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**Learn to solve equations using multiplication and division.**

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**Vocabulary Division Property of Equality**

Multiplication Property of Equality

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**DIVISION PROPERTY OF EQUALITY**

You can solve a multiplication equation using the Division Property of Equality. DIVISION PROPERTY OF EQUALITY Words Numbers Algebra You can divide both sides of an equation by the same nonzero number, and the equation will still be true. 4 • 3 = 12 x = y 4 • 3 = 12 x = y 2 2 z z 12 = 6 2

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**Additional Example 1: Solving Equations Using Division**

Solve 8x = 32. 8x = 32 8x = 32 Divide both sides by 8. 8 8 1x = 4 1 • x = x x = 4 Check 8x = 32 8(4) = 32 ? Substitute 4 for x. 32 = 32 ?

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**Try This: Example 1 Solve 9x = 36. 9x = 36 9x = 36**

Divide both sides by 9. 9 9 1x = 4 1 • x = x x = 4 Check 9x = 36 9(4) = 36 ? Substitute 4 for x. 36 = 36 ?

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**MULTIPLICATION PROPERTY OF EQUALITY**

You can solve a division equation using the Multiplication Property of Equality. MULTIPLICATION PROPERTY OF EQUALITY Words Numbers Algebra You can multiply both sides of an equation by the same number, and the statement will still be true. 2 • 3 = 6 x = y 4 • 2 • 3 = z x = y 8 • 3 = 24

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**Additional Example 2: Solving Equations Using Multiplication**

7 Solve = 7. n 7 = 7 • 7 • Multiply both sides by 7. n = 49 Check n 7 = 7 49 7 = 7 ? Substitute 49 for n. 7 = 7 ?

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**Try This: Example 2 n 4 Solve = 16 n 4 = 16 4 • 4 •**

= 4 • 4 • Multiply both sides by 4. n = 64 Check n 4 = 16 64 4 = 16 ? Substitute 64 for n. 16 = 16 ?

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**Additional Example 3: Food Application**

Joe has enough flour to bake one sheet cake but would rather make cookies. How many dozen cookies can he make? Dessert Apple crisp Bread pudding Cookies (1 doz.) Pumpkin pie Sheet cake Tiramisu Cups of Flour 1.5 4 2 1 8 3

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**= = • Additional Example 3 Continued**

cups of flour for 1 dozen cookies number of dozen cookies cups of flour in 1 cake = • = 2 c 8 2c = 8 Write the equation. 2c = 8 Divide both sides by 2. 2 2 c = 4 Joe can make 4 dozen cookies with the same amount of flour that he would need for 1 sheet cake.

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**= = • Try This: Example 3 cups of flour for bread pudding**

number of bread pudding desserts cups of flour in 1 cake = • = 4 b 8 4b = 8 Write the equation. 4b = 8 Divide both sides by 4. 4 4 c = 2 Joe can make 2 bread pudding desserts with the same amount of flour that he would need for 1 sheet cake.

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**= = • Additional Example 4: Money Application 1 4 x 50 x = 50 1 4**

Meg has saved $50, which is one-fourth of the amount she needs for a school trip. What is the total amount she needs? fraction of amount raised so far total amount needed = amount raised so far 1 4 • = x 50 x = 50 1 4 Write the equation. x = 1 4 Multiply both sides by 4. 4 • 4 • x = 200 Meg needs $200 total.

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**= = • Try This: Example 4 1 x 750 8 1 x = 750 8 1 8 • x = 750 8 • 8**

The school library needs money to complete a new collection. So far, the library has raised $750, which is only one-eighth of what they need. What is the total amount needed? fraction of total amount raised so far total amount needed = amount raised so far 1 8 • = x 750 x = 750 1 8 Write the equation. x = 1 8 8 • 8 • Multiply both sides by 8. x = 6000 The library needs to raise a total of $6000.

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Sometimes it is necessary to solve equations by using two inverse operations. For instance, the equation 6x 2 = 10 has multiplication and subtraction. Variable term 6x 2 = 10 Multiplication Subtraction To solve this equation, add to isolate the term with the variable in it. Then divide to solve.

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**Additional Example 5: Solving a Simple Two-Step Equation**

Solve 3y – 7 = 20. Step 1: 3y – 7 = 20 Add 7 to both sides to isolate the term with y in it. + 7 + 7 3y = 27 Step 2: 3y = 27 Divide both sides by 3. 3 3 y = 9

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Try This: Example 5 Solve 4y + 5 = 29. Step 1: 4y + 5 = 29 Subtract 5 from both sides to isolate the term with y in it. – 5 – 5 4y = 24 Step 2: 4y = 24 Divide both sides by 4. 4 4 y = 6

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Solving Algebraic Equations. Equality 3 = 3 3+4 = 3+4 3+4 = 7 For what value of x is: x + 4 = 7 true?

Solving Algebraic Equations. Equality 3 = 3 3+4 = 3+4 3+4 = 7 For what value of x is: x + 4 = 7 true?

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