2-5
Equations and Problem Solving
2-5
1. Plan
Lesson Preview
What You’ll Learn
Check Skills You’ll Need
(For help, go to Lesson 1-1.)
OBJECTIVE
1
To define a variable in
terms of another
variable
Write a variable expression for each situation.
To model distancerate-time problems
3. number of miles traveled at 34 mi/h in h hours 34h
4. weight of 5 crates if each crate weighs x kilograms 5x
1. value in cents of q quarters 25q
OBJECTIVE
2
. . . And Why
2. twice the length O 2/
2-1
Lesson 1-1: Example 1
Exercises 1–8
Extra Practice, p. 702
New Vocabulary • consecutive integers • uniform motion
Lesson Resources
Interactive lesson includes instant
self-check, tutorials, and activities.
OBJECTIVE
✓Check Skills You’ll Need
Using Variables
5. cost of n items at $3.99 per item 3.99n
To solve real-world problems
involving distance, rate, and
time, as in Examples 3–5
Lesson Preview
Teaching Resources
Practice, Reteaching, Enrichment
Defining
Part 1 Variables
Using Equations to Solve Problems
Reaching All Students
Some problems contain two or more unknown quantities. To solve such problems,
first decide which unknown quantity the variable will represent. Then express the
other unknown quantity or quantities in terms of that variable.
1
EXAMPLE
Defining One Variable in Terms of Another
Geometry The length of a rectangle is 6 in. more than its width. The perimeter of
the rectangle is 24 in. What is the length of the rectangle?
Relate The length is 6 in. more than the width.
Problem Solving Hint
For Example 1, drawing
a diagram will help you
understand the
problem.
Then w + 6 = the length.
w⫹6
w
Computer Test Generator CD
P = 2O + 2w
Use the perimeter formula.
24 = 2(w + 6) + 2w
Substitute 24 for P and w ± 6 for O.
24 = 2w + 12 + 2w
Use the Distributive Property.
24 = 4w + 12
Combine like terms.
24 - 12 = 4w + 12 - 12
3=w
Technology
Resource Pro® CD-ROM
Computer Test Generator CD
Prentice Hall Presentation Pro CD
Subtract 12 from each side.
12 = 4w
12 = 4w
4
4
www.PHSchool.com
Simplify.
Divide each side by 4.
Simplify.
The width of the rectangle is 3 in. The length of the rectangle is 6 in. more than the
width. So the length of the rectangle is 9 in.
Check Understanding
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Transparencies
• Check Skills You’ll Need 2-5
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• Lesson Quiz 2-5
PH Presentation Pro CD 2-5
The length is described in terms of the width.
So define a variable for the width first.
Define Let w = the width.
Write
Practice Workbook 2-5
Spanish Practice Workbook 2-5
Basic Algebra Planning Guide 2-5
1 The width of a rectangle is 2 cm less than its length. The perimeter of the rectangle
is 16 cm. What is the length of the rectangle? 5 cm
Student Site
• Teacher Web Code: aek-5500
• Reasoning & Puzzles pp. 38,
39, 46
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Plus
Lesson 2-5 Equations and Problem Solving
103
✓ Ongoing Assessment and Intervention
Before the Lesson
During the Lesson
After the Lesson
Diagnose prerequisite skills using:
• Check Skills You’ll Need
Monitor progress using:
• Check Understanding
• Additional Examples
• Standardized Test Prep
Assess knowledge using:
• Lesson Quiz
• Computer Test Generator CD
103
2. Teach
Consecutive integers differ by 1. The integers 50 and 51 are consecutive integers,
and so are -10, -9, and -8. For consecutive integer problems, it may help to define a
variable before describing the problem in words. Let a variable represent one of the
unknown integers. Then define the other unknown integers in terms of the first one.
Math Background
Time-motion relationships occur
in the real world in situations
ranging from the atomic to the
astronomic. Students will use the
concepts of the lesson in much
future work in mathematics.
2
Consecutive Integer Problem
EXAMPLE
The sum of three consecutive integers is 147. Find the integers.
Define Let n = the first integer.
Then n + 1 = the second integer,
and n + 2 = the third integer.
OBJECTIVE
1
1
Teaching Notes
Relate
first
integer
plus
second
integer
plus
third
integer
is
147
Math Tip
Write
n
+
n+ 1
+
n+ 2
=
147
EXAMPLE
Defining the variable is a key step
in interpreting the solution of the
equation.
n + n + 1 + n + 2 = 147
3n + 3 = 147
Combine like terms.
3n + 3 - 3 = 147 - 3
2
EXAMPLE
Visual Learners
Make sure students understand
consecutive integers. Have them
look at 4 and 5 on a number line.
Ask students to give their own
definition of consecutive integers
using the number line. On a
number line, consecutive
integers do not have any other
integers between them. Ask
students to name three
consecutive integers. Answers
may vary. Sample: 7, 8, 9
Subtract 3 from each side.
3n = 144
Simplify.
3n = 144
3
3
Divide each side by 3.
n = 48
Simplify.
If n = 48, then n + 1 = 49, and n + 2 = 50. The three integers are 48, 49, and 50.
Check Is the solution correct? Yes; 48 + 49 + 50 = 147.
Check Understanding
2 The sum of three consecutive integers is 48.
a. Define a variable for one of the integers. Let x ≠ the first integer.
b. Write expressions for the other two integers. x ± 1 is the second integer
and x ± 2 is
c. Write and solve an equation to find the three integers.
3x + 3 = 48; 15, 16, 17
the third integer.
OBJECTIVE
2
Part
1 2 Distance-Rate-Time Problems
An object that moves at a constant rate is said to be in uniform motion.
The formula d = rt gives the relationship between distance d, rate r, and time t.
Uniform motion problems may involve objects going the same direction, opposite
directions, or round trips.
In the diagram below, the two vehicles are traveling the same direction at different
rates. The distances the vehicles travel are the same.
40 mi/h • 5 h
200 mi
50 mi/h • 4 h
Since the distances are equal, the products of rate and time for the two cars are
equal. For the vehicles shown, 40 ? 5 = 50 ? 4.
104
Chapter 2 Solving Equations
Reaching All Students
104
Below Level Emphasize how helpful it
is for students to organize the
information from a distance-rate-time
problem into a table before attempting
to solve the problem.
Advanced Learners Remind
students that the units of distance
used to solve a problem must be
consistent.
Visual Learners
See note on page 104.
Tactile Learners
See note on page 105.
Additional Examples
A table can also help you understand relationships in distance-rate-time
problems.
3
EXAMPLE
Same-Direction Travel
Engineering A train leaves a train station at 1 P.M. It travels at an average rate
of 60 mi/h. A high-speed train leaves the same station an hour later. It travels at
an average rate of 96 mi/h. The second train follows the same route as the first
train on a track parallel to the first. In how many hours will the second train
catch up with the first train?
1 The width of a rectangle is
3 in. less than its length. The
perimeter of the rectangle is
26 in. What is the width of the
rectangle? 5 in.
2 The sum of three consecutive
integers is 72. Find the integers.
23, 24, 25
OBJECTIVE
Define Let t = the time the first train travels.
2
Then t - 1 = the time the second train travels.
Relate
Write
Train
Rate
Time
Distance Traveled
1
60
t
60t
2
96
t⫺1
96(t ⫺ 1)
60t = 96(t - 1)
The distances traveled by the trains are equal.
60t = 96t - 96
Use the Distributive Property.
60t - 60t = 96t - 96 - 60t
Subtract 60t from each side.
0 = 36t - 96
Combine like terms.
0 + 96 = 36t - 96 + 96
Add 96 to each side.
96 = 36t
96
36t
36 = 36
t = 2 23
t - 1 = 1 23
3
EXAMPLE
Tactile Learners
Since each distance is represented
by a different expression, some
students may not understand how
the distances can be the same.
Have two students demonstrate.
Both students stand at the same
starting place. One student starts
walking very slowly. The second
student waits a few moments.
Then the second student starts
walking quickly to catch up with
the first student. Show how they
have traveled the same distance
but at different speeds or rates.
Simplify.
Divide each side by 36.
Simplify.
Find the time the second train travels.
Additional Example
The second train will catch up with the first train in 1 23 h.
Check Understanding
Teaching Notes
3 A group of campers and one group leader left a campsite in a canoe. They traveled
at an average rate of 10 km/h. Two hours later, the other group leader left the
campsite in a motorboat. He traveled at an average rate of 22 km/h.
a. How long after the canoe left the campsite did the motorboat catch up with it?
b. How long did the motorboat travel?
3 23 h
3
For uniform motion problems that involve a round trip, it is important to
remember that the distance going is equal to the distance returning.
3 An airplane left an airport
flying at 180 mi/h. A jet that flies
at 330 mi/h left 1 hour later. The
jet follows the same route as the
airplane at a different altitude.
How many hours will it take the
jet to catch up with the airplane?
115 h
20 mi/h • 3 h
60 mi
30 mi/h • 2 h
60 mi
Since the distances are equal, the products of rate and time for traveling in both
directions are equal. That is, 20 ? 3 = 30 ? 2.
Lesson 2-5 Equations and Problem Solving
105
105
Additional Examples
4
4 Suppose you hike up a hill at
4 km/h. You hike back down at
6 km/h. Your hiking trip took
3 hours. How much time did it
take you to hike up the hill? 1.8 h
EXAMPLE
Noya drives into the city to buy a software program at a computer store. Because
of traffic conditions, she averages only 15 mi/h. On her drive home she averages
35 mi/h. If the total travel time is 2 hours, how long does it take her to drive to the
computer store?
5 Two jets leave Dallas at the
same time and fly in opposite
directions. One is flying west
50 mi/h faster than the other.
After 2 hours, the jets are
2500 miles apart. Find the speed
of each jet.
jet flying east: 600 mi/h;
jet flying west: 650 mi/h
Define Let t = time of Noya’s drive to the computer store.
Reading Math
2 - t = the time of Noya’s drive home.
The total travel time is
for a round trip. If it
takes x out of a 2-hour
round trip to get to
the store, then 2 – x is
the time it will take for
the drive home.
Relate
Write
4
EXAMPLE
Error Prevention
EXAMPLE
Tactile Learner
Rate
Time
Distance
To the computer store
15
t
15t
Return home
35
2⫺t
35(2 ⫺ t)
Noya drives
15t miles to the
computer store and
35(2 ⴚ t) miles back.
15t = 35(2 - t)
The distances traveled to and from the store are equal.
15t = 70 - 35t
Use the Distributive Property.
Add 35t to each side.
50t = 70
Combine like terms.
50t = 70
50
50
Divide each side by 50.
t = 1.4
Simplify.
It took Noya 1.4 h to drive to the computer store.
Check Understanding
Make a number line on the floor
with masking tape. Have two
students stand beside each other
at 0. Instruct the students to
walk in opposite directions for
2 seconds. Have one student walk
very slowly and one walk briskly.
Lead students to note that the
two distances are not equal, but
combine to make a total distance
walked.
4 On his way to work from home, your uncle averaged only 20 miles per hour. On his
drive home, he averaged 40 miles per hour. If the total travel time was 1 12 hours,
how long did it take him to drive to work? 1h
For uniform motion problems involving two objects
moving in opposite directions, you can write
equations using the fact that the sum of their
distances is the total distance.
5
EXAMPLE
9 mi
3 mi
6 mi
Opposite-Direction Travel
Jane and Peter leave their home traveling in opposite directions on a straight road.
Peter drives 15 mi/h faster than Jane. After 3 hours, they are 225 miles apart. Find
Peter’s rate and Jane’s rate.
Closure
Ask: What are two ways to help
set up distance problems like the
ones in this lesson? Answers may
vary. Sample: draw a diagram
and use a table
Define Let r = Jane’s rate.
Then r + 15 = Peter’s rate.
Relate
Write
106
106
Part of Noya’s Travel
15t + 35t = 70 - 35t + 35t
Students may write t - 2 instead
of 2 - t. Ask: Which is greater,
the total travel time or the time
to drive to the computer store?
total travel time Which makes
more sense, t - 2 or 2 - t? 2 – t
5
Round-Trip Travel
Chapter 2 Solving Equations
Person
Rate
Time
Distance
Jane
r
3
3r
Peter
r ⫹ 15
3
3(r ⫹ 15)
3r + 3(r + 15) = 225
Jane’s distance is 3r. Peter’s
distance is 3(r ⴙ 15).
The sum of Jane’s and Peter’s distances is the
total distance, 225 miles.
3. Practice
3r + 3(r + 15) = 225
3r + 3r + 45 = 225
6r + 45 = 225
Use the Distributive Property.
Combine like terms.
Assignment Guide
6r + 45 - 45 = 225 - 45 Subtract 45 from each side.
6r = 180
Simplify.
6r = 180
6
6
Divide each side by 6.
r = 30
1
Objective
A B Core 1–9, 16–20,
25, 28–31
C Extension 32, 34
2
Objective
A B Core 10–15, 21–24,
Simplify.
Jane’s rate is 30 mi/h, and Peter’s rate is 15 mi/h faster, which is 45 mi/h.
Check Understanding
5 Sarah and John leave Perryville traveling in opposite directions on a straight road.
Sarah drives 12 miles per hour faster than John. After 2 hours, they are 176 miles
apart. Find Sarah’s speed and John’s speed. John: 38 mi/h; Sarah: 50 mi/h
26–27
C
EXERCISES
For more practice, see Extra Practice.
Extension 33
Standardized Test Prep 35–39
Practice
Practiceand
andProblem
ProblemSolving
Solving
Mixed Review 40–50
Practice by Example
Example 1
(page 103)
1. The length of a rectangle is 3 in. more than its width. The perimeter of the
c. 2w ± 2(w ± 3) ≠ 30; 6
rectangle is 30 in.
a. Define a variable for the width. Let w ≠ width
b. Write an expression for the length in terms of the width. /≠ w ± 3
c. Write an equation to find the width of the rectangle. Solve your equation.
See above.
d. What is the length of the rectangle? 9 in.
2. The length of a rectangle is 8 in. more than its width. The perimeter of the
rectangle is 24 in. What are the width and length of the rectangle? 2 in.; 10 in.
3. The width of a rectangle is one half its length. The perimeter of the rectangle is
54 cm. What are the width and length of the rectangle? 9 cm; 18 cm
Exercise 6 Suggest to students
that they first think of any even
whole number and the next
greater even whole number for
part b.
Enrichment 2-5
Reteaching 2-5
Practice 2-5
Name
Example 2
(page 104)
4. The length of a rectangular garden is 3 yd more than twice its width. The
perimeter of the garden is 36 yd. What are the width and length of the garden?
5 yd; 13 yd
5. The sum of the two consecutive integers is -35. If n = the first integer, which
equation best models the situation? C
A. n(n + 1) = -35
B. n + 2n = -35
C. n + (n + 1) = -35
D. n + (2n + 1) = -35
6. The sum of two consecutive even integers is 118.
a. Define a variable for the smaller integer. Let n ≠ the first integer.
b. What must you add to an even integer to get the next greater even integer? 2
c. Write an expression for the second integer. n ± 2
d. Write and solve an equation to find the two even integers.
n ± n ± 2 ≠ 118; 58, 60
7. The sum of two consecutive odd integers is 56.
a. Define a variable for the smaller integer. Let n ≠ the first integer.
b. What must you add to an odd integer to get the next greater odd integer? 2
c. Write an expression for the second integer. n ± 2
d. Write and solve an equation to find the two odd integers.
n ± n ± 2 ≠ 56; 27, 29
8. The sum of three consecutive integers is 915. What are the integers?
304, 305, 306
9. The sum of two consecutive even integers is -298. What are the integers?
–148, –150
Lesson 2-5 Equations and Problem Solving
Class
Date
Practice 2-5
Equations and Problem Solving
Write and solve an equation for each situation.
1. A passenger train’s speed is 60 mi/h, and a freight train’s speed is
40 mi/h. The passenger train travels the same distance in 1.5 h less time
than the freight train. How long does each train take to make the trip?
2. Lois rode her bike to visit a friend. She traveled at 10 mi/h. While she
was there, it began to rain. Her friend drove her home in a car traveling
at 25 mi/h. Lois took 1.5 h longer to go to her friend’s than to return
home. How many hours did it take Lois to ride to her friend’s house?
3. May rides her bike the same distance that Leah walks. May rides her
bike 10 km/h faster than Leah walks. If it takes May 1 h and Leah 3 h to
travel that distance, how fast does each travel?
4. The length of a rectangle is 4 in. greater than the width. The perimeter
of the rectangle is 24 in. Find the dimensions of the rectangle.
5. The length of a rectangle is twice the width. The perimeter is 48 in. Find
the dimensions of the rectangle.
6. At 10:00 A.M., a car leaves a house at a rate of 60 mi/h. At the same
time, another car leaves the same house at a rate of 50 mi/h in the
opposite direction. At what time will the cars be 330 miles apart?
7. Marla begins walking at 3 mi/h toward the library. Her friend meets
her at the halfway point and drives her the rest of the way to the
library. The distance to the library is 4 miles. How many hours did
Marla walk?
© Pearson Education, Inc. All rights reserved.
A
8. Fred begins walking toward John’s house at 3 mi/h. John leaves his
house at the same time and walks toward Fred’s house on the same
path at a rate of 2 mi/h. How long will it be before they meet if the
distance between the houses is 4 miles?
9. A train leaves the station at 6:00 P.M. traveling west at 80 mi/h. On a
parallel track, a second train leaves the station 3 hours later traveling
west at 100 mi/h. At what time will the second train catch up with
the first?
10. It takes 1 hour longer to fly to St. Paul at 200 mi/h than it does to return
at 250 mi/h. How far away is St. Paul?
11. Find three consecutive integers whose sum is 126.
12. The sum of four consecutive odd integers is 216. Find the four integers.
13. A rectangular picture frame is to be 8 in. longer than it is wide. Dennis
uses 84 in. of oak to frame the picture. What is the width of the frame?
14. Each of two congruent sides of an isosceles triangle is 8 in. less than
twice the base. The perimeter of the triangle is 74 in. What is the length
of the base?
Algebra 1 Chapter 2
Lesson 2-5 Practice
5
107
107
Exercise 10 Suggest to students
that they use 0.75 instead of 34 for
the time.
Example 3
(page 105)
Exercises 18, 19 Help students
see that odd integers can be
represented the same way as
even integers here. Ask: What is
the difference between two
consecutive odd integers? 2
What is the difference between
two consecutive even integers? 2
10. A moving van leaves a house traveling at an average rate of 35 mi/h. The family
leaves the house 34 hour later following the same route in a car. They travel at
an average rate of 50 mi/h.
a. Define a variable for the time traveled by the moving van. Let t ≠ time for
b. Write an expression for the time traveled by the car. t – 3 the moving van.
4
c. Copy and complete the table.
Vehicle
Moving van
Car
Rate
Time
■ 35
■t
■ 50
3
4
t■
–
Distance Traveled
■ 35t
■ 50 Qt – 34 R
d. Write and solve an equation to find out how long it will take the car to catch
up with the moving van. 35t ≠ 50Qt – 43 R; t ≠ 2 21 , 2 12 – 43 ≠ 1 34 h
11. Air Travel A jet leaves the Charlotte, North Carolina, airport traveling at an
average rate of 564 km/h. Another jet leaves the airport one half hour later
traveling at 744 km/h in the same direction. How long will the second jet take
to overtake the first? 1 17
30 h
Example 4
(page 106)
12b. 22x ≠ 72 – 32x; 113 h
12. Juan drives to work. Because of traffic conditions, he averages 22 miles per
hour. He returns home averaging 32 miles per hour. The total travel time
is 2 14 hours.
a. Define a variable for the time Juan takes to travel to work. Write an
expression for the time Juan takes to return home. x; 2 14 – x
b. Write and solve an equation to find the time Juan spends driving to work.
13. Air Travel An airplane flies from New Orleans, Louisiana, to Atlanta, Georgia,
at an average rate of 320 miles per hour. The airplane then returns at an
average rate of 280 miles per hour. The total travel time is 3 hours.
a. Define a variable for the flying time from New Orleans to Atlanta. Write an
expression for the travel time from Atlanta to New Orleans. x; 3 – x
b. Write and solve an equation to find the flying time from New Orleans
to Atlanta. 320x ≠ 840 – 280x; 125 h
Example 5
(page 106)
B
Apply Your Skills
14. John and William leave their home traveling in opposite directions on a straight
road. John drives 20 miles per hour faster than William. After 4 hours they are
250 miles apart.
x; x – 20
a. Define a variable for John’s rate. Write an expression for William’s rate.
b. Write and solve an equation to find John’s rate. Then find William’s rate.
4x ± 4x – 80 ≠ 250; 4114 mi/h; 2114 mi/h
15. Two bicyclists ride in opposite directions. The speed of the
first bicyclist is 5 miles per hour faster than the second.
After 2 hours they are 70 miles apart. Find their rates. 15 mi/h; 20 mi/h
16 a. Which of the following numbers is not the sum of three
consecutive integers? 61
I. 51
II. 61
III. 72
IV. 81
b. Critical Thinking What common trait do the other numbers share?
They are all multiples of three.
17. Geometry The length of a rectangle is 8 cm more than twice the width. The
perimeter of the rectangle is 34 cm. What is the length of the rectangle? 14 cm
18. The sum of four consecutive even integers is 308. Write and solve an equation
to find the four integers. x ± x ± 2 ± x ± 4 ± x ± 6 ≠ 308; 74, 76, 78, 80
19. The sum of three consecutive odd integers is -87. What are the integers?
–27, –29, –31
108
108
Chapter 2 Solving Equations
20. The tail of a kite is 1.5 ft plus twice the length of the kite. Together, the kite and
tail are 15 ft 6 in. long.
a. Write an expression for the length of the kite and tail together. 1.5 ± 2x ± x
b. Write 15 ft 6 in. in terms of feet. 15.5 ft
c. Write and solve an equation to find the length of the tail. 5
3x ± 1.5 ≠ 15.5; 10 6 ft or 10 ft 10 in.
21. Travel A bus traveling at an average rate of 30 miles per hour left the city at
11:45 A.M. A car following the bus at 45 miles per hour left the city at noon. At
what time did the car catch up with the bus? 12:30 P.M.
22. Ellen and Kate raced on their bicycles to the library after school. They both
left school at 3:00 P.M. and bicycled along the same path. Ellen rode at a speed
of 12 miles per hour and Kate rode at 9 miles per hour. Ellen got to the library
15 minutes before Kate.
a. How long did it take Ellen to get to the library? 45 min
b. At what time did Ellen get to the library? 3:45
23. At 1:30 P.M., Tom leaves in his boat from a dock and heads south. He travels at
a rate of 25 miles per hour. Ten minutes later, Mary leaves the same dock in her
speedboat and heads after Tom. If she travels at a rate of 30 miles per hour,
when will she catch up with Tom? 2:30 P.M.
24. Air Travel Two airplanes depart from an airport traveling in opposite
directions. The second airplane is 200 miles per hour faster than the first. After
2 hours they are 1100 miles apart. Find the speeds of the airplanes.
175 mi/h; 375 mi/h
25. Three friends were born in consecutive years. The sum of their birth years is
5961. Find the year in which each person was born. 1986, 1987, 1988
26. Two boats leave a ramp traveling in opposite directions. The second boat is
10 miles per hour faster than the first. After 3 hours they are 150 miles apart.
Find the speeds of the boats. first boat: 20 mi/h; second boat: 30 mi/h
27. Travel A truck traveling 45 miles per hour and a train traveling 60 miles per
hour cover the same distance. The truck travels 2 hours longer than the train.
How many hours does each travel? truck: 8 h; train: 6 h
31b. Yes; if n is the
middle integer,
n – 1 is the
previous integer
and n ± 1 is the
next integer. The
three integers
would be
consecutive.
C
Challenge
28. Electricity A group of ten
Batteries in Series
6- and 12-volt batteries are
wired in series as shown at
the right. The sum of their
voltages is 84 volts. How
many of each type of
battery are used?
6 6-V; 4 12-V
29. Writing Describe the steps you would use to solve consecutive integer
problems. See margin.
30. Open-Ended Write a word problem that could be solved using
the equation 35(t - 1) = 20t. See margin.
31. a. Write and solve an equation to find three consecutive integers with
a sum of 126. Let n = the first integer. n ± n ± 1 ± n ± 2 ≠ 126; 41, 42, 43
b. Critical Thinking In part (a), could you solve the problem by letting n = the
middle integer, n - 1 = the smallest integer, and n + 1 = the largest integer?
See left.
32. Geometry A triangle has a perimeter of 165 cm. The first side is 65 cm less
than twice the second side. The third side is 10 cm less than the second side.
Write and solve an equation to find the length of each side of the triangle.
x ± 2x – 65 ± x – 10 ≠ 165; 60; 55 cm, 60 cm, 50 cm
Lesson 2-5 Equations and Problem Solving
pages 107–110 Exercises
29. Answers may vary.
Sample: Define a variable
to represent the first
integer. Use this variable
to write expressions for
the other integers. Write
an equation that describes
how the integers are
related. Solve this equation
to find the integers.
30. Answers may vary. Sample:
Jeff and Anne both left
school for the city at the
4. Assess
Lesson Quiz 2-5
1. The sum of three
consecutive integers is 117.
Find the integers. 38, 39,
40
2. You and your brother
started biking at noon
from places that are
52 mi apart. You rode
toward each other and
met at 2:00 P.M. Your
brother’s average speed
was 4 mi/h faster than
your average speed. Find
both speeds.
your speed: 11 mi/h;
brother’s speed: 15 mi/h
3. Joan ran from her home to
the lake at 8 mi/h. She ran
back home at 6 mi/h. Her
total running time was
32 minutes. How much
time did it take Joan to run
from her home to the lake?
about 13.7 minutes
Alternative Assessment
Organize students in groups of 5.
In each group, assign each
student one of the examples from
the lesson. Instruct each student
to write a problem similar to the
assigned example on a card. Have
students in each group exchange
cards and solve the problem given
to them. Repeat until each
student in a group has solved
each problem. Have students
compare answers and strategies.
109
same time. Jeff drove
35 mi/h and Anne drove
20 mi/h. Jeff arrived 1 h
before Anne. How long
did each drive?
109
Standardized Test Prep
33. At 9:00 A.M., your friends begin hiking at 2 mi/h. You begin from the same
place at 9:25 A.M. You hike at 3 mi/h.
5
a. How long will you have hiked when you catch up with your friends? 6 h
b. At what time will you catch up with your friends? 10:15 A.M.
Resources
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p. 129
• Test-Taking Strategies, p. 124
• Test-Taking Strategies with
Transparencies
34. Find five consecutive odd integers such that the sum of the first and the fifth is
one less than three times the fourth. –9, –7, –5, –3, –1
Standardized
Test Prep Test Prep
Standardized
Multiple Choice
35. Solve 3n - 7 + 2n = 8n + 11. A
B. 11
A. -6
3
C. 33
5
D. 9
36. Which expression represents the sum of 3 odd integers of which n is the
least integer? H
F. n + 3
G. 3n + 3
H. 3n + 6
I. 3n + 7
37. Which equation does NOT have -2 as its solution? B
A. 2x + 5 = 5x + 11
B. 7n + 9 = 3 - 9n
C. 3k + 6 - 4k = k + 10
D. 4 + 3q = 7q + 12
38. A truck traveling at an average rate of 45 miles per hour leaves a rest stop.
Fifteen minutes later a car traveling at an average rate of 60 miles per hour
leaves the same rest stop traveling the same route. How long will it take
for the car to catch up with the truck? G
F. 15 minutes
G. 45 minutes
H. 1 hour 15 minutes
I. 3 hours
Take It to the NET
Online lesson quiz at
www.PHSchool.com
39. The perimeter of the triangle at
the right is 22.6 in. What is the
value of n?
A
A. 3.5
B. 4.6
C. 7.8
D. 9.4
n in.
(n ⫹ 5.2) in.
(2n ⫹ 3.4) in.
Web Code: aea-0205
Mixed Mixed
ReviewReview
Lesson 2-4
Solve each equation. If the equation is an identity, write identity. If it has no
solution, write no solution.
40. 2x = 7x + 10 –2
41. 2q + 4 = 4 - 2q 0
42. 0.5t + 3.6 = 4.2 - 1.5t 0.3
43. 2x + 5 + x = 2(3x + 3) –13
44. 4 + x + 3x = 2(2x + 5) no solution 45. 8z + 2 = 2(z - 5) - z –157
Lesson 2-3
46. Brendan earns $8.25 per hour at his job. He also makes $12.38 per hour for
any number of hours over 40 that he works in one week. He worked 40 hours
last week, plus some overtime, and made $385.71. How many overtime hours
did he work? 4 12 h
Lesson 1-5
Simplify.
47. -8 - 4 –12
110
110
Chapter 2 Solving Equations
48. 2 - 12 –10
49. 45 - (-9) 54
50. 18 - 15 3