solving mulTiplicaTion and division EquaTions lesson 3

A balance scale is another way to represent how two sides of an equation are equal in amount. Solving equations is like keeping a scale balanced. Imagine that the equals sign is the balancing point on the scale. Any operation performed on one side of the scale must also be performed on the other side to keep the scale balanced.

So far in this block you have learned how to solve addition and subtraction equations. In this lesson, you will work with multiplication and division equations. Multiplication and division are inverse operations. To solve a multiplication equation, you will divide each side of the equation by a number to get the variable by itself. To solve a division equation, you will use multiplication.

step 1: If you do not have an equation mat, draw one like the one seen on the right on a blank sheet of paper.

step 2: On the equation mat, place two variable cubes on one side. On the other side of the mat, place 6 chips. This represents the equation 2x = 6.

step 3: Divide the chips into two equal groups since there are two variable cubes. How many chips are equal to one variable cube? This is the value of x.

step 4: Clear the mat and place counters on the mat to represent the equation 4x = 8. Draw this on your paper.

step 5: How can you determine how many chips are equal to one variable cube?

step 6: Create a multiplication equation on the mat. Record the algebraic equation on your paper.

step 7: Solve the equation. What does the variable equal?

step 8: Write a few sentences that describe how to solve a multiplication equation using the equation mat, chips and variable cubes.

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x x 1 1 1 1 1 1

=

ExplorE! mulTiplicaTion EquaTions

=xx

84 Lesson 3.5 ~ Solving Multiplication and Division Equations

solving mulTiplicaTion and division EquaTions

lesson 3.5

solve each equation. show your work and check your solution.a. m __ 8 = 3 b. 6.2 =

y __ 3

c. 11p = 99 d. 12x = 60

a. m __ 8 = 3 b. 6.2 = y _ 3

8 ∙ m __ 8 = 3 ∙ 8 3 ∙ 6.2 = y __ 3 ∙ 3

m = 24 18.6 = y

☑ 24 __ 8 = 3 ☑ 6.2 = 18.6 ___ 3

c. 11p

___ 11 = 99 __ 11 d. 12x ___ 12 = 60 __ 12 p = 9 x = 5

☑ 11(9) = 99 ☑ 12(5) = 60

Peter is four times older than his son, matthew. Peter is 52 years old. write a multiplication equation and solve the equation to determine matthew’s age.

Let m represent Matthew’s age. Write the equation that represents this situation. 4m = 52

Solve the equation by using division (the inverse operation of multiplication). 4m ___ 4 = 52 __ 4

m = 13Matthew is 13 years old.

ExamplE 1

solutions

ExamplE 2

solution

Lesson 3.5 ~ Solving Multiplication and Division Equations 85

ExErcisEs

determine which operation (multiplication or division) would be used to solve each equation.

1. h __ 7 = 12 2. 3x = 57 3. 14 = 4m solve each equation using inverse operations. show all work necessary to justify your answer.

4. 6y = 12 5. 2p = 24 6. m __ 3 = 7

7. h __ 10 = 8 8. 42 = 7x 9. 5p = 35

10. 14j = 28 11. 4 = b __ 11 12. y _

6 = 0.5

13. 2.5m = 10 14. c __ 20 = 8 15. 0.9x = 7.2

write an algebraic equation for each sentence. solve each equation. 16. Six times a number y is sixty-six. 17. A number x divided by four is eight. 18. The quotient of x divided by seven is five. 19. Thirty-six is nine times a number x.

20. Oscar is three times older than James. Oscar is 42 years old. a. Let j represent James’ age. Write a multiplication equation for this situation. b. Solve the equation. How old is James?

21. Hannah is five times as old as her sister. Hannah is 15 years old. Write a multiplication equation. Solve the equation to determine her sister’s age.

22. Three friends share a pizza. Each friend eats 4 pieces. a. Let p represent the total number of pieces of pizza. Write a division equation

for this situation. b. Solve the equation. How many pieces of pizza were there in all?

23. Five people purchased a piece of farm land together. Ten years later they sold the land. Each person received $45,217. What was the total amount the land was sold for? Explain how you know your answer is correct.

24. Dora weighs 5 1 _ 2 times as much as her baby brother, Jeremie. Dora weighs 60 1 _ 2 pounds. How much does Jeremie weigh? Show all work necessary to justify your answer.

86 Lesson 3.5 ~ Solving Multiplication and Division Equations

rEviEw

match each expression with an equivalent expression.

25. 2(x − 7) a. 7x + 6 26. 4x + 3 + 8x + 1 B. 12x + 4 27. 7(x + 1) − 1 c. 7x + 8 28. 2(x + 3) + 5x + 2 D. 2x − 14

tic-tAc-toe ~ mis si ng pr ice tAgs Kiera went to the store and purchased six items. All items were on sale but the original price tags had been removed. Use the information shown below to help Kiera determine the original price of each item she purchased. Use the given percent equation. Percents must be converted to decimal form.

original price ∙ percent savings = amount saved

1. Determine the original cost of each item.2. Determine the total cost for all six items before they went on sale.3. Determine the total amount Kiera spent on the sale items.

15% offSave$4.50

20% offSave$1.60

40% off

Save$7.20

60% off

Save$5.40

25% offSave$4.00

30% offSave$6.90

Lesson 3.5 ~ Solving Multiplication and Division Equations 87

tic-tAc-toe ~ l i k e te r ms

Some equations have like terms that must be combined before you can use inverse operations to isolate the variable. There are two types of equations you may deal with that have like terms:1) Equations that have like terms on the same side of the equals sign.2) Equations that have like terms on opposite sides of the equals sign.

type 1: When equations have like terms on the same side of the equals sign, you must first combine the like terms before using inverse operations to isolate the variable.

For example: 5x + 2x − 3x = 20 4x __ 4 = 20 __ 4 4x = 20 x = 5

type 2: Equations with like terms on each side of the equals sign require you to move the variables to one side of the equation and the constants to the other side of the equation. Whenever a term is moved from one side of the equals sign to the other, the inverse operation must be used.

step 1: Move the variable term with the smaller coefficient to the other side of the equals sign using inverse operations. step 2: Move all constants away from the variable. step 3: Use division to solve.

For example: 9x − 5 = 6x + 7 − 6x −6x 3x − 5 = 7 +5 + 5 3x __ 3 = 12 __ 3

x = 4

solve each equation. show all work necessary to justify your answer.

1. 8x − 3x + 4x = 72 2. 2x + 22 = 6x + 2 3. 4 + 11 + 15 = 5x

4. 12 + 3x = x + 42 5. 10x − 41 = 2x + 7 6. 27 = 8x + 2x − 7x 7. 3x + 18 = 9x − 3 8. x + 2x + 3x = 4 + 7 + 1 9. 2x = 8x − 24

88 Lesson 3.5 ~ Solving Multiplication and Division Equations