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Chapter 5: Expressions Study Guide Writing Algebraic Expressions: Define your variable Fewer than, less than, more than -> flip the order 3 less than a number x x –3 Quantity of = put what follows in parentheses 2 times the quantity of 5 minus a number n 2(5 – n) Evaluating Algebraic Expressions: Evaluate = Solve Plug your value in the expression using parentheses Use PEMDAS to solve Evaluate: 3x – 2w for x = 8 and w = 4 3(8) – 2(4) ** Plug in values in parentheses ** 24 – 8 16 Properties: Commutative Property of Addition: 2 + 5 = 5 + 2 Change order of numbers (or variables, if applicable) Commutative Property of Multiplication: 3∙7 = 7∙3 Change order of numbers (or variables, if applicable) Chapter 5: Expressions Study Guide Associative Property of Addition: 2 + (3 + 4) = (2 + 3) + 4 Same order of numbers, different groupings (or variables, if applicable) Associative Property of Multiplication: 6(xy) = (6x)y Same order of numbers, different groupings (or variables, if applicable) Identity Property of Multiplication (Multiplicative Identity): 7∙1 = 7 Identity Property of Addition (Additive Identity): 6 + 0 = 6 Zero Property of Multiplication (Multiplicative Property of Zero): 5∙0 =0 Distributive Property Distributive Property = multiply each term inside the parenthesis by the number outside the parenthesis Example: Use the distributive property to simplify 7(3 + 12) * Distribute 7∙3 + 7∙12 * Multiply 21 + 84 * Clean-up 105 Example: Use the distributive property to simplify 11(x - 5) * Distribute 11∙x - 11∙5 * Multiply 11x - 55 * Clean-up Chapter 5: Expressions Study Guide Example: Use the distributive property to simplify -3(x - 12) * Distribute (-3)∙x – (-3)∙12 * Multiply **Use Integer rules LCO -3x + 36 * Clean-up Example: Use the distributive property to simplify (4x - 10)7 * Distribute 7(4x) – (7)∙10 * Multiply 28x - 70 * Clean-up Simplifying Algebraic Expressions 3x + 5 + 2x -3 Terms: parts of an algebraic expression separated by a + or – Negative signs must be included (positive is assumed) If the term is x the coefficient is 1 If the term is –x the coefficient is -1 Example: 3x, 5, 2x, -3x Like Terms: terms that contain the same variable or just numbers Example: 3x, 2x and 5, -3 Coefficients: number in front of the variable (letter) ~ similar to an adjective it describes the variable Example: 3, 2 (because it’s 3x and 2x) Constants: a number without a variable, a plain number! Example: 5, -3 Chapter 5: Expressions Study Guide Simplifying algebraic expressions = no like terms or parentheses Combine the like terms If the term is x the coefficient is 1 If the term is –x the coefficient is -1 Replace with the combined like terms Do not add/subtract terms together that are not alike Example: 2x + 7 ≠ 9 Example: Write in simplest form 7u + u + 12 7u + u = 9u * Combine like terms 12 9u + 12 Example: Write in simplest form 10z + z -5 - 9z + 2 10z + z – 9z = 11z – 9z = 2z * Combine like terms -5 + 2 = -3 * Combine like terms 2z + (-3) = 2z -3 Example: Write in simplest form 5a + 3b +6 – 3a + 2 5a -3a = 2a * Combine like terms 6+2=8 * Combine like terms 3b 2a + 3b + 8 Adding Linear Expressions Combine like terms using integer rules Use the Distributive Property, if applicable and then combine like terms Chapter 5: Expressions Study Guide Example 1: Add. (7x + 12) + (5x + 2) Two different ways to solve, same result Horizontally (7x + 12) + (5x + 2) 7x + 5x = 12x * Combine like terms 12 + 2 = 14 * Combine like terms 12x + 14 Verically (7x + 12) + (5x + 2) 7x + 12 * Write first expression 5x + 2 * Write second expression, lining up like terms and combine like terms 12x + 14 Example 2: Add. (12a + 5) + (-5a - 4) Two different ways to solve, same result Horizontally Vertically (12a + 5) + (-5a - 4) (12a + 5) + (-5a - 4) 12a + -5a = 7a 12a + 5 5 + -4 = 1 7a + 1 + -5a - 4 7a + 1 Example 3: Add. 5(6b - 3) + (-4b + 12) * Distribute first Horizontally Vertically 5(6b - 3) + (-4b + 12) 5(6b - 3) + (-4b + 12) (30b – 15) + (-4b +12) (30b – 15) + (-4b + 12) Chapter 5: Expressions Study Guide 30b +-4b = 26b 30b -15 -15 + 12 = -3 26b -3 -4b + 12 26b -3 Remember: Perimeter = add all sides of the shape Subtracting Linear Expressions Combine like terms using integer rules Use the Distributive Property, if applicable and then combine like terms Example 1: Subtract. (8x + 15) - (5x + 3) Two different ways to solve, same result Horizontally (8x + 15) - (5x + 3) 8x - 5x = 3x * Combine like terms 15 - 3 = 12 * Combine like terms 3x + 12 Verically (8x + 15) - (5x + 3) 8x + 15 + -5x - 3 3x + 12 * Write first expression * Add second expression using additive inverse, lining up like terms and combine like terms Example 2: Subtract. (20a + 16) - (-5a - 4) Two different ways to solve, same result Horizontally Vertically Chapter 5: Expressions Study Guide (20a + 16) - (-5a - 4) (20a + 16) - (-5a - 4) 20a - -5a = 20a + 5a = 25a 16 - -4 = 16 + 4 = 20 20a +16 + 25a + 20 5a + 4 25a + 20 Example 3: Subtract. 4(5a - 3) - (-2a - 6) First distribute, then solve Horizontally Vertically (20a - 12) - (-2a -6) (20a - 12) - (-2a - 6) 20a - -2a = 20a + 2a = 22a -12 - -6 = -12 + 6 = -6 20a -12 + 22a – 6 2a + 6 22a - 6 Greatest Common Factor: GCF Use upside down division for numbers Match variables (common variables) are included in the GCF Example1: Find the GCF of each pair of monomials: 12 and 20 2 2 GCF = 4 | 12 20 | 6 3 10 5 Chapter 5: Expressions Study Guide Example2: Find the GCF of each pair of monomials: 9ab and 15abc 3 | 9ab 3ab 15abc 5abc GCF = 3ab Factoring Linear Expressions using GCF Find the GCF of each term Common variables are included in the GCF To check your factoring, use the distributing property Examples: Factor each linear expression using GCF: 8x + 12 5x -32 8xy + 56x GCF = 4 GCF = 1 GCF = 8x 4(2x + 3) cannot be factored 8x(y + 7) Use distributive to check your answers: √ 8x + 12 √ 5x -32 √ 8xy + 56x