Lesson 9-1 Properties

Lesson 9-2 Solving Addition Equations

Lesson 9-3 Solving Subtraction Equations

Lesson 9-4 Solving Multiplication Equations

Lesson 9-5 Solving Two-Step Equations

Lesson 9-6 Functions

Lesson 9-7 Graphing Functions

Example 1 Use the Distributive Property

Example 2 Apply the Distributive Property

Example 3 Identify Properties

Example 4 Identify Properties

Example 5 Apply Properties

Find mentally using the Distributive Property.

Write 43 as

Distributive Property

Multiply 6 and 40 mentally.

Add 240 and 18 mentally.

Answer: 258

Find mentally using the Distributive Property.

Answer: 231

THEME PARKS Suppose admission to a theme park costs $35 per person and meals cost $20 per person. What is the cost for a family of 5 people?

Method 1 Find the cost of 5 admissions and 5 meals. Then add.

cost of 5 admissions cost of 5 meals

Method 2 Find the cost for 1 person. Then multiply by 5.

cost for 1 person

Evaluate either expression.

Answer: The total cost is $275.

Distributive Property

Multiply.

Add.

BOWLING Suppose the cost of bowling three games at a local bowling alley is $6.50 and the cost of shoe rental is $1.50. What is the cost for a group of 6 friends to each rent a pair of shoes and bowl three games?

Answer: $48

Answer: This is the Commutative Property of Addition.

The order in which the numbers are added changes.

Identify the property shown by the equation

Answer: Commutative Property of Multiplication

Identify the property shown by the equation

Answer: This is the Associative Property of Multiplication.

The grouping of the numbers to be multiplied changes.

Identify the property shown by the equation

Answer: Associative Property of Addition

Identify the property shown by the equation

Find mentally.

Since you can easily multiply 2 and 4, change the order.

Commutative Property

Now group the numbers. The parentheses tell you which to perform first.

Associative Property

Multiply 2 and 4 mentally.

Multiply 8 and 12 mentally.

Answer: 96

Find mentally.

Answer: 117

Example 1 Solve an Equation by Subtracting

Example 2 Solve an Equation Using Zero Pairs

Method 1 Use models.

Method 2 Use symbols.

Write the equation.

Answer: The solution is 1.

Subtract 4 from each side to “undo” the addition of 4 on the left.

Answer:

Method 1 Use models.

Check your solution.

Method 2 Use symbols.Write the equation.

Subtract 11 from each side to undo x plus 11.

CheckWrite the equation.Replace x with –4.This sentence is true.

Answer: The solution is –4.

Subtract 11 from each side.

Answer: –6

Example 1 Solve an Equation by Adding

Example 2 Solve a Subtraction Equation

Example 3 Use an Equation to Solve a Problem

Method 1 Use models.

Method 2 Use symbols.

Write the equation.

Add 5 to each side.Simplify.

Add 5 to each side to undo the subtraction of 5 on the left.

Answer: The solution is 15.

Answer: 12

Check your solution.

Write the equation.Add 5 to each side.Simplify.

Answer: The solution is 4.

CheckWrite the original equation.

Replace x with 4.

This sentence is true.

Check your solution.

Answer: 3

GRID-IN TEST ITEM The difference between the record high and the record low temperatures in Oregon is 173F. The record low temperature is –54F. What is the record high temperature in degrees Fahrenheit?Read the Test Item

The record high temperature is 119F.

Write the equation.

Definition of subtractionSubtract 54 from each side.Simplify.

Solve the Test Item

You need to find the record high temperature. Write and solve an equation. Let x represent the high temperature.

Answer:

GRID-IN TEST ITEM The difference between the age of Julie’s mother and Julie’s age is 27 years. Julie’s age is 6. What is the age of Julie’s mother?

Answer:

Example 1 Solve a Multiplication Equation

Example 2 Solve a Multiplication Equation

Example 3 Use an Equation to Solve a Problem

Check your solution.

Check

Write the original equation.

Replace x with 3.

This sentence is true.

Answer: The solution is 3.

Check your solution.

Answer: 5

Write the equation.

Divide each side by –5.

Answer: The solution is –3. Check this solution.

Answer: –7

GEOMETRY The area of a rectangle is 144 square inches and the width is 4 inches. Write an equation to find the length of the rectangle and use it to solve the problem.

The area of a rectangle is equal to its length times its width.

144

Write the equation.

Divide each side by 4.

Simplify.

Answer: The length of the rectangle is 36 inches.

Check

GEOMETRY The area of a rectangle is 126 square feet and the width is 7 feet. Write an equation to find the length of the rectangle and use it to solve the problem.

Answer: 18 feet

Example 1 Solve a Two-Step Equation

Example 2 Solve a Two-Step Equation

Example 3 Use an Equation to Solve a Problem

Answer: The solution is –3.

Answer: 2

Check your solution.

Write the equation.Add 2 to each side.Simplify.

Divide each side by 4.

Simplify.

Answer: The solution is 2. Check this solution.

Answer: 4

MOVIE NIGHT Three friends went to the movies. The tickets cost $6 each. They bought 2 large popcorns to share. If they spent a total of $24 at the movies, how much did a large popcorn cost?

The cost of 2 large popcorns plus 3 tickets is $24.

Two popcorns at $p each plus tickets costs $24

Words

Variable

Equation

2p

Write the equation.Subtract 18 from each side.

Simplify.

Divide each side by 2.

Simplify.

Answer: A large popcorn cost $3. Is this answer reasonable?

SHOPPING Jen went to her favorite store at the mall. She bought 3 t-shirts which cost $12 each. She also bought 2 pairs of the same jeans. If Jen spent a total of $80 at the store, how much did each pair of jeans cost?

Answer: $22

Example 1 Complete a Function Table

Example 2 Find a the Rule for a Function Table

Example 3 Solving a Problem Using a Function

Complete the function table.

The function rule is 3n. Multiply each input by 3.

Input Output –1 –3

0 0

1 31

0

–1

Output (3n)

Input (n)

Answer:

–3

1

0

Complete the function table.

3

2

1

Output

InputAnswer:

3

5

4

Find the rule for the function table.

25

58

710

Output

InputInput Output10 7

8 5

5 2

The output is three less than the input.

Answer: So, the function rule is n – 3.

Study the relationship between each input and output.

Find the rule for the function table.

29

310

411

Output

Input

Answer:

COFFEE Nina buys a refillable mug for $4.50 on her first day at a new job. Starting with her second day, she gets a refill of coffee costing $2.00 every day on the way to work. How much does she spend on coffee in her first 8 workdays?

First, determine the function rule.

The function rule is

Then, replace d in the rule with the number of workdays after Nina’s first day, 7.

Replace d with 7.

Multiply 2 and 7.

Add 14 and 4.50.

Answer: Nina spends $18.50 on coffee in her first 8 workdays.

MOVIE RENTAL A video store offers a deal where the first movie rented costs $5.25 and each movie rented after the first costs $2.50. Find the total cost to rent 6 movies.

Answer: $17.75

Example 1 Graph a Function

Example 2 Make a Function Table for a Graph

Step 1 Record the input and output in a function table. List the input and output as ordered pairs.

Make a function table for the rule Use input values of –4, 0, and 4. Then graph the function.

4

0

–4

(x, y)(y)( )(x)

Ordered Pairs

Output Function Rule Input

–2 (–4, –2)

2 (0, 2)

6 (4, 6)

Step 2 Graph the ordered pairs on the coordinate plane.

Step 3 The points appear to lie on a line. Draw the line that contains these points. The line is the graph of For any point on this line

Answer:

Make a function table for the rule Use input values of –2, 1, and 5. Then graph the function.

(5, 2)25 – 35

(1, –2)–21 – 31

(–2, –5)–5–2 – 3–2

(x, y)(y)(x – 3)(x)

Ordered Pairs

Output Function Rule Input

Answer:

Answer:

Make a function table for the graph. Then determine the function rule.

Use the ordered pairs to make a function table.

Study the input and output. Look for a rule.

Input (x) Output (y) (x, y)

–4 8 (–4, 8)

–2 4 (–2, 4)

0 0 (0, 0)

2 –4 (2, –4)

Input Output

–4 8

–2 4

0 0

2 –4

Each input is multiplied by –2 to get the output.

Answer: The function rule is

Make a function table for the graph. Then determine the function rule.

Answer: