Download Equations and Mental Math

Page 1 of 5
Equations and Mental Math
BEFORE
Now
WHY?
You evaluated numerical and
variable expressions.
You’ll solve equations using
mental math.
So you can find how long to save
money, as in Ex. 44.
In the Real World
Word Watch
equation, p. 36
solution, p. 36
solve, p. 37
Camping You are going camping with
friends. You fill your backpack until it
weighs 13 pounds. Your friend adds
another item from the list at the right,
and your backpack weighs 20 pounds.
What item did your friend add?
EXAMPLE
1
clothes
3 pounds
food
6 pounds
sleeping bag 4 pounds
tent
7 pounds
Guess, Check, and Revise
To answer the question above about backpacking, you can use the
problem solving strategy guess, check, and revise.
1 Try an item on
the list.
2 Try the food, which
is heavier.
13 4 17
This total weight is
under 20 pounds.
3 Try the tent. It
weighs 7 pounds.
13 7 20
13 6 19
This total weight is
still one pound low.
This total weight
equals 20 pounds.
ANSWER Your friend added the tent to your backpack.
Equations You can use equations to answer questions like the one
above. An equation is a mathematical sentence formed by placing an
equal sign () between two expressions. A solution of an equation is a
number that, when substituted for a variable, makes the equation true.
EXAMPLE
2
Checking a Possible Solution
Tell whether the given number is a solution of the equation.
a. 3y 21; 6
3(6) 21
18 21
b. x 3 7; 10
10 3 7
77
The symbol can be read “is not equal to.”
ANSWER 6 is not a solution.
36
Chapter 1
Number Sense and Algebraic Thinking
ANSWER 10 is a solution.
Page 2 of 5
Solving Equations To solve an equation, you find all the solutions
of the equation. To solve simple equations using mental math, you can
think of the equation as a question.
EXAMPLE
with
Notetaking
In your notes on
Equations and Mental
Math, you may want to
include examples showing
equations rewritten as
questions, like the ones in
Example 3.
3
Using Mental Math to Solve Equations
Equation
Question
Solution
Check
a. y 8 4
What number minus 8
equals 4?
12
12 8 4
b. 10x 90
10 times what number
equals 90?
9
10 p 9 90
c. n 4 7
What number divided
by 4 equals 7?
28
28 4 7
Mental Math When you do mental math, keep in mind the following
rules that will help you to solve some equations that involve a 0 or a 1.
Operations Involving 0 and 1
Adding 0 The sum of any number and 0 is that number.
Multiplying by 0 The product of any number and 0 is 0.
Multiplying by 1 The product of any number and 1 is that
number.
EXAMPLE
4
Mental Math with 0 and 1
Equation
Question
Solution
Check
a. y 2 2
What number plus 2
equals 2?
0
022
b. 6 p x 6
6 times what number
equals 6?
1
6p16
Your turn now
Solve the equation using mental math.
1. x 10 24 14
2. 13n 13 1
3. 35y 0 0
4. 9x 54 6
5. y 6 11 17
6. 18 n 9 2
Lesson 1.6
Equations and Mental Math
37
Page 3 of 5
EXAMPLE
5
Solving Problems Using Mental Math
Salmon Migration You are 100 miles upstream from the sea. Salmon
migrating upstream have been spotted 41 miles from the sea. Use mental
math to solve the equation d 41 100 to find the distance d, in miles,
between you and the salmon.
Solution
Think of the equation as a question.
Equation
d
41
Question
What number
plus
41
equals 100?
Solution
59
plus
41
equals 100.
100
ANSWER The distance between you and the salmon is 59 miles.
INTERNET
Exercises
eWorkbook Plus
CLASSZONE.COM
More Practice, p. 708
Getting Ready to Practice
1. Vocabulary The number 9 is the _?_ of the equation x 5 4. solution
Tell whether 5 is a solution of the equation.
2. t 2 7
5 is a solution.
3. 16 k 11
5 is a solution.
4. 15 m 5
5. 4y 20
5 is not a solution. 5 is a solution.
Solve the equation using mental math.
6. 4 z 12 8
7. x 2 1 3
8. 6r 24 4
9. w 2 5 10
10. 17 y 9 8 11. c 8 21 13 12. 56 m 7 8 13. 3t 27 9
14. Sports The table shows three major league baseball players and their
home run totals as of June 11 of a recent season. Solve the equations in
the table to find the number of home runs, x, that each player needed
to hit in order to reach 40 home runs.
Player
Barry Bonds
Luis Gonzalez
Manny Ramirez
38
Chapter 1
Number Sense and Algebraic Thinking
Home Runs
32
26
21
Equation
x 32 40
x 26 40
x 21 40
8
14
19
Page 4 of 5
Practice and Problem Solving
with
Example
1
2
3
4
5
Homework
Exercises
44
15–23
24–39
40–43
54, 55
Tell whether the given number is a solution of the equation.
A
15. 7 x 10; 3
3 is a solution.
18. m 2 6; 4
4 is a solution.
21. 10s 77; 7
7 is not a solution.
16. y 3 7; 4
4 is not a solution.
19. 4c 32; 9
9 is not a solution.
22. 30 x 10; 10
10 is not a solution.
17. 9 r 8; 2
2 is not a solution.
20. 4y 4; 1
1 is a solution.
23. 48 r 6; 8
8 is a solution.
Match the equation with the question.
Online Resources
CLASSZONE.COM
• More Examples
• eTutorial Plus
24. 6 n 2 D
A. What number minus 2 equals 6?
25. n 2 6 A
B. 2 times what number equals 12?
26. 2n 12 B
C. 12 times what number equals 6?
27. 12n 6 C
D. 6 minus what number equals 2?
Solve the equation using mental math.
28. 8 y 10 2
29. s 5 9 14
30. 14 z 1 13
31. d 3 11 8
32. 2w 16 8
33. 27 r 9 3
34. m 6 7 42
35. 5w 45 9
36. c 6 13 7
37. 23 t 10 13
38. 22 n 17 5
39. x 9 20 11
40. 36h 36 1
41. 35 a 35 0
42. 3f 0 0
43. Find the Error Describe and
correct the error in solving the
equation 5x 5.
The 0 in the second line should be 1.
5x 5
5(1) 5 So, 1 is the solution.
B
5x 5
5(0) 5
So, 0 is the solution.
44. Guess, Check, and Revise You are saving money to buy a camera that
costs $195. So far, you have saved $87. If you save $9 more each week,
in how many weeks will you have exactly $195? 12 weeks
Choose a Method Tell whether the given number is a solution of
the equation. Then tell whether you used mental math, paper and
pencil, or a calculator to get each answer. Methods will vary.
45. y 9 2; 12
46. 10 x 25; 15
47. 14t 122; 8
8 is not a solution.
12 is not a solution.
15 is a solution.
48. 3x 2 32; 9
49. 16y 6 300; 18 50. 32 4r 8; 6
9 is not a solution.
18 is not a solution.
6 is a solution.
51. 9r 63; 7
52. 110n 1980; 19
53. 3380 w 13; 26
7 is a solution.
19 is not a solution.
26 is not a solution.
54. Computer Games You earned 300 points playing a computer game.
Your goal is to reach 500 points. Solve the equation p 300 500
to find the number of points, p, you need to reach your goal. 200 points
Lesson 1.6
Equations and Mental Math
39
Page 5 of 5
55. Baby-sitting You make $7 an hour baby-sitting.
You need to earn $420. Solve the equation
7h 420 to find the number of hours, h, you
need to baby-sit. 60 h
C
56. Volleyball Your gym class is split up into teams of
6 students for a volleyball tournament. There are
48 students in the class. Choose the equation you can
use to find how many teams, t, you have. Then solve
the equation using mental math. B; 8
A. t 6 48
B. 6t 48
C. t 48 6
D. t 6 48
Challenge Tell whether the equation has no
solution, one solution, or many solutions. Use
examples to explain your answer.
57. many solutions; 0 p 4 0;
0 p 9 0; 0 p 26 0
58. no solution; 6 p 0 0,
23 p 0 0, 74 p 0 0
59. many solutions; 1 p 6 6,
1 p 9 9, 1 p 25 25
60. one solution; 0 0 0
57. 0 p x 0
58. x p 0 8
59. 1 p x x
60. 0 x 0
Mixed Review
61. You know that 717 children and 489 adults went to your school fair.
Estimate the total number of people who went to the fair. (Lesson 1.2)
about 1200 people
Evaluate the expression when y 3. (Lesson 1.5)
62. 15 y 12
63. 8 p y 24
64. y 9 12
65. 18 y 6
Basic Skills Tell which expression has the greater value.
66. 8 4 or 20 4
84
67. 15 6 or 20 2
20 2
68. 3 8 or 9 9 3 8
Test-Taking Practice
INTERNET
State Test Practice
CLASSZONE.COM
69. Multiple Choice You earn $15 each time you mow your neighbor’s
lawn. Solve the equation d 4 15 to find the amount of money d, in
dollars, you earn by mowing the lawn 4 times. D
A. 11
B. 19
C. 21
D. 60
70. Multiple Choice Your friend can read 2 pages per minute in her library
book. She read 20 pages last night. Solve the equation m p 2 20 to
find the number of minutes, m, she spent reading. F
F. 10
40
Chapter 1
Number Sense and Algebraic Thinking
G. 20
H. 22
I. 40