California Standards Preview of Grade 7 AF1.3 Simplify numerical expressions by applying properties of rational numbers (e.g., identity, inverse, distributive, associative, and commutative) and justify the process used. Also covered: Preview of Algebra 1 5.0

Additional Example 1: Combining Like Terms to Solve Equations Solve 12 – 7b + 10b = 18. 12 – 7b + 10b = 18 12 + 3b = 18 Combine like terms. – 12 – 12 Subtract 12 from both sides. 3b = 6 3b = 6 Divide both sides by 3. 3 3 b = 2

Check It Out! Example 1 Solve 14 – 8b + 12b = 62. 14 – 8b + 12b = 62 14 + 4b = 62 Combine like terms. – 14 – 14 Subtract 14 from both sides. 4b = 48 4b = 48 Divide both sides by 4. 4 4 b = 12

You may need to use the Distributive Property to solve an equation that has parentheses. Multiply each term inside the parentheses by the factor that is outside the parentheses. Then combine like terms.

Additional Example 2: Using the Distributive Property to Solve Equations Solve 5(y – 2) + 6 = 21. 5(y – 2) + 6 = 21 5(y) – 5(2) + 6 = 21 Distribute 5 on the left side. 5y – 10 + 6 = 21 Simplify. 5y – 4 = 21 Combine like terms. + 4 + 4 Add 4 to both sides. 5y = 25 Divide both sides by 5. 5 5 y = 5

The Distributive Property states that a(b + c) = ab + ac. For instance, 2(3 + 5) = 2(3) + 2(5). Remember!

Check It Out! Example 2 Solve 3(x – 3) + 4 = 28. 3(x – 3) + 4 = 28 3(x) – 3(3) + 4 = 28 Distribute 3 on the left side. 3x – 9 + 4 = 28 Simplify. 3x – 5 = 28 Combine like terms. + 5 + 5 Add 5 to both sides. 3x = 33 Divide both sides by 3. 3 3 x = 11

Additional Example 3: Problem Solving Application Troy has three times as many trading cards as Hillary. Subtracting 8 from the combined number of trading cards Troy and Hillary have gives the number of cards Sean has. If Sean owns 24 trading cards, how many trading cards does Hillary own?

Understand the Problem Additional Example 3 Continued 1 Understand the Problem Rewrite the question as a statement. • Find the number of trading cards that Hillary owns. List the important information: • Troy owns 3 times as many trading cards as Hillary. • Subtracting 8 from the combined number of trading cards Troy and Hillary own gives the number cards Sean owns. • Sean owns 24 trading cards.

Additional Example 3 Continued 2 Make a Plan Let c represent the number of trading cards Hillary owns. Then 3c represents the number Troy owns. Troy’s cards + Hillary’s cards – 8 = Sean’s cards 3c + c – 8 = 24 Solve the equation 3c + c – 8 = 24 for c.

Additional Example 3 Continued Solve 3 3c + c – 8 = 24 4c – 8 = 24 Combine like terms. + 8 + 8 Add 8 to both sides. 4c = 32 4c = 32 4 Divide both sides by 4. c = 8 Hillary owns 8 cards.

Additional Example 3 Continued 4 Look Back Make sure that your answer makes sense in the original problem. Hillary owns 8 cards. Troy owns 3(8) = 24 cards. Sean owns 24 + 8 – 8 = 24.

Check It Out! Example 3 John is twice as old as Hiro. Subtracting 4 from the combined age of John and Hiro gives William’s age. If William is 29, how old is Hiro?

Understand the Problem Check It Out! Example 3 Continued 1 Understand the Problem Rewrite the question as a statement. • Find Hiro’s age. List the important information: • John is 2 times as old as Hiro. • Subtracting 4 from the combined age of John and Hiro gives William’s age. • William is 29 years old.

Check It Out! Example 3 Continued 2 Make a Plan Let h represent Hiro’s age. Then 2h represents John’s age. John’s age + Hiro’s age – 4 = William’s age 2h + h – 4 = 29 Solve the equation 2h + h – 4 = 29.

Check It Out! Example 3 Continued Solve 3 2h + h – 4 = 29 3h – 4 = 29 Combine like terms. + 4 + 4 Add 4 to both sides. 3h = 33 3h = 33 3 Divide both sides by 3. h = 11 Hiro is 11 years old.

Check It Out! Example 3 Continued 4 Look Back Make sure that your answer makes sense in the original problem. Hiro is 11 years old, then John is 2(11) = 22 years old. William is 22 + 11 – 4 = 29 years old.