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**Columbus State Community College**

Chapter 2 Section 1 Introduction to Variables

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**Introduction to Variables**

Identify variables, constants, and expressions. Evaluate variable expressions for given replacement values. Write properties of operations using variables. Use exponents with variables.

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**Expressions, Variables, and Constants**

EXAMPLE Writing an Expression and Identifying the Variable and Constant Write an expression for each rule. Identify the variable and the constant. (a) Maria increased her test average by 12 points. Variable a Constant (b) The price of a game dropped by $20 Variable p – 20 Constant

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**Evaluating an Expression**

EXAMPLE Evaluating an Expression Use this rule for finding the price of a game: The price of a game dropped by $20. The expression is p – 20. (a) Evaluate the expression when the original price is $90. p – 20 Replace p with 90. 90 – 20 Follow the rule. Subtract to find 90 – 20. 70 The new price will be $70.

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**Evaluating an Expression**

EXAMPLE Evaluating an Expression Use this rule for finding the price of a game: The price of a game dropped by $20. The expression is p – 20. (b) Evaluate the expression when the original price is $78. p – 20 Replace p with 78. 78 – 20 Follow the rule. Subtract to find 78 – 20. 58 The new price will be $58.

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**Numerical Coefficients**

The number part in a multiplication expression is called the numerical coefficient, or just the coefficient. 3x –4m –d n 3x –4m –1d n The numerical coefficients are 3, –4, –1, and 1 respectively.

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CAUTION CAUTION If an expression involves adding, subtracting, or dividing, then you do have to write +, –, or ÷. It is only multiplication that is understood without writing an operation symbol. 5 + x – x ÷ x x Add x Subtract x Divide by x Multiply by x

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**The Perimeter of a “STOP” Sign**

The shape of a common “STOP” sign is called an “Octagon.” An octagon has 8 equal sides as shown in the diagram below. To find the distance around an object, called the perimeter, simply add the outside edges together. The expression (rule) can be written in shorthand form as shown below. s s s STOP s s 8 s s s s

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**Evaluating an Expression with Multiplication**

EXAMPLE Evaluating an Expression with Multiplication The expression (rule) for finding the perimeter of an octagon is 8s. Evaluate the expression when the length of one side of the “STOP” sign is 15 inches. See the diagram below. 15 in. 8 s Replace s with 15 inches. 15 in. 15 in. STOP 8 • 15 inches Multiply. 15 in. 15 in. 120 inches 15 in. 15 in. The total distance around this “STOP” sign (perimeter) is 120 inches. 15 in.

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**Evaluating an Expression with Several Steps**

EXAMPLE Evaluating an Expression with Several Steps A car rental company charges a flat fee of $50 plus $30 per day to rent a certain car. The expression (rule) for finding the amount to charge a customer is shown below. Evaluate the given expression for a person who rents this car for 6 days. 30d Replace d with 6, the number of days. 30 ( 6 ) Follow the order of operations. Multiply first. Add. 230 The cost of renting the car for 6 days is $230.

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A Rectangular Garden Suppose you wanted to put a fence around a rectangular-shaped flower garden. The length of the garden is 24 feet and the width is 16 feet. How much fencing material would you need to finish the job? 24 feet 16 feet 16 feet 24 feet 24 feet + 16 feet + 24 feet + 16 feet = 80 feet of fencing

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Rectangles In general, the expression (rule) for finding the amount of fencing needed to surround a rectangular garden can be found as follows. l w w l l + w + l + w = 2l + 2w = amount of fencing

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**Evaluating an Expression with Two Variables**

EXAMPLE Evaluating an Expression with Two Variables (a) The expression (rule) for finding the perimeter of a rectangle is 2l + 2w. Evaluate the expression of a rectangular table that has a length, l, of 5 feet and a width, w, of 2 feet. 2 l w Replace l with 5 feet and w with 2 feet. 2 ( 5 feet ) ( 2 feet ) There is no operation between the 2 and the l and there is no operation between the 2 and the w, so it is understood to be multiplication. 10 feet feet Add. 14 feet The perimeter of this table is 14 feet.

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**Evaluating an Expression with Two Variables**

EXAMPLE Evaluating an Expression with Two Variables (b) Complete the table below to show how to evaluate each expression. Expression ( Rule ) Value of a Value of b a – b a • b 1. 5 7 5 – 7 is –2 5 • 7 is 35 2. –4 9 –4 – 9 is –13 –4 • 9 is –36 3. 6 –3 6 – –3 is 9 6 • –3 is –18 4. –2 –8 –2 – –8 is 6 –2 • –8 is 16

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**Writing Properties of Operations Using Variables**

EXAMPLE Writing Properties of Operations Using Variables Use the variable n to state this property: When any number is divided by 1, the quotient is the number. Use the letter n to represent any number. n 1 =

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**Understanding Exponents Used with Variables**

EXAMPLE Understanding Exponents Used with Variables Rewrite each expression without exponents. (a) m5 can be written as m • m • m • m • m m is used as a factor 5 times. (b) 8 x y4 can be written as • x • y • y • y • y Coefficient is 8. y4 (c) –5 b3 c2 can be written as –5 • b • b • b • c • c The exponent applies only to y. Coefficient is –5. b3 c2

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**Evaluating Expressions with Exponents**

EXAMPLE Evaluating Expressions with Exponents Evaluate each expression. (a) g2 when g is –4 g means g • g Replace each g with –4. –4 • –4 Multiply –4 times –4. 16 So g2 becomes ( –4 )2, which is ( –4 ) ( –4 ), or 16.

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**Evaluating Expressions with Exponents**

EXAMPLE Evaluating Expressions with Exponents Evaluate each expression. Replace m with –3, and replace n with –2. Multiply two factors at a time. (b) m3 n2 when m is –3 and n is –2 So m3 n2 becomes ( –3 )3 ( –2 )2, which is ( –3 ) ( –3 ) ( –3 ) ( –2 ) ( –2 ), or –108. m3 n2 means m • m • m • n • n –3 • –2 • –3 • –2 • –2 – • –2 • –2 • –2 –108

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**Evaluating Expressions with Exponents**

EXAMPLE Evaluating Expressions with Exponents Evaluate each expression. Replace w with –4 and replace v with 3. Multiply two factors at a time. (c) –2 w v2 when w is –4 and v is 3 So –2 w v2 becomes –2 ( –4 ) ( 3 )2, which is ( –2 ) ( –4 ) ( 3 ) (3 ), or 72. –2 w v2 means –2 • w • v • v –2 • –4 3 • • 3 • 3 72

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**Introduction to Variables**

Chapter 2 Section 1 – Completed Written by John T. Wallace

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