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Math 101 Study Session Quiz 2 Chapter 5 Sections 1 through 3 July 28, 2016 Chapter 5 Section 1: Evaluating Variable Expressions A variable is a letter of the alphabet that is used to stand for a quantity that is unknown or that can change or vary. An expression that contains one or more variables is called a variable expression. Note: terms in an expression are separated by plus signs (+) and minus signs (−) that are NOT contained in grouping symbols. Terms that contain variables are called variable terms. Terms that do not contain variables are called constant terms or simply constants. Replacing each variable in a variable expression by its value and then simplifying the resulting numerical expression is called evaluating a variable expression. Chapter 5 Section 2 Simplifying Variable Expressions Like terms of a variable expression are terms with the same variable part. The Distributive Property If a, b, and c are real numbers, then a(b + c) = ab + ac or (b + c)a = ba + ca. If If If If If If The Associative Property of Addition a, b, and c are real numbers, then (a + b) + c = a + (b + c). The Commutative Property of Addition a and b are real numbers then a + b = b + a. The Addition Property of Zero a is a real number, then a + 0 = a and 0 + a = a. The Inverse Property of Addition a is areal number, then a + (−a) = 0 and (−a) + a = 0. The Associative Property of Multiplication a, b, and c are real numbers, then (ab)c = a(bc). The Commutative Property of Multiplication a and b are real numbers, then ab = ba. 1 The Multiplication Property of One If a is a real number, then a · 1 = a and 1 · a = a. The Inverse Property of Multiplication If a is a real number and a is not equal to 1 1 1 1 is called the reciprocal of a. is also called the zero, then a · = 1 and · a = 1. a a a a multiplicative inverse of a. The product of a number and its reciprocal is 1. Chapter 5 Section 3 Translating Verbal Expressions into Variable Expressions Words or Phrases for Addition added to 6 added to y y+6 more than 8 more than x x+8 the sum of the sum of x and z x + z increased by t increased by 9 t+9 the total of the total of 5 and d 5 + d plus b plus 17 b + 17 Words or Phrases for Subtraction minus x minus 2 less than 7 less than t less 7 less t subtracted from 5 subtracted from d decreased by m decreased by 3 the difference between the difference between y and 4 Words or Phrases for Multiplication times of the product of multiplied by twice x−2 t−7 7−t d−5 m−3 y−4 10 times t 10t 1 one-half of x x 2 the product of x and y xy b multiplied by 11 11b twice n 2n Phrases for Division divided by x divided by 12 the quotient of the quotient of y and z the ratio of the ratio of t to 9 x 12 y z t 9 2 Phrases for Power the square of the square of x the cube of the cube of a x2 a3 Below are some examples for us to try with solutions at the end of the study guide: Solve each of the following: 1. Simplify 2 3 11 − a− − a − a 5 10 15 2. Simplify 3 (6x − 10y + 12) 4 3. Simplify −7 [4x + 2(7 − x)] 4. Evaluate the variable expression when a = −2, b = 6, c = −5, and d = 4. 1 −5 b + (ac + bd) 6 2 5. Translate the verbal expression “the sum of one-sixth of a number and four-fifths of the number” into a variable expression using the variable x to represent number. Then simplify the variable expression. 3 Solutions to the Study Guide Questions Solution to 1: The LCD of 5, 10, and 15 is 30. 2 3 11 2 3 11 − a− − a − a=− a+ a− a 5 10 15 5 10 15 2 6 3 3 11 2 =− · a+ · a− · a 5 6 10 3 15 2 12 9 22 =− a+ a− a 30 30 30 −12 + 9 − 22 = a 30 −34 + 9 = a 30 −25 = a 30 −5 = a 6 −5a = 6 Solution to 2: 3 3 3 3 (6x − 10y + 12) = (6x) + (−10y) + (12) 4 4 4 4 3 · 6x 3 · −10y 3 · 12 = + + 4 4 4 18x −30y 36 = + + 4 4 4 9x 15y = − +9 2 2 Solution to 3: −7[4x + 2(7 − x)] = −7[4x + 2(7) + 2(−x)] = −7[4x + 14 − 2x] = −7(4x) + −7(14) + −7(−2x) = −28x − 98 + 14x = −14x − 98 Solution to 4: To evaluate the variable expression, replace each a with -2, each b with 6, each c with -5, and each d with 4. 4 1 −5 1 −5 b + (ac + bd) = · (6) + ((−2)(−5) + (6)(4)) 6 2 6 2 −30 1 = + (10 + 24) 6 2 1 = −5 + (34) 2 34 = −5 + 2 = −5 + 17 = 12 Solution to 5: Variable Expression: 1 4 x+ x 6 5 Simplified Variable Expression: 1 4 5 1 64 x+ x= · x+ x 6 5 5 6 65 5 24 = x+ x 30 30 29 = x 30 Thus, the simplified variable expression is 29 x. 30 Questions for You to Try Below are some questions for you to try on your own. 1. Evaluate the variable expression when a = −2, b = 4, c = −2, and d = 3. 2 1 − d − (bd − ac) 3 4 2. Simplify x2 − 6x + (−3x2 ) + 3x 3. Simplify −6(−9x2 + 6x − 9) 4. Simplify 2x − 9[x − (2 − x)] 5 5. Translate the verbal expression “the quotient of five more than twice a number and the number” into a variable expression using x to represent number. Then simplify the variable expression. Solutions 1. -4 2. −2x2 − 3x 3. 54x2 − 36x + 54 4. −16x + 18 5. Variable Expression: 2x + 5 5 Simplified Variable Expression: 2 + x x 6