Evaluate and Simplify Algebraic Expressions


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1 TEKS 1.2 a.1, a.2, 2A.2.A, A.4.B Evaluate and Simplify Algebraic Expressions Before You studied properties of real numbers. Now You will evaluate and simplify expressions involving real numbers. Why? So you can estimate calorie use, as in Ex. 60. Key Vocabulary power variable term coefficient identity A numerical expression consists of numbers, operations, and grouping symbols. An expression formed by repeated multiplication of the same factor is a power. A power has two parts: an exponent and a base. The exponent represents the number of times the base is used as a factor. In the power shown below, the base 7 is used as a factor 3 times. exponent base p 7 p 7 power You do not usually write the exponent when it is 1. For instance, you can write 8 1 simply as 8. E XAMPLE 1 Evaluate powers a. (25) 4 5 (25) p (25) p (25) p (25) b (5 p 5 p 5 p 5) In Example 1, notice how parentheses are used in part (a) to indicate that the base is 25. In part (b), the base of the power is 5, not 25. An order of operations helps avoid confusion when evaluating expressions. KEY CONCEPT For Your Notebook Order of Operations Steps Example STEP 1 First, do operations that occur within grouping symbols p (5 2 3) STEP 2 Next, evaluate powers p 2 STEP 3 STEP 4 Then, do multiplications and divisions from left to right. Finally, do additions and subtractions from left to right p Chapter 1 Equations and Inequalities
2 VARIABLES Avariable is a letter that is used to represent one or more numbers. An expression involving variables is called an algebraic expression. When you substitute a number for each variable in an algebraic expression and simplify, you are evaluating the algebraic expression. E XAMPLE 2 Evaluate an algebraic expression Evaluate 24x 2 2 6x 1 11 when x x 2 2 6x (23) 2 2 6(23) 1 11 Substitute 23 for x. 524(9) 2 6(23) 1 11 Evaluate power Multiply. 527 Add. at classzone.com E XAMPLE 3 Use a verbal model to solve a problem CRAFT FAIR You are selling homemade candles at a craft fair for $3 each. You spend $120 to rent the booth and buy materials for the candles. Write an expression that shows your profit from selling c candles. Find your profit if you sell 75 candles. Solution STEP 1 Write a verbal model. Then write an algebraic expression. Use the fact that profit is the difference between income and expenses. Price per candle (dollars/candle) p Number of candles sold (candles) 2 Expenses (dollars) 3 p c An expression that shows your profit is 3c STEP 2 Evaluate the expression in Step 1 when c c (75) Substitute 75 for c Multiply Subtract. c Your profit is $105. GUIDED PRACTICE for Examples 1, 2, and 3 Evaluate the expression (22) x(x 2 2) when x y 2 2 4y when y (z 1 3) 3 when z WHAT IF? In Example 3, find your profit if you sell 135 candles. 1.2 Evaluate and Simplify Algebraic Expressions 11
3 KEY CONCEPT For Your Notebook Terms and Coefficients In an expression that can be written as a sum, the parts added together are called terms. A term that has a variable part is called a variable term. A term that has no variable part is called a constant term. When a term is a product of a number and a power of a variable, the number is called the coefficient of the power. variable terms constant term 3x 2 1 5x 1 (27) coefficients SIMPLIFYING An expression is simplified if it contains no grouping symbols and all like terms are combined. Like terms are terms that have the same variable parts. (Constant terms are also considered like terms.) The distributive property allows you to combine like terms by adding coefficients. E XAMPLE 4 Simplify by combining like terms a. 8x 1 3x 5 (8 1 3)x Distributive property 5 11x Add coefficients. AVOID ERRORS The terms 3p 2 and p are not like terms. They use the same variable but different exponents, so the terms cannot be combined. b. 5p 2 1 p 2 2p 2 5 (5p 2 2 2p 2 ) 1 p Group like terms. 5 3p 2 1 p Combine like terms. c. 3(y 1 2) 2 4(y 2 7) 5 3y y 1 28 Distributive property 5 (3y 2 4y) 1 (6 1 28) Group like terms. 52y 1 34 Combine like terms. d. 2x 2 3y 2 9x 1 y 5 (2x 2 9x) 1 (23y 1 y) Group like terms. 527x 2 2y Combine like terms. IDENTITIES Two algebraic expressions are equivalent expressions if they have the same value for all values of their variable(s). For instance, in part (a) of Example 4, the expressions 8x 1 3x and 11x are equivalent. A statement such as 8x 1 3x 5 11x that equates two equivalent expressions is called an identity. GUIDED PRACTICE for Example 4 8. Identify the terms, coefficients, like terms, and constant terms in the expression 2 1 5x 2 6x 2 1 7x 2 3. Then simplify the expression. Simplify the expression m 2 9m 10. 2n n p 3 1 5p 2 2 p q 2 1 q 2 7q 2 5q (x 2 3) 2 2(x 1 6) y 2 x 1 10x 1 y 12 Chapter 1 Equations and Inequalities
4 E XAMPLE 5 Simplify a mathematical model DIGITAL PHOTO PRINTING You send 15 digital images to a printing service that charges $.80 per print in large format and $.20 per print in small format. Write and simplify an expression that represents the total cost if n of the 15 prints are in large format. Then find the total cost if 5 of the 15 prints are in large format. Solution Write a verbal model. Then write an algebraic expression. Price of large print (dollars/print) p Number of large prints (prints) 1 Price of small print (dollars/print) p Number of small prints (prints) INTERPRET EXPRESSIONS The total number of prints is 15, so if n are in large format, then 15 2 n are in small format. 0.8 p n p (15 2 n) An expression for the total cost is 0.8n 1 0.2(15 2 n). 0.8n 1 0.2(15 2 n) 5 0.8n n Distributive property 5 (0.8n 2 0.2n) 1 3 Group like terms n 1 3 Combine like terms. c When n 5 5, the total cost is 0.6(5) $6. GUIDED PRACTICE for Example WHAT IF? In Example 5, write and simplify an expression for the total cost if the price of a large print is $.75 and the price of a small print is $ EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKEDOUT SOLUTIONS on p. WS1 for Exs. 21, 29, and 59 5 TAKS PRACTICE AND REASONING Exs. 24, 33, 51, 59, 64, and 65 5 MULTIPLE REPRESENTATIONS Ex VOCABULARY Copy 12 7 and label the base and the exponent. 2. WRITING Explain what it means for terms to be like terms. 3. ERROR ANALYSIS Describe and correct the error in evaluating the power shown at the right EXAMPLE 1 on p. 10 for Exs EVALUATING POWERS Evaluate the power (23) (24) (22) (28) Evaluate and Simplify Algebraic Expressions 13
5 EXAMPLE 2 on p. 11 for Exs ORDER OF OPERATIONS Evaluate the expression for the given value of the variable d 2 6 when d f 1 15 when f h h when h j 2 3j p 5 when j (k 1 2) 2 2 6k when k m 1 (2m 2 9) 3 when m n 3 2 4n 1 10 when n x 4 2 4x 3 when x 521 at classzone.com 24. MULTIPLE TAKS REASONING CHOICE What is the value of 2x 2 2 6x 1 15 when x 522? A 11 B 19 C 35 D 43 EXAMPLE 4 on p. 12 for Exs SIMPLIFYING EXPRESSIONS Simplify the expression x 2 4x y 2 1 2y 1 3y z 2 2 2z 1 8z w 2 2 4w 1 3w w 29. 7(m 2 3) 1 4(m 1 5) (n 2 1 n) 2 6(n 2 2 2) 31. 4p p 2 9p 2 1 3(4p 1 7) 32. 6(q 2 2) 2 2(q 2 1 6q) 33. MULTIPLE TAKS REASONING CHOICE Which terms are like terms? A 2x, 2y B 3x 2, 4x C x 2, y 2 D 10x 3, 2x 3 GEOMETRY Write a simplified expression for the perimeter of the figure. Then evaluate the expression for the given value(s) of the variable(s). 34. a 5 3, b n g 5 5, h 5 4 5a 5a 1 b 4n g 1 2h 2b n 1 12 EVALUATING EXPRESSIONS Evaluate the expression for the given values of x and y x 1 6y when x 5 16 and y x 1 11y when x 522 and y x 3 1 5y when x 5 4 and y (3x) 2 2 y 3 when x 5 4 and y 5 5 HINT Fraction bars are grouping symbols. 41. x 2 y } x 1 y when x 5 10 and y SIMPLIFYING EXPRESSIONS Simplify the expression. x 1 2y } 4x 2 y when x 523 and y c 2 10d 1 3d 2 5c 44. 9j 1 4k 2 2j 2 7k 45. 2m 2 2 5n 2 1 6n 2 2 8m 46. p 3 1 3q 2 2 q 1 3p m 2 1 3n m 2 2 3n y 2 1 5x 2 12x 1 9y (s 2 t) 1 16(t 2 s) 50. 3(x 2 2 y) 1 9(x 2 1 2y) 51. OPENENDED TAKS REASONING MATH Write an algebraic expression that includes three coefficients, two like terms, and one constant term. Then simplify the expression. 5 WORKEDOUT SOLUTIONS 14 Chapter 1 Equations p. WS1and Inequalities 5 TAKS PRACTICE AND REASONING 5 MULTIPLE REPRESENTATIONS
6 GROUPING SYMBOLS Add parentheses to make a true statement p p CHALLENGE Under what conditions are the expressions (x 1 y) 2 and x 2 1 y 2 equal? Are the expressions equivalent? Explain. PROBLEM SOLVING EXAMPLE 3 on p. 11 for Exs MOVIE COSTS In the United States, the average movie ticket price (in dollars) since 1974 can be modeled by 0.131x where x is the number of years since What values of x should you use to find the ticket prices in 1974, 1984, 1994, and 2004? Find the ticket prices for those years. 58. MILEAGE You start driving a used car when the odometer reads 96,882. After a typical month of driving, the reading is 97,057. Write an expression for the reading on the odometer after m months, assuming the amount you drive each month is the same. Predict the reading after 12 months. 59. SHORT TAKS REASONING RESPONSE A student has a debit card with a prepaid amount of $270 to use for school lunches. The cafeteria charges $4.50 per lunch. Write an expression for the balance on the card after buying x lunches. Does your expression make sense for all positive integer values of x? Explain. EXAMPLE 5 on p. 13 for Exs CROSSTRAINING You exercise for 60 minutes, spending w minutes walking and the rest of the time running. Use the information in the diagram below to write and simplify an expression for the number of calories burned. Find the calories burned if you spend 20 minutes walking. Walking burns 4 calories per minute. Running burns 10 calories per minute. 61. MULTIPLE REPRESENTATIONS A theater has 30 rows of seats with 20 seats in each row. Tickets for the seats in the n rows closest to the stage cost $45 and tickets for the other rows cost $35. a. Visual Thinking Make a sketch of the theater seating. b. Modeling Write a verbal model for the income if all seats are sold. c. Simplifying Write and simplify an expression for the income. d. Making a Table Make a table for the income when n 5 5, 10, and COMPUTERS A company offers each of its 80 workers either a desktop computer that costs $900 or a laptop that costs $1550. Write and simplify an expression for the cost of all the computers when n workers choose desktop computers. Find the cost if 65 workers choose desktop computers. 1.2 Evaluate and Simplify Algebraic Expressions 15
7 63. CHALLENGE You want to buy 25 fish for an aquarium. You decide to buy danios, tetras, and rainbowfish. danios tetras rainbowfish $1.50 each $2.00 each $8.00 each Write and simplify an expression for the total cost of x danios, y tetras, and the rest rainbowfish. You buy 8 danios, 10 tetras, and the rest rainbowfish. What is the total cost? MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Skills Review Handbook p. 984; TAKS Workbook REVIEW TAKS Preparation p. 902; TAKS Workbook 64. TAKS PRACTICE A roadside fruit stand sells three apples for a total of $0.79. The total cost, c, of purchasing n apples can be found by TAKS Obj. 10 A multiplying n by c C dividing n by c B multiplying n by the cost of 1 apple D dividing c by the cost of 1 apple 65. TAKS PRACTICE A rectangle has a length of 6 feet and a perimeter of 22 feet. What is the perimeter of a similar rectangle with a width of 20 feet? TAKS Obj. 8 F 52 ft G 82 ft H 88 ft J 100 ft QUIZ for Lessons Graph the numbers on a number line. (p. 2) 1. 25, 7 } 2, 1, 2 4 } , 5.4, Ï } 5, , 27.3, 2 2 } 5, 2Ï } 3 Identify the property that the statement illustrates. (p. 2) 4. 6(4 1 9) 5 6(4) 1 6(9) p p (25) (217) 5 0 Evaluate the expression for the given value of the variable. (p. 10) 7. 10m 1 32 when m (8 2 n) 3 when n p 3 2 3p 2 when p 522 Simplify the expression. (p. 10) 10. 8x 1 6x 2 2 9x 2 2 4x 11. 5(x 1 9) 2 2(4 2 x) x 2 6y 1 15y 2 18x 13. CD COSTS CDs are on sale for $8 each and you have a gift card worth $100. Write an expression for the amount of money left on the gift card after purchasing n CDs. Evaluate the expression to find the amount of money left after purchasing 6 CDs. (p. 10) 16 EXTRA PRACTICE for Lesson 1.2, p ONLINE QUIZ at classzone.com
8 Graphing Calculator ACTIVITY Use after Lesson Evaluate Expressions TEKS a.2, a.5, a.6, 2A.2.A TEXAS classzone.com Keystrokes QUESTION How can you use a calculator to evaluate expressions? You can use a scientific calculator or a graphing calculator to evaluate expressions. Keystrokes for evaluating several expressions are shown below. Note that to enter a negative number, you use the key on a scientific calculator or the key (not the key) on a graphing calculator. EXAMPLE Evaluate expressions EXPRESSION CALCULATOR KEYSTROKES RESULT a Scientific Graphing b. (24) Scientific (24) Graphing c. (39 4 3) 3 Scientific (39 4 3) 3 Graphing d p 8 } 4 Scientific p 8 } 4 Graphing PRACTICE Use a calculator to evaluate the expression (20) (28) (24) ( ) } Use a calculator to evaluate the expression when x 523, y 5 5, and z z 1 y 8. x y 2 z x } 2z (x 1 y) 2 1 3z 12. (24x 1 9) 4 (y 1 2) 13. ERROR ANALYSIS A student evaluated the expression 7 1 (24) 3 on a graphing calculator by pressing The calculator displayed an error message. Describe and correct the error. 1.2 Evaluate and Simplify Algebraic Expressions 17
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