Solving Equations, Part 2 - Diablo Valley...

Free Pre-Algebra Lesson 16 ! page 1

© 2010 Cheryl Wilcox

Lesson 16

Solving Equations, Part 2

The equations we’ve solved so far have had one operation on the variable. Once we’ve undone that one operation, the equation is solved. But equations come in more complicated forms, so let’s explore the next level. Simplify First Algebraic equations may have expressions that can be simplified on either side of the equals sign.

Example: Solve the equation 2x + 6x = 18 – 2.

2x + 6x = 18 ! 2

8x = 16

8x

8

=16

8

x = 2

Example: The cost of your crafts booth is your cost per item times the number of items, plus the rental fee. Your booth will have 600 items, and your cost for each is $5. The total cost of the booth is $4000. What is the rental fee?

The cost of your crafts booth = your cost per item • the number of items + the rental fee.

C = Y • N + R

$4000 = $5 / item • 600 items + R

4000 = 5 • 600 +R

4000 = 3000 +R

R + 3000 = 4000

!3000 = !3000

R = 1000

The rental fee is $1000.

More Than One Operation Once simplified, so far only one operation has been done with the variable in our equation. Equations get more complicated when two or more operations are involved. We undo one operation with the opposite operation. When you put on your shoe, the opposite operation is to take it off. When you put on a sock and then a shoe, there are two operations that must be removed. To take them off, first remove the shoe (which you put on last), then remove the sock. In equations with more than one operation on the variable, we undo first the operation that was done last.

Firs simplify each side. Then solve as usual.

Putting Them On

Taking Them Off



Example: Solve the equation 2x + 1 = 19.

According to the order of operations, the variable, x, has first been multiplied by 2, and then 1 has been added. When solving, we undo the operations in the opposite order they were done. First we undo adding 1 (by subtracting 1). Then we undo multiplying by 2 (by dividing by 2).

Write the problem: 2x + 1 = 19

1 Begin by answering these questions to orient yourself:

What are we trying to find? Where is the variable? State the operations in the order they are done to the variable.

“x is multiplied by 2, then 1 is added.”

2x+1

=

19

2 Undo the last operation with its opposite. 2x+1 – 1 2x

= =

19 – 1 18

3 Continue undoing operations in the opposite order they were done,

until the variable is alone on one side of the equation.

2x /2 x

= =

18 /2 9

4 Check back in the original equation to be sure the answer makes sense.

2(9) + 1

=

19 !

Example: Solve the equation 3(x + 5) = 21.

Here the order of operations requires that we first add 5 to x, and then multiply the result by 3. To solve, we undo the last operation first, so we first divide by 3. Then we’ll subtract the 5.

3(x + 5) = 21

3 (x + 5)

3=

21

3

x + 5 = 7

!5 = !5

x = 2

Some people prefer to simplify first, using the distributive property, then solve.

3(x + 5) = 21

3x + 15 = 21

!15 = !15

3x = 6

3x

3=

6

3

x = 2

As you can see, either method gives the same solution.



Example: Solve 3x/8 = 15.

Here the two operations are multiplication and division. In this situation, either operation can be undone first.

3x

8= 15

8 •3x

8

= 15 • 8

3x = 120

3x

3

=120

3

x = 40

There is a shortcut, in which we do both steps at once. You can use this if you like.

3x

8= 15

8

3

•3x

8

= 15 •8

3

x =15

5

1•

8

3

= 40

Example: The cost of a flower arrangement containing N flowers is $5 per flower plus a $45 arrangement fee. If the arrangement you ordered cost $120, how many flowers did it contain?

cost of flower arrangement = price per flower • number of flowers + arrangement fee

C = P • N + F

$120 = $5 / flower • N flowers + $45

120 = 5N + 45

5N + 45 = 120

!45 = !45

5N = 75

5N

5

=75

5

N = 15

The arrangement contains 15 flowers.



Simplifying and Solving

First simplify when possible, then solve, undoing operations in the reverse order that they were done.

Example: Solve the equation 2(4x +3) + 5(2x + 1) = 5 • 13.

Simplify each side first:

2(4x + 3) + 5(2x + 1) = 5 • 13

8x + 6 + 10x + 5 = 65

18x + 11= 65

Then solve the equation:

18x + 11= 65

!11= !11

18x = 54

18x

18

=54

18

x = 3

Example: The perimeter of a rectangle is 44 inches. The length is 9 inches. What is the width?

Put the numbers you know into the formula,

P = 2L + 2W

44 = 2(9) + 2W

2W + 18 = 44

Solve the equation:

2W + 18 = 44

!18 = !18

2W = 26

2W

2

=26

2

W = 13

The width is 13 inches.



The Answer Might Be a Fraction If the numbers in the equation do not divide evenly, the answer is written as a fraction (in lowest terms). In algebra problems, leave improper fractions alone – do not convert them to mixed numbers. In word problems, the answer should be written in the most meaningful way for the context of the situation.

Example: Solve 5x + 2 = 20.

5x + 2 = 20

!2 = !2

5x = 18

5x

5

=18

5

x =18

5

Example: The bus traveled 252 miles at an average rate of 56 mph. How long did the trip take?

d = rt

252 = 56t

56t

56=

252

56

t =252 ÷ 28

56 ÷ 28=

9

2= 4

1

2

The trip took 41/2 hours.

!



Lesson 16: Solving Equations, Part 2

Worksheet Name _______________________________________

Fill in the blank, then solve the equation.

Fill in the formula with the numbers you know. Then solve the equation to solve the problem.

1. 5x + 7x = 36

Begin by simplifying: first __________________________.

2. 5x + 6 = 36

Here x is multiplied by 5, then 6 is added. To undo in the reverse order we must first _______________________.

3. 5(x + 1) = 35

4.

5x

3= 35

5.

x ! 2

3= 35

Starting with x, we first subtract 2, then divide by 3. To undo in the reverse order, we first undo the division by ________________________.

6. 5(x + 2) + 3(x + 7) = 39

1. The Cost of a crafts booth is the Price per item times the Number of items plus the Rental fee. Find the number of items if the cost per item is $4, the rental fee is $1500, and the total cost is $5000.

C = P • N + R

2. The perimeter of a rectangle is 18 feet and the width is 3 feet. What is the length?



Make up your own equations by following these steps. Then challenge your partner by exchanging equations and solving.

Instructions Example

1 Pick a variable. x

2 Decide what number the variable will be equal to. This will be the solution to your equation. x = 5

3 Write some operations with the number that you can undo later. 3 • 5 + 9

4 Figure out the answer when you do your operations with the number. 3 • 5 + 9 = 24

5 Remove the number, and write the variable in its place. 3 • x + 9 = 24

6 Your equation is ready for someone else to solve. (You already know the solution is x = 5.) 3x + 9 = 24

Let each person in your group make up two equations with solutions on scratch paper. Write your equations on a group member’s worksheet (without the solutions) and have them write their equations on yours. Solve, then check your answer with the person who created the equation. Equation 1: Solution: Equation 2: Solution:



Lesson16: Solving Equations, Part 2

Homework 16A Name _____________________________________

1. Working with Fractions.

a. Write

208

312 in lowest terms.

b. Write

45ab

48bin lowest terms.

c. Write 5 fractions equivalent to

6

7.

d. Change

82

5 to an improper fraction.

e. Change

67

9 to a mixed number.

f. Multiply

22a

5•

b

110a

g. Multiply

15 •1

5

2. Simplifying Algebraic Expressions.

a. Multiply (14a)(3ab)

b. Multiply (5x 2 )(8x)

c. Combine like terms to simplify 4x + 2x + 7x + 1

d. Combine like terms to simplify 10N + 5M + 6N +M

e. Use the distributive property to simplify 9(2M + 1)

f. Use the distributive property to simplify 2 b + 3( )

g. Simplify 8a + 3(2a + 9)

h. Simplify 18 + 7(5x + 11)

i. Simplify 3(3x + 3) + 5(5x + 4)



3. Solve the equations.

a. 17x = 102

b. x ! 19 = 16

c.

12 =x

6

d. a + 99 = 100

e. 4(x + 3) = 20

f. 17 = 2 + 3n

g.

7x

2= 14

h.

x + 6

5= 2

4. Use a formula to solve the problem.

a. The area of a rectangle is 630 square feet, and the length is 35 feet. What is the width?

b. The perimeter of a rectangle is 234 centimeters, and the length is 56 centimeters. What is the width?

c. The volume of a box is 11,594 cubic inches. The length is 17 inches, and the width is 22 inches. What is the height?

d. The superhero traveled 201 feet in 3 seconds. What was her speed?

e. The hero’s evil nemesis traveled 200 feet at a rate of 50 feet per second. How long did that take?




Homework 16A Answers


a. Write

208


208

312=

2 • 2 • 2 • 2 • 13

2 • 2 • 2 • 3 • 13

=2

3

b. Write

45ab

48bin lowest terms.

45ab

48b=

3 • 3 • 5 • a • b

2 • 2 • 2 • 2 • 3 • b

=15a

16


6

7.

6

7=

12

14=

18

21=

24

28=

30

35=

36

42

d. Change

82


82

5=

40

5+

2

5=

42

5

e. Change

67


67 ÷ 9 = 7r 4

74

9

f. Multiply

22a

5•

b

110a

2 • 11 • a

5•

b

2 • 5 • 11 • a

=b

25

g. Multiply

15 •1

5

15

3

1•

1

5

=3

1= 3


a. Multiply (14a)(3ab)

(14a)(3ab) = 14 • 3 • a • a • b = 42a2b

b. Multiply (5x 2 )(8x)

(5x 2 )(8x) = 5 • 8 • x • x • x = 40x 3

c. Combine like terms to simplify 4x + 2x + 7x + 1

13x + 1

d. Combine like terms to simplify 10N + 5M + 6N +M

16N + 6M


18M + 9

f. Use the distributive property to simplify 2 b + 3( )

2b + 6

g. Simplify 8a + 3(2a + 9)

8a + 6a + 27 = 14a + 27

h. Simplify 18 + 7(5x + 11)

18 + 35x + 77 = 35x + 95

i. Simplify 3(3x + 3) + 5(5x + 4)

9x + 9 + 25x + 20 = 34x + 29




a. 17x = 102

17x

17

=102

17x = 6

b. x ! 19 = 16

x ! 19 + 19 = 16 + 19 x = 35

c.

12 =x

6

6 • 12 =x

6

• 6 x = 72

d. a + 99 = 100

a + 99 ! 99 = 100 ! 99 a = 1

e. 4(x + 3) = 20

4(x + 3)

4=

20

4x + 3 = 5

x + 3 ! 3 = 5 ! 3 x = 2

f. 17 = 2 + 3n

3n + 2 ! 2 = 17 ! 2 3n = 15

3n

3

=15

3n = 5

g.

7x

2= 14

2

7•

7x

2= 14

2

•2

7

x = 4

h.

x + 6

5= 2

5 •x + 6

5

= 2 • 5 x + 6 = 10

x + 6 ! 6 = 10 ! 6 x = 4



A =LW 630 = 35W

35W

35

=630

35W = 18 feet


P = 2L + 2W 234 = 2(56) + 2W

2W + 112 ! 112 = 234 ! 112 2W = 122

2W

2=

122

2W = 61 cm

c. The volume of a box is 11,594 cubic inches. The length is 17 inches, and the width is 22 inches. What is the height?

V =LWH 11594 = (17)(22)H

374H

374=

11594

374H = 31 inches

d. The superhero traveled 201 feet in 3 seconds. What was her speed?

d = rt 201= r • 3

3r

3=

201

3r = 67 feet per second

e. The hero’s evil nemesis traveled 200 feet at a rate of 50 feet per second. How long did that take?

d = rt 200 = 50t

50t

50

=200

50t = 4 seconds




Homework 16B Name _____________________________________


a. Write

90


b. Write

63x

54x2

in lowest terms.


2

13.

d. Change

113


e. Change

74


f. Multiply

15x

28y•

8y

30

g. Multiply

40 •3

8


a. Multiply (8n)(13mn)

b. Multiply (15a2 )(10a2 )

c. Combine like terms to simplify 3M + 2M + 4M + 3

d. Combine like terms to simplify 2x + 9y + 7y + 4x


f. Use the distributive property to simplify 6(x + 1)

g. Simplify 22x + 7(10x + 9)

h. Simplify 60 + 10(40x + 80)

i. Simplify 2(3x + 4) + 9(5x + 6)




a. a + 7 = 29

b. 3x = 12

c.

22 =x

8

d. b ! 99 = 100

e. 3(2x + 1) = 33

f. 47 = 5 + 3n

g.

5x

3= 10

h.

x ! 5

2= 15




c. The volume of a box is 2184 cubic inches. The length is 14 inches, and the width is 12 inches. What is the height?

d. The riverboat traveled 90 miles in 5 hours. What was her speed?

e. The Coast Guard patrol boat traveled 90 miles at a rate of 45mph. How long did that take?

Solving Equations, Part 2 - Diablo Valley...

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