Download Equations and Mental Math

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Equations and Mental Math
BEFORE
Now
WHY?
You used mental math to add,
subtract, multiply, and divide.
You’ll use mental math to solve
an equation.
So you can find the flying time for a
homing pigeon, as in Example 3.
You can use chips to find the value of a variable.
Word Watch
equation, p. 26
solution, p. 26
solving an equation, p. 27
1 Use chips to model the statement
n 4 7. Let each chip represent 1.
n
2 Replace n with chips until you have
the same number of chips on each
side of the equal sign.
3 Replacing n with 3 chips gives a total
of 7 chips on each side, so n 3.
1. 6 x 10
2. y 2 9
3. 8 m 3
In the activity, you solved equations by modeling with chips. An
equation is a mathematical sentence formed by setting two expressions
equal. A solution of an equation is a number that you can substitute
for a variable to make the equation true.
EXAMPLE
1
NO PRINT ZONE
Model with chips to help you find the value of the variable.
Checking Possible Solutions
Tell whether the value of the variable is a solution of n 5 12.
a. n 9
b. n 7
Solution
with
Symbol
9 5 12
Reading
Meaning
is equal to
is equal to?
is not equal to
14 12
b. n 5 12
7 5 12
12 12 ✓
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Chapter 1
Write original equation.
Substitute 9 for n.
The equation is not true, so 9 is not a solution.
Write original equation.
Substitute 7 for n.
The equation is true, so 7 is a solution.
Number Sense, Patterns, and Algebraic Thinking
PAGE 26
a. n 5 12
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Your turn now
Tell whether the value of the variable is a solution of
the equation.
1. 3x 12; x 4
2. 7 13 n; n 5
3. 6 y 3; y 2
Solving an Equation Finding all solutions of an equation is called
solving the equation . You can use mental math to solve simple
equations by thinking of the equation as a question.
EXAMPLE
2
NO PRINT ZONE
Equation
Using Mental Math to Solve Equations
Question
Solution
Check
a. 9 x 12
9 plus what number
equals 12?
3
9 3 12
b. n 5 10
What number minus
5 equals 10?
15
15 5 10
c. 4t 20
4 times what number
equals 20?
5
4(5) 20
d. m 3 12
What number divided
by 3 equals 12?
36
36 3 12
Your turn now
4. 7x 35
Solve the equation using mental math.
5. 15 n 6
6. 12 a 32
7. 24 n 6
Distance Problems Some distance problems can be solved using an
equation that relates distance, speed, and time. For example, if you travel
on a moving walkway at a speed of 2 feet per second for 30 seconds, you
can use the following formula to find the distance you travel.
Distance, Speed, and Time
Words Distance traveled is equal to the speed (rate of travel)
times the travel time.
PAGE 27
Algebra d rt
Numbers distance 2 feet per second p 30 seconds 60 feet
Lesson 1.5
Equations and Mental Math
27
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EXAMPLE
3
Using Mental Math to Solve an Equation
Homing Pigeons A homing pigeon is a bird trained to fly back to its
home. Homing pigeons can fly at a speed of about 50 miles per hour.
About how long would it take for a homing pigeon to fly 300 miles?
Solution
d rt
Write formula for distance.
300 50t
Substitute the values you know.
300 50 p 6
Use mental math to solve equation.
ANSWER It would take a homing pigeon about 6 hours to fly 300 miles.
Your turn now
Solve the following problem.
8. A car travels on a highway at a constant speed. If it takes the car
INTERNET
Exercises
eWorkbook Plus
CLASSZONE.COM
More Practice, p. 705
NO PRINT ZONE
2 hours to drive 100 miles, at what speed is the car traveling?
Getting Ready to Practice
1. Vocabulary What is the difference between an expression and an
equation? Give an example of each.
Tell whether the given value of the variable is a solution of the
equation.
2. 5x 35; x 7
3. 16 y 22; y 8
4. 9 z 12; z 20
Mental Math Solve the equation using mental math.
5. 4a 24
6. b 6 7
8. 18 9 y
9. 8 25 t
z
7. 7
3
10. 10x 120
mountain in only 5 minutes. If the distance the ski lift travels to the top
of the mountain is 3000 feet, about how many feet per minute does the
ski lift travel?
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Chapter 1
Number Sense, Patterns, and Algebraic Thinking
PAGE 28
11. Skiing A ski lift is advertised as being able to bring you to the top of a
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Practice and Problem Solving
with
Example
1
2
3
Homework
Exercises
12–14
15–27, 31–32
28–30, 35–36
Online Resources
CLASSZONE.COM
• More Examples
• eTutorial Plus
Tell whether the given value of the variable is a solution of the
equation.
12. 40 8z; z 5
13. s 5 11; s 7
14. 24 a 13; a 9
Match the equation with the corresponding question. Then solve.
15. t 3 12
A. 12 divided by what number equals 3?
t
16. 3
12
B. What number plus 3 equals 12?
17. 12 t 3
C. What number divided by 12 equals 3?
18. 3t 12
D. 3 times what number equals 12?
Mental Math Solve the equation using mental math.
NO PRINT ZONE
Recreation
19. x 8 15
36
20. 6
y
21. 7t 56
22. 20 b 3
23. 10 x 7
24. 29 19 p
25. a 6 8
26. 11n 44
27. 54 9y
Use the formula for distance to find the unknown value.
28. d 100 miles, r 25 miles per hour, t _?_
29. d 72 miles, r _?_, t 8 hours
30. d 9 feet, r 3 feet per second, t _?_
31. Log Flumes You ride a log flume at an amusement park for 6 rotations
in a row. According to your watch, you spent a total of 18 minutes on
the ride. Solve the equation 6n 18 to find how long it takes to
complete one rotation of the ride.
32. St. Louis Arch The elevator train inside the St. Louis arch can carry a
total of 40 people, with 5 people in each of the cars. Solve the equation
40
5 to find the number of cars in the elevator train.
c
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■
Log Flumes
Some log flumes hold as
much as 90,000 gallons of
water. An Olympic-size pool
holds about 750,000 gallons
of water. About how many
log flumes could be filled
with the water from an
Olympic-size pool?
Critical Thinking Tell whether the equations have the same solution.
33. x 4 5 and x 2 3
34. 5y 45 and 6y 48
35. Cheetahs A cheetah can run at a top speed of about 100 feet per
second. At this speed, find the approximate time it would take a
cheetah to run the length of a football field that is 300 feet long.
36. Biking You bike at approximately the same speed for 2 hours. If you
travel 24 miles, what is your speed?
Lesson 1.5
Equations and Mental Math
29
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Challenge Solve the equation using mental math.
37. 2p 0 6
38. 4 2a 0
39. 10 3y 1
40. 2x 1 5
Mixed Review
Evaluate the expression when x 12 and y 3. (Lessons 1.3, 1.4)
41. 2y 3
42. x 2 y
43. xy 2
44. (x y)2
47. 11 3 p 8
48. 12 10 p 3
Evaluate the expression. (Lesson 1.4)
45. 18 9 5
46. 42 6 3
Basic Skills Find the perimeter.
4 in.
5 ft
50.
4 in.
2 ft
4 in.
2 ft
5 ft
4 in.
Test-Taking Practice
INTERNET
State Test Practice
CLASSZONE.COM
21
51. Multiple Choice What is the solution of the equation 3?
a
A. 3
B. 7
C. 18
D. 63
NO PRINT ZONE
49.
52. Multiple Choice Bats can fly at a speed of about 50 feet per second.
Solve the equation 350 50t to estimate the time is takes a bat to fly
350 feet.
F. 6 seconds
G. 7 seconds
H. 70 seconds
I. 300 seconds
Shape Equations
10 2
PAGE 30
Find the number that each shape
represents. Each square represents
the same number and each triangle
represents the same number.
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Chapter 1
Number Sense, Patterns, and Algebraic Thinking