Lesson
Notes
Solving Equations
Why?
1
Then
FOCUS
You used properties of
real numbers to evaluate
expressions. (Lesson 1-2)
Vertical Alignment
Now
Before Lesson 1-3
Use properties of real numbers to
evaluate expressions.
Translate verbal
expressions into
algebraic expressions
and equations, and vice
versa.
Solve equations using
the properties of
equality.
Lesson 1-3
Translate verbal expressions into
algebraic expressions and
equations, and vice versa.
Solve equations using the
properties of equality.
2
Reinforcement of
MA.912.A.3.1 Solve linear
equations in one variable
that include simplifying
algebraic expressions.
New Vocabulary
open sentence
equation
solution
TEACH
FL Math Online
glencoe.com
Scaffolding Questions
1 kilometer
m miles × __
≈ k kilometers
0.62137 mile
m
_
≈ k kilometers
0.62137
Verbal Expressions and Algebraic Expressions Verbal expressions can be translated
into algebraic expressions by using the language of algebra.
NGSSS
After Lesson 1-3
Solve systems of equations.
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.
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 p 3 + 4p
1B. three times the difference of a number and 8 3(x - 8)
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
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
are
on Resources
All of the Less
ents who are
leveled for stud
vel,l
vel,l on grade le
below grade le
r
e level, and fo
and above grad
e English
students who ar
rs.
language learne
2A, 2B. See margin.
2A. g - 5 = -2
2B. 2c = c2 - 4
Personal Tutor glencoe.com
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.
18 Chapter 1 Equations and Inequalities
Lesson 1-3 Resources
Resource
Teacher Edition
Approaching-Level
18
On-Level
• Differentiated Instruction, p. 20
• Differentiated Instruction, p. 21
Chapter
Resource
Masters
• 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
018_025_C01_L03_892265.indd
Beyond-Level
• Differentiated Instruction, p. 21
Study Guide and Intervention, pp. 17–18 • Practice, p. 20
Skills Practice, p. 19
• Word Problem Practice, p. 21
Practice, p. 20
• Enrichment, p. 22
Word Problem Practice, p. 21
Enrichment, p. 22
Graphing Calculator Activity, p. 23
English Learners
11/14/08
3:09:52 PM
• Differentiated Instruction, p. 20
• Study Guide and Intervention,
pp. 17–18
• Skills Practice, p. 19
• Practice, p. 20
• Word Problem Practice, p. 21
18 Chapter 1 Equations and Inequalities
0018-0025_C01L03_892270.indd 18
12/12/08 1:31:46 PM
Properties of Equality To solve equations, we can use properties of equality. Some of
Verbal Expressions and
Algebraic Expressions
these properties are listed below.
Key Concept
Property
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.
Example 1 shows how to translate
verbal expressions into algebraic
expressions. Example 2 shows how to
translate algebraic expressions into
verbal expressions.
Properties of Equality
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.
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.
Substitution
✓ Formative Assessment
Use the Guided Practice exercises after
each example to determine students’
understanding of concepts.
Additional Examples
Identify Properties of Equality
EXAMPLE 3
1
Name the property illustrated by each statement.
a. If 3a - 4 = b, and b = a + 17, then 3a - 4 = a + 17.
Transitive Property of Equality
a. 7 less than a number n - 7
b. If 2g - h = 62, and h = 24, then 2g - 24 = 62.
Substitution Property of Equality
b. the square of a number
decreased by the product of
5 and the number x2 - 5x
✓Guided Practice
3. If -11a + 2 = -3a, then -3a = -11a + 2. Symmetric
2
Personal Tutor glencoe.com
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.
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.
b. 7y - 2 = 19 Seven times a
number minus 2 is 19.
Addition and Subtraction Properties of Equality
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.
Additional Examples also in
Interactive Classroom PowerPoint®
Presentations
Multiplication and Division Properties of Equality
Symbols
INTERACTIVE
IWB WHITEBOARD
READY
For any real numbers, a, b, and c, c ≠ 0, if a = b,
a
b
_
then a · c = b · c and _
c = c.
Examples
m
m
If _ = -7, then 8 · _ = 8 · (-7).
8
8
Properties of Equality
-2y
12
If -2y = 12, then _ = _.
-2
Write a verbal sentence to
represent each equation.
a. 6 = –5 + x Six is equal to
–5 plus a number.
Key Concept
Symbols
Write an algebraic expression to
represent each verbal expression.
Example 3 shows how to identify
properties of equality.
-2
Lesson 1-3 Solving Equations
19
Additional Example
Additional
Answers (Guided Practice)
19
018_025_C01_L03_892265.indd
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.
11/14/08
3:10:07 PM
3
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
Lesson 1-3 Solving Equations
0018-0025_C01L03_892270.indd 19
19
12/12/08 1:31:54 PM
Examples 4 and 5 show how to solve
one-step and multi-step equations.
Solve One-Step Equations
EXAMPLE 4
Solve each equation. Check your solution.
a. n - 3.24 = 42.1
n - 3.24 = 42.1
Additional Examples
4
Add 3.24 to each side.
n = 45.34
Solve each equation. Check your
solution.
Simplify.
The solution is 45.34.
n - 3.24 = 42.1
CHECK
a. m - 5.48 = 0.02 5.5
Original equation
Substitute 45.34 for n.
45.34 - 3.24 42.1
b. 18 = _t 36
5
Original equation
n - 3.24 + 3.24 = 42.1 + 3.24
1
2
Solve
53 = 3(y - 2) - 2(3y - 1).
–19
b.
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
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.
42.1 = 42.1 ✔
5
-_
x = 20
8
5
-_
x = 20
Original equation
8
_( )
_
_
5
- 8 -_
x = - 8 (20)
5
5
8
Multiply each side by - 8 .
5
x = -32
Simplify.
The solution is -32.
5
-_
x = 20
CHECK
5
the equation by -_
Original equation
8
5
-_
(-32) 20
8
is the same as
multiplying each side
8
by -_
.
5
Simplify.
Replace x with -32.
8
20 = 20 ✔
Simplify.
✓Guided Practice
2
4B. _y = -18 -27
4A. x - 14.29 = 25 39.29
3
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.
Personal Tutor glencoe.com
To solve an equation with more than one operation, undo operations by working
backward.
r
Study Tips offe
l
fu
lp
students he
t the
ou
ab
n
informatio
e
ar
topics they
studying.
EXAMPLE 5
Solve a Multi-Step Equation
Solve 5(x + 3) + 2(1 - x) = 14.
5(x + 3) + 2(1 - x) = 14
5x + 15 + 2 - 2x = 14
3x + 17 = 14
3x = -3
x = -1
Tips for New Teachers
Original equation
Apply the Distributive Property.
Simplify the left side.
Subtract 17 from each side.
Divide each side by 3.
✓Guided Practice
Checking Solutions Explain that
checking solutions to discover possible
errors is a vital procedure when you
use math on the job.
Solve each equation.
5B. 2(2x - 1) - 4(3x + 1) = 2 -1
5A. -10x + 3(4x - 2) = 6 6
Personal Tutor glencoe.com
20 Chapter 1 Equations and Inequalities
Differentiated Instruction
018_025_C01_L03_892265.indd
20
AL
ELL
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.
11/14/08
3:10:24 PM
20 Chapter 1 Equations and Inequalities
0018-0025_C01L03_892270.indd 20
12/15/08 9:15:19 AM
You can use properties to solve an equation for a variable.
Solve for a Variable
EXAMPLE 6
GEOMETRY The formula for the area A of a trapezoid is
A = 1 h(b1 + b2), where h represents the height, and b1
2
and b2 represent the measures of the bases. Solve the
formula for b2.
b1
_
1
A=_
h(b1 + b2)
b2
1
2A = 2_
h(b1 + b2)
Multiply each side by 2.
2A = h(b1 + b2)
Simplify.
2
h
h
Area formula
2
2A
_
=
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.
h(b1 + b2)
_
Additional Examples
6
Divide each side by h.
h
2A
_
= b1 + b2
Simplify.
h
2A
_
- b1 = b1 + b2 - b1
Subtract b1 from each side.
2A
_
- b1 = b2
Simplify.
h
h
✓Guided Practice
S - πr2
7
_
2
6. h = S - 2πr
what is the value of 4g - 2? C
59
41
C –_
A –_
36
9
67
41
D –_
B –_
7
9
Personal Tutor glencoe.com
Using Properties
There are often many
ways to solve a
problem. Using the
properties of equality
can help you find a
simpler way.
NGSSS PRACTICE EXAMPLE 7
formula for . = _
πr
STANDARDIZED TEST
4
PRACTICE If 4g + 5 = _,
9
2πr
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.
Test-TakingTip
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
912.A.3.5
d Homework
e leveled D. 41
A. 5
B. 11
C.ov
35id
Options pr
any of the
assignments. M
d,
cises are paire
Read the Test Item
homework exer
e
th
do
n
ents
stud
You are asked to find the value of 6xso+th
5.at
Note
you
doca
not have to find the value
thethat
on
s
en
ev
of x. Instead, you can use the Addition Property
of Equality
d the to make the left side of
odds one day an
the equation 6x + 5.
y.
da
the next
rentiate
fe5?
If 6x - 12 = 18, what is the value
TheofD6xif+
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
includes one or
Every chapter
ed
out Standardiz
more workedr
ila
sim
e
ar
that
Test Examples
e
at
st
und on
to problems fo
.
ts
en
assessm
Simplify.
The answer is C.
✓Guided Practice
BLOG Have students write a
8
7. If 5y + 2 = _
, what is the value of 5y - 6? G
blog entry to summarize how to
solve one-step equations. Make
sure that students use the
concept of inverse operations in
their explanations.
3
-20
F. _
3
-16
G. _
3
16
H. _
32
I. _
3
3
Personal Tutor glencoe.com
Lesson 1-3 Solving Equations
Differentiated Instruction
018_025_C01_L03_892265.indd
21
OL
BL
21
11/14/08
3:10:29 PM
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 in 2
Lesson 1-3 Solving Equations
0018-0025_C01L03_892270.indd 21
21
12/12/08 1:32:03 PM
✓ Check Your Understanding
nderstanding
Check Your U
tended to be
exercises are in
ass. Example
completed in cl
students where
references show
r review.
to look back fo
Example 1
p. 18
Write an algebraic expression to represent each verbal expression.
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 - x 2
Example 2
p. 18
Write a verbal sentence to represent each equation. 3–6. See margin.
5. 5y -
3
Example 3
PRACTICE
p. 19
✓ Formative Assessment
4. x2 - 9 = 27
3. 5x + 7 = 18
y3
x
6. _ + 8 = -16
= 12
4
Name the property illustrated by each statement.
8. If a = -3 and -3 = d, then a = d.
7. (8x - 3) + 12 = (8x - 3) + 12
Reflexive Property
Examples 4 and 5
p. 20
Use Exercises 1–21 to check for
understanding.
Transitive Property
Solve each equation. Check your solution.
9. z - 19 = 34 53
12. -6x = 42 -7
Use the chart at the bottom of this page
to customize assignments for your
students.
Example 6
p. 21
Additional Answers
Example 7
4. The difference between the square
of a number and 9 is 27.
11. -y = 8 -8
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.
8r - 3
Pv
19. 8r - 5q = 3, for q q =
20. Pv = nrt, for n
=n
rt
5
_
_
y
5
y
5
21. MULTIPLE CHOICE If _ + 8 = 7, what is the value of _ - 2? B
p. 21
3. The sum of five times a number
and 7 equals 18.
10. x + 13 = 7 -6
A -10
B -3
C 1
= Step-by-Step Solutions begin on page R20.
Extra Practice begins on page 947.
Practice and Problem Solving
5. The difference between five times
a number and the cube of that
number is 12.
Example 1
p. 18
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.
6. Eight more than the quotient of a
number and four is -16.
Example 2
29. n = number of p. 18
home runs Jacobs hit;
n + 6 = number of
home runs Cabrera hit;
2n + 6 = 46; Jacobs:
20 home runs,
Cabrera: 26 home
Example 3
runs.
p. 19
D 5
Write an algebraic expression to represent each verbal expression.
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 8x 2
24. fifteen less than the cube of a number x 3 - 15
25. five more than the quotient of a number and 4
26. Four less than 8 times a number is 16.
_x + 5
4
Write a verbal sentence to represent each equation.
26. 8x - 4 = 16
29
x+3
27. _ = 5
28. 4y2 - 3 = 13
4
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. (=)
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
22 Chapter 1 Equations and Inequalities
Differentiated Homework Options
018_025_C01_L03_892265.indd
Level
22
Assignment
11/14/08
3:10:35 PM
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)
22 Chapter 1 Equations and Inequalities
0018-0025_C01L03_892270.indd 22
12/12/08 1:32:13 PM
Study Guide and Intervention
pp. 17–18 AL OL ELL
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?
1-3
Study Guide and Intervention
Solving 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.
n = number of rides;
2(7.50) + n(2.50) = 32.50; 7
Entranc
e Fe
Rides: $ e: $7.50
2.50 ea
ch
Word Expression
Operation
and, plus, sum, increased by, more than
addition
minus, difference, decreased by, less than
subtraction
(
)
1
of a number
times, product, of as in −
multiplication
divided by, quotient
division
2
Example 1
Example 2
Write a verbal sentence to
represent 6(n - 2) = 14.
Write an algebraic
expression to represent 18 less than
the quotient of a number and 3.
Six times the difference of a number and two
is equal to 14.
n
−
- 18
3
Exercises
Solve each equation. Check your solution.
Write an algebraic expression to represent each verbal expression.
35. 3y + 4 = 19 5
36. -9x - 8 = 55 -7
37. 7y - 2y + 4 + 3y = -20 -3
38. 5g + 18 - 7g + 4g = 8 -5
39
5(-2x - 4) - 3(4x + 5) = 97 -6
2
3
41. _(6c - 18) + _
(8c + 32) = -18
3
4
1. the sum of six times a number and 25 6n + 25
40. -2(3y - 6) + 4(5y - 8) = 92 8
3
1
42. _(15d + 20) - _
(18d - 12) = 38
5
6
-3
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.
2. four times the sum of a number and 3 4(n + 3)
3. 7 less than fifteen times a number 15n - 7
4. the difference of nine times a number and the quotient of 6 and the same number
6
9n - −
n
5. the sum of 100 and four times a number 100 + 4n
Lesson 1-3
p. 20
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Examples 4 and 5
6. the product of 3 and the sum of 11 and a number 3(11 + n)
7. four times the square of a number increased by five times the same number 4n2 + 5n
8. 23 more than the product of 7 and a number 7n + 23
Write a verbal sentence to represent each equation. Sample answers are given.
9. 3n - 35 = 79 The difference of three times a number and 35 is equal to 79.
10. 2(n3 + 3n2) = 4n Twice the sum of the cube of a number and three times the
square of the number is equal to four times the number.
44. x = the number of
days she takes 2 pills;
4 + 2x = 28; 12 days
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?
Example 6
Solve each equation or formula for the specified variable.
f+d
E
45. E = mc2, for m m =
46. c(a + b) - d = f, for a a =
-b
c
c2 z
x
+
y
47. z = πq3h, for h
h= 3
48. _
a
=
b,
for
y
y
=
z(a
+
b)
-x
z
πq y - bx - c
bc - wx
_
2
49. y = ax + bx + c, for a a =
50. wx + yz = bc, for z z =
y
x2
5n
=n-8
11. −
17
Chapter 1
Practice
p. 20
005_042_A2CRMC01_890526.indd 17
B
_
_
_
1-3
AL
OL
BL
Practice
Solving Equations
1. 2 more than the quotient of a number and 5
2. the sum of two consecutive integers
y
5
−+2
n + (n + 1)
3. 5 times the sum of a number and 1
4. 1 less than twice the square of a number
5(m + 1)
2y 2 - 1
Write a verbal sentence to represent each equation.
51. GEOMETRY The formula for the volume of a cylinder with
radius r and height h is π times the radius times the radius
times the height.
r
6. 3y = 4y3
5. 5 - 2x = 4
The difference of 5 and twice a
number is 4.
5–8. Sample answers
are given.
Three times a number is 4 times
the cube of the number.
m
8. −
= 3(2m + 1) The quotient
7. 3c = 2(c - 1)
5
Three times a number is twice the
difference of the number and 1.
h
_
of a number and 5 is 3 times the
sum of twice the number and 1.
Name the property illustrated by each statement.
9. If t - 13 = 52, then 52 = t - 13.
10. If 8(2q + 1) = 4, then 2(2q + 1) = 1.
Symmetric (=)
Division (=)
11. If h + 12 = 22, then h = 10.
12. If 4m = -15, then -12m = 45.
Subtraction (=)
Multiplication (=)
13. 14 = 8 - 6r -1
14. 9 + 4n = -59 -17
1
1
3
5 −
1
15. −
-−
n=−
5
3
11 −
16. −
s+−
=−
17. -1.6r + 5 = -7.8 8
18. 6x - 5 = 7 - 9x −
4
8 4
2
6
3
7
19. 5(6 - 4v) = v + 21 −
12 5
4
4
5
Solve each equation or formula for the specified variable.
E
c
3c - 1
2
2d + 1
22. c = −, for d d = −
21. E = mc2, for m m = −2
Solve each equation. Check your solution.
1
6
20. 6y - 5 = -3(2y + 1) −
3
h + gt 2
t
23. h = vt - gt2, for v v = −
1 2
24. E = −
Iw + U, for I
2
I = 2 (E - U)
−
w2
53. 5x - 9 = 11x + 3 -2
1
7
_1 _
54. _
3
x+ =
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.
55. 5.4(3k - 12) + 3.2(2k + 6) = -136 -4
117
4
7
57. _y + 5 = -_
y-8 9
9
11
56. 8.2p - 33.4 = 1.7 - 3.5p 3
3
2
1 32
1
58. _z - _
=_
z+_
4
3
5 5
3
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. g = green fees per person; 6(2) + 4g = 76; $16
12
w = width; 2(2w) + 2w = 60; 10 cm
_
59. FINANCIAL LITERACY Benjamin spent $10,734 on his
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?
Expense
Annual Cost
Electric
x = the cost of rent each month; 622 + 428 +
240 + 144 + 12x = 10,734; $775 per month
$622
Gas
$428
Water
$240
Renter’s Insurance
$144
Lesson 1-3 Solving Equations
Enrichment
p. 22 OL
1-3
20
Chapter 1
Word Problem Practice
p. 21 AL OL BL
1-3
4/11/08 12:49:55 AM
ELL
Word Problem Practice
Solving 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.
United 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?
15
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).
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?
3:10:38 PM
BL
Enrichment
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?
F = 3R + 5.
23
11/14/08
t
3t
20 suitcases
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.
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?
321 = 4N + 1.
w
b. How many dominoes did Nancy have
in her hallway?
w + 10
80 dominoes
w = 6 units
10.5 = 6.3 + 1 + G - 1. Therefore G = 4.2 trillion dollars.
Chapter 1
21
Glencoe Algebra 2
C
2. In 2001, the U.S. trade deficit remain at 1 trillion dollars, investments also remain
Glencoe Algebra 2
005_042_A2CRMC01_890526.indd 20
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
_
4
005_042_A2CRMC01_890526.indd 21
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Lesson 1-3 Solving Equations
0018-0025_C01L03_892270.indd 23
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Solve each equation. Check your solution.
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
23
4/11/08 12:49:43 AM
ELL
Write an algebraic expression to represent each verbal expression.
_
a. Write this as an algebraic expression. V = π × r × r × h
V
b. Solve the expression in part a for h. h =
πr 2
018_025_C01_L03_892265.indd
Glencoe Algebra 2
Lesson 1-3
p. 21
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.
n+3
23
12/12/08 1:32:18 PM
C
Multiple Representations In
Exercise 61, students use a number line
and a table of values to illustrate the
workings of absolute values.
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
Watch Out!
c. Is this answer reasonable? Explain. See margin.
Error Analysis In Exercise 62,
encourage students to give a complete
explanation of the error. For example,
“Steven should have subtracted b from
2A
the entire left side, _, in the last step
h
of the equation.”
61
MULTIPLE REPRESENTATIONS The absolute value of a number describes the distance
of the number from zero. a.
-5 -4 -3 -2 -1 0 1 2 3 4 5
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.
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.
exercises help
Find the Error
y and address
students identif
before they
common errors
of
s in the margin
occur. Prompt
u
yo
lp
he
ition
the Teacher Ed
ward
to
ts
en
ud
guide st
understanding.
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
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.
1
62. ERROR ANALYSIS Steven and Jade are solving A = _
h(b1 + b2) for b2. Is either of them
2
correct? Explain your reasoning.
61b.
Integer
–5
–4
–3
–2
–1
0
1
2
3
4
5
Distance
from Zero
5
4
3
2
1
0
1
2
3
4
5
A = _12 h(b1 + b2)
1
2A = (b + b )
_
1
2
2A
_
= (b1 + b2)
h
2A – b1
_
=b
h
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.
Jade
Steven
A=_
h(b1 + b2)
2
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.
Use Higher-Order Thinking Skills
63. CHALLENGE Solve d =
2
h
2A – b = b
_
1
2
h
(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:
(10x - 30)
• Take any number.
+ 6 = 2x
5
• Multiply it by ten.
(10x - 30)
• Subtract 30 from the result.
= 2x - 6
5
• Divide the new result by 5.
• Add 6 to the result.
(2x - 6) + 6 = 2x
• 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.
24 Chapter 1 Equations and Inequalities
y
61c.
018_025_C01_L03_892265.indd
8
24
4
-8
-4
0
4
8x
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.
11/14/08
3:10:44 PM
-4
-8
24 Chapter 1 Equations and Inequalities
0018-0025_C01L03_892270.indd 24
12/12/08 1:32:28 PM
NGSSS PRACTICE
4
912.A.3.6, 912.G.2.4, 912.A.2.13
67. The graph shows the solution of which
inequality? D
y
69. GEOMETRY Which
of the following
describes the
transformation of
ABC to ABC? A
y
"
Yesterday’s News Have each student
write how yesterday’s concepts helped
with today’s new material.
"'
#
x
$ #'
✓ Formative Assessment
$'
0
2
A. y < _
x+4
3
2
x+4
B. y > _
3
x
2
3
D. y > _
x+4
2
1
68. SAT/ACT What is 1_
subtracted from its
3
reciprocal? F
7
F. -_
12
_
G. - 1
12
1
H. _
Check for student understanding of
concepts in Lesson 1-3.
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
3
C. y < _
x+4
70.
4
_
I. 3
4
ASSESS
Quiz 2, p. 45
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%
Spiral Review
71. Simplify 3x + 8y + 5z - 2y - 6x + z. (Lesson 1-2) -3x + 6y + 6z
1
72. BAKING Tamera is making two types of bread. The first type of bread needs 2_
cups of
2
3
_
flour, and the second needs 1 cups of flour. Tamera wants to make 2 loaves of the first
4
recipe and 3 loaves of the second recipe. How many cups of flour does she need? (Lesson 1-2) 10 1 c
4
73. LANDMARKS Suppose the Space Needle in Seattle,
Washington, casts a 220-foot shadow at the same time a
1
feet
nearby tourist casts a 2-foot shadow. If the tourist is 5_
2
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
h ft
5.5 ft
2 ft
220 ft
Skills Review
Identify the additive inverse for each number or expression. (Lesson 1-2)
1 1
75. -4_
4
76. 3.5 -3.5
77. -2x 2x
5 5
2
2
_
79. 3
-3
80. -1.25 1.25
81. 5x -5x
3
3
_
_
78. 6 - 7y -6 + 7y
82. 4 - 9x -4 + 9x
Lesson 1-3 Solving Equations
018_025_C01_L03_892265.indd
25
25
11/14/08
3:10:52 PM
Lesson 1-3 Solving Equations
0018-0025_C01L03_892270.indd 25
25
12/12/08 1:32:32 PM
Mid-Chapter
Quiz
Mid-Chapter Quiz
NGSSS
912.A.2.13, 912.A.3.5, 912.G.2.5, 912.G.7.5
Lessons 1-1 through 1-3
1
1. Evaluate 3c - 4(a + b) if a = -1, b = 2 and c = _
.
✓ Formative Assessment
(Lesson 1-1)
Use the Mid-Chapter Quiz to assess
students’ progress in the first half of
the chapter.
3
-3
3. Evaluate (5 - m)3 + n(m - n) if m = 6 and n = -3.
(Lesson 1-1)
-28
(Lesson 1-1)
5
46
47
F. _m - _
n
15
15
G. 46m - 47n
mn
H. -_
15
5
9
I. _m - _
n
4
8
14. Identify the additive inverse and the multiplicative
6
7
7
inverse for _
. (Lesson 1-2) additive: - ; mult.:
6
7
6
_
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?
Customize and
create multiple
versions of your Mid-Chapter Quiz and
their answer keys.
2
1
(4m - 5n) + _
(2m + n)? (Lesson 1-2) F
to _
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
For problems answered incorrectly,
have students review the lessons
indicated in parentheses.
13. NGSSS PRACTICE Which expression is equivalent
73.54 units2
_
15. Write a verbal sentence to represent the equation
a
_
= 1. (Lesson 1-3) The quotient of a number a and
a-3
the difference of a number a and 3 is equal to 1.
16. Solve 6x + 4y = -1 for x. (Lesson 1-3) x = - 2 y - 1
_ _
3
Follow-Up
z
x
A. 4n - 13
B. 4(n - 13)
y
Before students complete the
Mid-Chapter Quiz, encourage them to
review the information for Lessons 1-1
through 1-3 in their Foldables.
6
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) B
q2
+ rt
5. NGSSS PRACTICE What is the value of _ if
qr - 2t
q = -4, r = 3, and t = 8? (Lesson 1-1) C
17
A. -_
6
1
B. -_
6
_
C. - 10
7
2
D. -_
7
4
C. _
n - 13
4n
D. _
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
15.5
Name the sets of numbers to which each number
belongs. (Lesson 1-2)
25
6. _ Q, R
7. -_ Z, Q, R
8. √
50 I, R
9. -32.4 Q, R
h
6
128
32
11
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)
-27a + 14b
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) +
" = 80.625
20. GEOMETRY The formula for the surface area of a
sphere is SA = 4πr2, and the formula for the volume
4 3
of a sphere is V = _
πr . (Lesson 1-3)
3
a. Find the volume and surface area of a sphere
with radius 2 inches. Write your answers in
terms of π. 32 π in3; 16π in2
3
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
_
3(26.50)
26 Chapter 1 Equations and Inequalities
Intervention Planner
Tier
Tier
1
On Level
If
students miss about 25% of the exercises or less,
Then
SE
CRM
TE
2
026_026_C01_MCQ_892265.indd
choose a resource:
Lessons 1-1, 1-2, and 1-3
Skills Practice, pp. 7, 13, and 19
If
Then
CRM
Strategic Intervention
Tier
approaching grade level
3
students miss about 50% of the exercises,
If
26
Intensive Intervention
2 or more grades below level
11/14/08
3:11:21 PM
students miss about 75% of the
exercises,
choose a resource:
Study Guide and Intervention, Chapter 1,
pp. 5, 11, 17
Then
use Math Triumphs, Alg. 2,
Ch. 1 and 2
Chapter Project, p. 2
FL Math Online
Self-Check Quiz
FL Math Online
Homework Help
Extra Examples, Personal Tutor,
FL Math Online Extra Examples,
Personal Tutor, Homework Help,
Review Vocabulary
26 Chapter 1 Equations and Inequalities
0026_C01MCQ_892270.indd 26
12/12/08 1:30:13 PM