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Chapter 3 Solving Equations and Problem Solving Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 3.1 Simplifying Algebraic Expressions Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Constant and Variable Terms A term that is only a number is called a constant term, or simply a constant. A term that contains a variable is called a variable term. 3y2 + (–4y) + 2 x+3 Constant terms Variable terms Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra, 6ed 3 Coefficients The number factor of a variable term is called the numerical coefficient. A numerical coefficient of 1 is usually not written. 5x x or 1x Numerical coefficient is 5. –7y 3y2 Numerical coefficient is –7. Understood numerical coefficient is 1. Numerical coefficient is 3. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra, 6ed 4 Like Terms Terms that are exactly the same, except that they may have different numerical coefficients are called like terms. Unlike Terms 5x, x 2 7x, 7y 5y, 5 6a, ab Like Terms 3x, 2x –6y, 2y, y –3, 4 2ab2, –5b2a The order of the variables does not have to be the same. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra, 6ed 5 Distributive Property A sum or difference of like terms can be simplified using the distributive property. Distributive Property If a, b, and c are numbers, then ac + bc = (a + b)c Also, ac – bc = (a – b)c Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra, 6ed 6 Distributive Property By the distributive property, 7x + 5x = (7 + 5)x = 12x This is an example of combining like terms. An algebraic expression is simplified when all like terms have been combined. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra, 6ed 7 Addition and Multiplication Properties The commutative and associative properties of addition and multiplication help simplify expressions. Properties of Addition and Multiplication If a, b, and c are numbers, then Commutative Property of Addition a+b=b+a Commutative Property of Multiplication a∙b=b∙a The order of adding or multiplying two numbers can be changed without changing their sum or product. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra, 6ed 8 Associative Properties The grouping of numbers in addition or multiplication can be changed without changing their sum or product. Associative Property of Addition (a + b) + c = a + (b + c) Associative Property of Multiplication (a ∙ b) ∙ c = a ∙ (b ∙ c) Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra, 6ed 9 Helpful Hint Examples of Commutative and Associative Properties of Addition and Multiplication 4+3=3+4 Commutative Property of Addition 6∙9=9∙6 Commutative Property of Multiplication (3 + 5) + 2 = 3 + (5 + 2) Associative Property of Addition (7 ∙ 1) ∙ 8 = 7 ∙ (1 ∙ 8) Associative Property of Multiplication Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra, 6ed 10 Multiplying Expressions We can also use the distributive property to multiply expressions. The distributive property says that multiplication distributes over addition and subtraction. 2(5 + x) = 2 ∙ 5 + 2 ∙ x = 10 + 2x or 2(5 – x) = 2 ∙ 5 – 2 ∙ x = 10 – 2x Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra, 6ed 11 Simplifying Expressions To simply expressions, use the distributive property first to multiply and then combine any like terms. Simplify: 3(5 + x) – 17 3(5 + x) – 17 = 3 ∙ 5 + 3 ∙ x + (–17) Apply the Distributive Property = 15 + 3x + (–17) Multiply = 3x + (–2) or 3x – 2 Combine like terms Note: 3 is not distributed to the –17 since –17 is not within the parentheses. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra, 6ed 12 Finding Perimeter 7z feet 3z feet 9z feet Perimeter is the distance around the figure. Perimeter = 3z + 7z + 9z = 19z feet Don’t forget to insert proper units. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra, 6ed 13 Finding Area (2x – 5) meters 3 meters A = length ∙ width = 3(2x – 5) = 6x – 15 square meters Don’t forget to insert proper units. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra, 6ed 14 Helpful Hint Don’t forget . . . Area: • surface enclosed • measured in square units Perimeter: • distance around • measured in units Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra, 6ed 15