CHAPTER OUTLINE 3 Solving Equations Slide 2 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 3.1Simplifying Expressions and Combining Like Terms 3.2Addition and Subtraction Properties of Equality 3.3Multiplication and Division Properties of Equality 3.4Solving Equations with Multiple Steps 3.5Applications and Problem Solving

Section Objectives 3.1 Simplifying Expressions and Combining Like Terms Slide 3 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1.Identifying Like Terms 2.Commutative, Associative, and Distributive Properties 3.Combining Like Terms 4.Simplifying Expressions

Section 3.1 Simplifying Expressions and Combining Like Terms 1.Identifying Like Terms Slide 4 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. A term is a number or the product or quotient of numbers and variables. An algebraic expression is the sum of one or more terms.

Section 3.1 Simplifying Expressions and Combining Like Terms 1.Identifying Like Terms (continued) Slide 5 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Variable terms will change when different numbers are substituted for the variables. A constant term has a value that will never change.

Section 3.1 Simplifying Expressions and Combining Like Terms 1.Identifying Like Terms Slide 6 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The coefficient of the term is the numerical factor of the term. Terms are said to be like terms if they each have the same variables, and the corresponding variables are raised to the same powers.

Example 1Identifying Terms, Coefficients, and Like Terms Slide 7 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example Solution: 1Identifying Terms, Coefficients, and Like Terms Slide 8 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Section 3.1 Simplifying Expressions and Combining Like Terms 2.Commutative, Associative, and Distributive Properties Slide 9 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Section 3.1 Simplifying Expressions and Combining Like Terms 2.Commutative, Associative, and Distributive Properties Slide 10 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Section 3.1 Simplifying Expressions and Combining Like Terms 2.Commutative, Associative, and Distributive Properties Slide 11 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Section 3.1 Simplifying Expressions and Combining Like Terms 2.Commutative, Associative, and Distributive Properties Slide 12 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Section 3.1 Simplifying Expressions and Combining Like Terms 2.Commutative, Associative, and Distributive Properties Slide 13 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. *Note that the distributive property of multiplication over addition is sometimes referred to as the distributive property.

Example 2Applying the Commutative Properties Slide 14 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Apply the commutative property of addition or multiplication to rewrite the expression.

Example Solution: 2Applying the Commutative Properties Slide 15 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 3Applying the Associative Properties Slide 16 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Use the associative property of addition or multiplication to rewrite each expression. Then simplify the expression.

Example Solution: 3Applying the Associative Properties Slide 17 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Apply the associative property of addition. Simplify. Apply the associative property of multiplication. Simplify.

Example 4Applying the Distributive Property Slide 18 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example Solution: 4Applying the Distributive Property Slide 19 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 5Applying the Distributive Property Slide 20 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example Solution: 5Applying the Distributive Property Slide 21 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Section 3.1 Simplifying Expressions and Combining Like Terms 3.Combining Like Terms Slide 22 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Two terms may be combined if they are like terms.

Example 6Using the Distributive Property to Add and Subtract Like Terms Slide 23 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example Solution: 6Using the Distributive Property to Add and Subtract Like Terms Slide 24 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 7Adding and Subtracting Like Terms Slide 25 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Simplify by combining like terms.

Example Solution: 7Adding and Subtracting Like Terms Slide 26 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Section 3.1 Simplifying Expressions and Combining Like Terms 4.Simplifying Expressions Slide 27 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. For expressions containing parentheses, it is necessary to apply the distributive property before combining like terms. Notice that when we apply the distributive property, the parentheses are dropped. This is often called clearing parentheses.

Example 8Clearing Parentheses and Combining Like Terms Slide 28 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Simplify by clearing parentheses and combining like terms.

Example Solution: 8Clearing Parentheses and Combining Like Terms (continued) Slide 29 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The order of operations indicates that we must perform multiplication before subtraction. It is also important to understand that a factor of –3 (not 3) will be multiplied to all terms within the parentheses. To see why, we can rewrite the subtraction in terms of addition of the opposite.

Example Solution: 8Clearing Parentheses and Combining Like Terms Slide 30 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Section Objectives 3.2 Addition and Subtraction Properties of Equality Slide 31 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1.Definition of a Linear Equation in One Variable 2.Addition and Subtraction Properties of Equality

Section 3.2 Addition and Subtraction Properties of Equality 1.Definition of a Linear Equation in One Variable Slide 32 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. An equation is a statement that indicates that two quantities are equal. A solution to an equation is a value of the variable that makes the equation a true statement.

Avoiding Mistakes Slide 33 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. It is important to distinguish between an equation and an expression. An equation has an equal sign, whereas an expression does not. For example: 2x + 4 = 16 equation 7x – 9 expression

Example 1Determining Whether a Number Is a Solution to an Equation Slide 34 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Determine whether the given number is a solution to the equation.

Example Solution: 1Determining Whether a Number Is a Solution to an Equation (continued) Slide 35 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The right-hand side equals the left-hand side. Thus, 6 is a solution to the equation 2x – 9 = 3. Substitute 6 for x. Simplify.

Example Solution: 1Determining Whether a Number Is a Solution to an Equation Slide 36 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Substitute –2 for p. Simplify. The right-hand side does not equal the left-hand side. Thus, –2 is not a solution to the equation 20 = 8p – 4.

DEFINITIONLinear Equation in One Variable Slide 37 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Let a and b be numbers such that a 0. A linear equation in one variable is an equation that can be written in the form ax + b = 0

Section 3.2 Addition and Subtraction Properties of Equality 1.Definition of a Linear Equation in One Variable Slide 38 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Note: A linear equation in one variable contains only one variable and the exponent on the variable is 1.

Section 3.2 Addition and Subtraction Properties of Equality 2.Addition and Subtraction Properties of Equality Slide 39 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Two equations are called equivalent equations if they have the same solution.

PROPERTYAddition and Subtraction Properties of Equality Slide 40 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 2Applying the Addition Property of Equality Slide 41 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example Solution: 2Applying the Addition Property of Equality Slide 42 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. To solve an equation, the goal is to isolate the variable on one side of the equation. That is, we want to create an equivalent equation of the form x = number. To accomplish this, we can use the fact that the sum of a number and its opposite is zero.

Example Solution: 2Applying the Addition Property of Equality (continued) Slide 43 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. To isolate x, add 6 to both sides, because –6 + 6 = 0. Simplify. The variable is isolated (by itself) on the left-hand side of the equation. The solution is 24.

Example Solution: 2Applying the Addition Property of Equality (continued) Slide 44 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example Solution: 2Applying the Addition Property of Equality (continued) Slide 45 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example Solution: 2Applying the Addition Property of Equality Slide 46 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 3Applying the Subtraction Property of Equality Slide 47 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example Solution: 3Applying the Subtraction Property of Equality (continued) Slide 48 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. To isolate z, subtract 11 from both sides, because 11 – 11 = 0. Simplify. The solution is 3.

Example Solution: 3Applying the Subtraction Property of Equality Slide 49 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 4Applying the Addition and Subtraction Properties of Equality Slide 50 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example Solution: 4Applying the Addition and Subtraction Properties of Equality (continued) Slide 51 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example Solution: 4Applying the Addition and Subtraction Properties of Equality Slide 52 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Section Objectives 3.3 Multiplication and Division Properties of Equality Slide 53 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1.Multiplication and Division Properties of Equality 2.Comparing the Properties of Equality

Section 3.3 Multiplication and Division Properties of Equality 1.Multiplication and Division Properties of Equality Slide 54 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Adding or subtracting the same quantity on both sides of an equation results in an equivalent equation. The same is true when we multiply or divide both sides of an equation by the same nonzero quantity.

PROPERTYMultiplication and Division Properties of Equality Slide 55 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 1Applying the Division Property of Equality Slide 56 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example Solution: 1Applying the Division Property of Equality (continued) Slide 57 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example Solution: 1Applying the Division Property of Equality Slide 58 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 2Applying the Multiplication Property of Equality Slide 59 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example Solution: 2Applying the Multiplication Property of Equality (continued) Slide 60 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example Solution: 2Applying the Multiplication Property of Equality (continued) Slide 61 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example Solution: 2Applying the Multiplication Property of Equality Slide 62 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Section 3.3 Multiplication and Division Properties of Equality 2.Comparing the Properties of Equality Slide 63 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. It is important to determine which property of equality should be used to solve an equation.

Section 3.3 Multiplication and Division Properties of Equality 2.Comparing the Properties of Equality Slide 64 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. For example, compare equations: 4 + x = 12 and 4x = 12 In the first equation, the operation between 4 and x is addition. Therefore, we want to reverse the process by subtracting 4 from both sides. In the second equation, the operation between 4 and x is multiplication. To isolate x, we reverse the process by dividing by 4.

Example 3Solving Linear Equations Slide 65 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example Solution: 3Solving Linear Equations Slide 66 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The operation between m and 12 is division. To obtain a coefficient of 1 for the m term, multiply both sides by 12. Multiply both sides by 12. Regroup. Simplify both sides. The solution –36 checks in the original equation.

Example Solution: 3Solving Linear Equations (continued) Slide 67 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example Solution: 3Solving Linear Equations Slide 68 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Section Objectives 3.4 Solving Equations with Multiple Steps Slide 69 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1.Solving Equations with Multiple Steps 2.General Procedure to Solve a Linear Equation

Example 1Solving a Linear Equation Slide 70 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example Solution: 1Solving a Linear Equation Slide 71 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Remember that our goal is to isolate x. Therefore, in this equation, we will first isolate the term containing x. This can be done by adding 3 to both sides.

Section 3.4 Solving Equations with Multiple Steps 1.Solving Equations with Multiple Steps Slide 72 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Generally apply the addition (or subtraction) property of equality to isolate the variable term first. Then apply the multiplication (or division) property of equality to obtain a coefficient of 1 on the variable term.

Example 2Solving a Linear Equation Slide 73 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example Solution: 2Solving a Linear Equation (continued) Slide 74 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. We first isolate the term containing the variable by subtracting 10 from both sides.

Example Solution: 2Solving a Linear Equation Slide 75 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 4Solving a Linear Equation with Variables on Both Sides Slide 76 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example Solution: 4Solving a Linear Equation with Variables on Both Sides (continued) Slide 77 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. To isolate x, we must first “move” all x terms to one side of the equation. For example, suppose we add 2x to both sides. This would “remove” the x term from the right-hand side because –2x + 2x = 0.The term 2x is then combined with 4x on the left-hand side.

Example Solution: 4Solving a Linear Equation with Variables on Both Sides Slide 78 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

TIP: Slide 79 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. It should be noted that the variable may be isolated on either side of the equation.

PROCEDURESolving a Linear Equation in One Variable (continued) Slide 80 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Step 1 Simplify both sides of the equation. Clear parentheses if necessary. Combine like terms if necessary. Step 2 Use the addition or subtraction property of equality to collect the variable terms on one side of the equation..

PROCEDURESolving a Linear Equation in One Variable Slide 81 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.. Step 3 Use the addition or subtraction property of equality to collect the constant terms on the other side of the equation. Step 4 Use the multiplication or division property of equality to make the coefficient of the variable term equal to 1. Step 5 Check the answer in the original equation.

Example 5Solving a Linear Equation Slide 82 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example Solution: 5Solving a Linear Equation (continued) Slide 83 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Step 1: Simplify both sides of the equation. Clear parentheses. Step 2: Add 4y to both sides to collect the variable terms on the left.

Example Solution: 5Solving a Linear Equation (continued) Slide 84 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Simplify. Step 3: Subtract 20 from both sides to collect the constants on the right. Simplify.

Example Solution: 5Solving a Linear Equation (continued) Slide 85 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Step 4: Divide both sides by 6 to obtain a coefficient of 1 on the y term The solution is –2.

Example Solution: 5Solving a Linear Equation Slide 86 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Step 5: Check the solution in the original equation. Substitute 2 for y. The solution checks.

Section Objectives 3.5 Applications and Problem Solving Slide 87 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1.Problem-Solving Flowchart 2.Translating Verbal Statements into Equations 3.Applications of Linear Equations

Section 3.5 Applications and Problem Solving 1.Problem-Solving Flowchart (continued) Slide 88 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Section 3.5 Applications and Problem Solving 1.Problem-Solving Flowchart (continued) Slide 89 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Section 3.5 Applications and Problem Solving 1.Problem-Solving Flowchart (continued) Slide 90 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Section 3.5 Applications and Problem Solving 1.Problem-Solving Flowchart Slide 91 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Section 3.5 Applications and Problem Solving 2.Translating Verbal Statements into Equations (continued) Slide 92 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Section 3.5 Applications and Problem Solving 2.Translating Verbal Statements into Equations Slide 93 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 1Translating Sentences to Mathematical Equations Slide 94 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. A number decreased by 7 is 12. Find the number.

Example Solution: 1Translating Sentences to Mathematical Equations (continued) Slide 95 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Step 1: Read the problem completely. Step 2: Label the variable. Step 3: Write the equation in words Let x represent the number. A number decreased by 7 is 12.

Example Solution: 1Translating Sentences to Mathematical Equations (continued) Slide 96 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Step 4: Translate to a mathematical equation. A number decreased by 7 is 12.

Example Solution: 1Translating Sentences to Mathematical Equations Slide 97 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Step 5: Solve the equation. Step 6: Interpret the answer in words. The number is 19.

Example 2Translating Sentences to Mathematical Equations Slide 98 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Seven less than 3 times a number results in 11. Find the number.

Example Solution: 2Translating Sentences to Mathematical Equations (continued) Slide 99 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example Solution: 2Translating Sentences to Mathematical Equations Slide 100 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 4Representing Quantities Algebraically Slide 101 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. a. Kathleen works twice as many hours in one week as Kevin. If Kevin works for h hours, write an expression representing the number of hours that Kathleen works. b. At a carnival, rides cost $3 each. If Alicia takes n rides during the day, write an expression for the total cost. c. Josie made $430 less during one week than her friend Annie made. If Annie made D dollars, write an expression for the amount that Josie made.

Example Solution: 4Representing Quantities Algebraically (continued) Slide 102 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example Solution: 4Representing Quantities Algebraically Slide 103 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example 5Applying a Linear Equation to Carpentry Slide 104 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. A carpenter must cut a 10-ft board into two pieces to build a brace for a picnic table. If one piece is to be four times longer than the other piece, how long should each piece be?

Example Solution: 5Applying a Linear Equation to Carpentry (continued) Slide 105 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. We can let x represent the length of either piece. However, if we choose x to be the length of the shorter piece, then the longer piece has to be 4x (4 times as long). Let x = the length of the shorter piece. Then 4x = the length of the longer piece.

Example Solution: 5Applying a Linear Equation to Carpentry (continued) Slide 106 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Example Solution: 5Applying a Linear Equation to Carpentry Slide 107 Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Recall that x represents the length of the shorter piece. Therefore, the shorter piece is 2 ft. The longer piece is given by 4x or 4(2 ft) = 8 ft. The pieces are 2 ft and 8 ft.