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.

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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?

28

Chapter 1


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.

30

Chapter 1
