTATIONS The absolute value of a number describes the distance of the number from zero. a.

-2-3-4-5 543-1 210 a. GEOMETRIC Draw a number line. Label the integers from -5 to 5.

b. TABULAR Create a table of the integers on the number line and their distance from zero. b–d. See margin.

c. GRAPHICAL Make a graph of each integer x and its distance from zero y using the data points in the table.

d. VERBAL Make a conjecture about the integer and its distance from zero. Explain the reason for any changes in sign.

H.O.T. Problems Use Higher-Order Thinking Skills

62. ERROR ANALYSIS Steven and Jade are solving A = 1 _ 2 h(b1 + b2) for b2. Is either of them

correct? Explain your reasoning.

StevenA = 1 _ 2 h(b1 + b2)

2A

_ h = (b1 + b2)

2A – b1

_ h = b2

JadeA = 1 _ 2 h(b1 + b2)

2A _ h

= (b1 + b2)

2A _ h

– b1 = b2

63. CHALLENGE Solve d = √ �� (x2 - x1)2 + (y2 - y1)2 for y1. y1 = y2 - √ �� d 2 - (x2 - x1)2

64. REASONING Use what you have learned in this lesson to explain why the following

number trick works. Translating this number trick into an expression yields:

• Take any number.• Multiply it by ten.• Subtract 30 from the result.• Divide the new result by 5.• Add 6 to the result.• Your new number is twice your original.

65. OPEN ENDED Provide one example of an equation involving the Distributive Property that has no solution and another example that has infinitely many solutions.

Sample answer: 3(x - 4) = 3x + 5; 2(3x - 1) = 6x - 2 66. WRITING IN MATH Compare and contrast the Substitution Property of Equality and

the Transitive Property of Equality.

The Milliau Viaduct, located in southern France, is the world’s tallest vehicular bridge. The bridge stands 1122 feet tall, 1.5 miles long, and 4 lanes wide.Source: National Geographic Society

62. Sample answer: Jade; in the last step, when Steven subtracted b1 from each side, he mistakenly put the - b1 in the numerator instead of after the entire fraction.

66. Sample answer: The Transitive Property utilizes the Substitution Property. While the Substitution Property is done with two values, that is, one being substituted for another, the Transitive Property deals with three values, determining that since two values are equal to a third value, then they must be equal.

(10x - 30)

_ 5 + 6 = 2x

(10x - 30)

_ 5 = 2x - 6

(2x - 6) + 6 = 2x

C

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Yesterday’s News Have each student write how yesterday’s concepts helped with today’s new material.

Formative AssessmentCheck for student understanding of concepts in Lesson 1-3.

Quiz 2, p. 45

4 ASSESS4 ASSESS

✓ ✓


Spiral Review

71. Simplify 3x + 8y + 5z - 2y - 6x + z. (Lesson 1-2) -3x + 6y + 6z

72. BAKING Tamera is making two types of bread. The first type of bread needs 2 1 _ 2 cups of

flour, and the second needs 1 3 _

4 cups of flour. Tamera wants to make 2 loaves of the first

recipe and 3 loaves of the second recipe. How many cups of flour does she need? (Lesson 1-2) 10 1 _ 4 c

73. LANDMARKS Suppose the Space Needle in Seattle,

220 ft

h ft

2 ft

5.5 ft

Washington, casts a 220-foot shadow at the same time a

nearby tourist casts a 2-foot shadow. If the tourist is 5 1 _ 2 feet

tall, how tall is the Space Needle? (Lesson 0-6) 605 ft

74. Evaluate a - [c(b - a)], if a = 5, b = 7, and c = 2. (Lesson 1-1) 1

Skills Review

Identify the additive inverse for each number or expression. (Lesson 1-2)

75. -4 1 _ 5 4 1 _

5 76. 3.5 -3.5 77. -2x 2x 78. 6 - 7y -6 + 7y

79. 3 2 _ 3 -3 2 _

3 80. -1.25 1.25 81. 5x -5x 82. 4 - 9x -4 + 9x

67. The graph shows the solution of which inequality? D

y

x

A. y < 2 _ 3 x + 4 C. y < 3 _

2 x + 4

B. y > 2 _ 3 x + 4 D. y > 3 _

2 x + 4

68. SAT/ACT What is 1 1 _ 3 subtracted from its

reciprocal? F

F. - 7 _

12 H. 1 _

4

G. - 1 _

12 I. 3 _

4

69. GEOMETRY Which y

x

'

'

'

of the following describes the transformation of �ABC to �A′B′C′? A

A. a reflection across the y-axis and a translation down 2 units

B. a reflection across the x-axis and a translation down 2 units

C. a rotation 90° to the right and a translation down 2 units

D. a rotation 90° to the right and a translation right 2 units

70. SHORT RESPONSE A local theater sold 1200 tickets during the opening weekend of a movie. On the following weekend, 840 tickets were sold. What was the percent decrease of tickets sold? 30%

912.A.3.6, 912.G.2.4, 912.A.2.13NGSSS PRACTICE

018_025_C01_L03_892265.indd 25 11/14/08 3:10:52 PM



Mid-Chapter Quiz

Formative AssessmentUse the Mid-Chapter Quiz to assess students’ progress in the first half of the chapter.

For problems answered incorrectly, have students review the lessons indicated in parentheses.

Customize and create multiple

versions of your Mid-Chapter Quiz and their answer keys.

Follow-Up

Before students complete the Mid-Chapter Quiz, encourage them to review the information for Lessons 1-1 through 1-3 in their Foldables.

✓ ✓

Intervention Planner

Tier 1 On Level Tier

2Strategic Interventionapproaching grade level

Tier 3

Intensive Intervention2 or more grades below level

If students miss about 25% of the exercises or less, If students miss about 50% of the exercises, If students miss about 75% of the exercises,

Then choose a resource: Then choose a resource:

SE

CRM

TE

Lessons 1-1, 1-2, and 1-3

Skills Practice, pp. 7, 13, and 19

Chapter Project, p. 2

CRM Study Guide and Intervention, Chapter 1, pp. 5, 11, 17

Then use Math Triumphs, Alg. 2, Ch. 1 and 2

FL Math Online Self-Check Quiz FL Math Online Extra Examples, Personal Tutor, Homework Help

FL Math Online Extra Examples, Personal Tutor, Homework Help, Review Vocabulary


Mid-Chapter QuizLessons 1-1 through 1-3

1. Evaluate 3c - 4(a + b) if a = -1, b = 2 and c = 1 _ 3 .

(Lesson 1-1) -3

2. TRAVEL The distance that Maurice traveled in 2.5 hours riding his bicycle can be found by using the formula d = rt, where d is the distance traveled, r is the rate, and t is the time. How far did Maurice travel if he traveled at a rate of 16 miles per hour? (Lesson 1-1) 40 m

3. Evaluate (5 - m)3 + n(m - n) if m = 6 and n = -3. (Lesson 1-1) -28

4. GEOMETRY The formula for the surface area of the rectangular prism below is given by the formula S = 2xy + 2yz + 2xz. What is the surface area of the prism if x = 2.2, y = 3.5, and z = 5.1? (Lesson 1-1) 73.54 units2

x

y

z

5. NGSSS PRACTICE What is the value of q2 + rt

_ qr - 2t

if q = -4, r = 3, and t = 8? (Lesson 1-1) C

A. - 17

_ 6

B. - 1 _

6

C. - 10

_ 7

D. - 2 _ 7

Name the sets of numbers to which each number belongs. (Lesson 1-2)

6. 25 _

11 Q, R 7. -

128 _

32 Z, Q, R

8. √ � 50 I, R 9. -32.4 Q, R

10. What is the property illustrated by the equation (4 + 15)7 = 4 · 7 + 15 · 7? (Lesson 1-2) Dist.

11. Simplify -3(7a - 4b) + 2(-3a + b). (Lesson 1-2)

12. CLOTHES Brittany is buying T-shirts and jeans for her new job. T-shirts cost $10.50, and jeans cost $26.50. She buys 3 T-shirts and 3 pairs of jeans. Illustrate the Distributive Property by writing two expressions representing how much Brittany spent. (Lesson 1-2) 3(10.50 + 26.50) or 3(10.50) + 3(26.50)

13. NGSSS PRACTICE Which expression is equivalent

to 2 _ 3 (4m - 5n) + 1 _

5 (2m + n)? (Lesson 1-2) F

F. 46

_ 15

m - 47 _

15 n

G. 46m - 47n H. -

mn _ 15

I. 5 _

4 m - 9 _

8 n

14. Identify the additive inverse and the multiplicative

inverse for 7 _ 6 . (Lesson 1-2) additive: - 7 _

6 ; mult.: 6 _

7

15. Write a verbal sentence to represent the equation a _

a - 3 = 1. (Lesson 1-3)

16. Solve 6x + 4y = -1 for x. (Lesson 1-3)

17. NGSSS PRACTICE Which algebraic expression represents the verbal expression, the product of 4 and the difference of a number and 13? (Lesson 1-3) BA. 4n - 13

B. 4(n - 13)

C. 4 _

n - 13

D. 4n _ 13

18. Solve -3(6x + 5) + 2(4x) = 20. (Lesson 1-3) - 7 _ 2

19. What is the height of the trapezoid below? (Lesson 1-3) 7.5 units

h

15.5

= 80.625

6

20. GEOMETRY The formula for the surface area of a sphere is SA = 4πr2, and the formula for the volume

of a sphere is V = 4 _ 3 πr3. (Lesson 1-3)

a. Find the volume and surface area of a sphere with radius 2 inches. Write your answers in terms of π. 32 _

3 π in3; 16π in2

b. Is it possible for a sphere to have the same numerical value for the surface area and volume? If so, find the radius of such a sphere. yes; 3 units

The quotient of a number a and the difference of a number a and 3 is equal to 1.

-27a + 14b

x = - 2 _ 3 y - 1 _

6

912.A.2.13, 912.A.3.5, 912.G.2.5, 912.G.7.5

NGSSS

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