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Solving Equations and Problem Solving

Chapter 3. Solving Equations and Problem Solving. Simplifying Algebraic Expressions. 3.1. 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. Constant and Variable Terms. 3 y 2 + ( – 4 y ) + 2.

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Solving Equations and Problem Solving

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  1. Chapter 3 Solving Equations and Problem Solving

  2. Simplifying Algebraic Expressions 3.1

  3. 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. Constant and Variable Terms 3y2 + (–4y) + 2 x + 3 Constant terms Variable terms

  4. The number factor of a variable term is called the numericalcoefficient. A numerical coefficient of 1 is usually not written. Coefficients 5xx or 1x–7y 3y2 Numerical coefficient is 5. Numerical coefficient is –7. Understood numerical coefficient is 1. Numerical coefficient is 3.

  5. Terms that are exactly the same, except that they may have different numerical coefficients are called like terms. Like Terms Unlike Terms Like Terms 3x, 2x –6y, 2y, y –3, 4 5x, x2 7x, 7y 5y, 5 6a, ab 2ab2, –5b2a The order of the variables does not have to be the same.

  6. A sum or difference of like terms can be simplified using the distributive property. Distributive Property Distributive Property If a, b, and c are numbers, then ac + bc = (a + b)c Also, ac–bc = (a–b)c

  7. 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.

  8. 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.

  9. 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)

  10. Helpful Hint Examples of Commutative and Associative Properties of Addition and Multiplication 4 + 3 = 3 + 4 6 ∙ 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

  11. We can also use the distributive property to multiplyexpressions. Multiplying Expressions 2(5 + x) = 2∙5 + 2∙x = 10 + 2x or 2(5 –x) = 2∙5 – 2∙x = 10 – 2x The distributive property says that multiplication distributes over addition and subtraction.

  12. To simply expressions, use the distributive property first to multiply and then combine any like terms. Simplifying Expressions 3(5 + x) – 17 = Simplify: 3(5 + x) – 17 Apply the Distributive Property 3 ∙5 + 3 ∙x+ (–17) =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.

  13. 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.

  14. Finding Area (2x – 5) meters 3meters A = length ∙width = 3(2x – 5) = 6x – 15 square meters Don’t forget to insert proper units.

  15. Helpful Hint Don’t forget . . . • Area: • surface enclosed • measured in square units • Perimeter: • distance around • measured in units

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