Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 3 Solving Equations and Problem Solving

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 3.1 Simplifying Algebraic Expressions

Martin-Gay, Prealgebra, 6ed 33 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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. x + 3 Constant terms Variable terms 3y 2 + ( – 4y) + 2 Constant and Variable Terms

Martin-Gay, Prealgebra, 6ed 44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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 – 7y 3y 2 Numerical coefficient is 5. Numerical coefficient is – 7. Numerical coefficient is 3. Understood numerical coefficient is 1. Coefficients

Martin-Gay, Prealgebra, 6ed 55 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Terms that are exactly the same, except that they may have different numerical coefficients are called like terms. Like Terms Unlike Terms 3x, 2x – 6y, 2y, y – 3, 4 7x, 7y 5y, 5 6a, ab The order of the variables does not have to be the same. 2ab 2, – 5b 2 a 5x, x 2 Like Terms

Martin-Gay, Prealgebra, 6ed 66 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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 Distributive Property

Martin-Gay, Prealgebra, 6ed 77 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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. Distributive Property

Martin-Gay, Prealgebra, 6ed 88 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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. Addition and Multiplication Properties

Martin-Gay, Prealgebra, 6ed 99 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 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) Associative Properties

Martin-Gay, Prealgebra, 6ed 10 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Examples of Commutative and Associative Properties of Addition and Multiplication = ∙ 9 = 9 ∙ 6 (3 + 5) + 2 = 3 + (5 + 2) (7 ∙ 1) ∙ 8 = 7 ∙ (1 ∙ 8) Commutative Property of Addition Commutative Property of Multiplication Associative Property of Addition Associative Property of Multiplication Helpful Hint

Martin-Gay, Prealgebra, 6ed 11 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. We can also use the distributive property to multiply expressions. 2(5 + x) = 2 ∙ ∙ x = x or 2(5 – x) = 2 ∙ 5 – 2 ∙ x = 10 – 2x The distributive property says that multiplication distributes over addition and subtraction. Multiplying Expressions

Martin-Gay, Prealgebra, 6ed 12 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. To simply expressions, use the distributive property first to multiply and then combine any like terms. 3(5 + x) – 17 = Simplify: 3(5 + x) 17 Simplify: 3(5 + x) – 17 = x + ( – 17) Apply the Distributive Property Multiply = 3x + ( – 2) or 3x – 2 Combine like terms Note: 3 is not distributed to the – 17 since – 17 is not within the parentheses. 3 ∙ ∙ x + ( – 17) Simplifying Expressions

Martin-Gay, Prealgebra, 6ed 13 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Finding Perimeter 3z feet 9z feet 7z feet Perimeter = 3z + 7z + 9z = 19z feet = 19z feet Don’t forget to insert proper units. Perimeter is the distance around the figure.

Martin-Gay, Prealgebra, 6ed 14 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Finding Area A = length ∙ width = 3(2x – 5) = 3(2x – 5) = 6x – 15 square meters = 6x – 15 square meters Don’t forget to insert proper units. 3 meters (2x – 5) meters

Martin-Gay, Prealgebra, 6ed 15 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Don’t forget... Area: surface enclosed measured in square units Perimeter: distance around measured in units Helpful Hint