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Chapter 3. Language of Algebra 3.1 3.2 3.3 3.4 3.5 3.6 Algebraic Expressions Evaluating Algebraic Expressions and Formulas Simplifying Algebraic Expressions and the Distribution Property Combine Like Terms Simplifying Expressions to Solve Equations Using Equations to Solve Application Problems 1 3.1 Algebraic Expressions 1. Translate word phrases to algebraic expressions 2. Write algebraic expressions to represent unknown quantities 2 3.1 Algebraic Expressions 1. Translate word phrases to algebraic expressions Variable: a letter (or symbol) that stands for a number. Algebraic Expressions Variables and/or numbers can be combined with operations of addition, subtraction, multiplication and division to created algebraic expression. e.g. 4a + 7, 10 − y 3 3 3.1 Algebraic Expressions 1. Translate word phrases to algebraic expressions Addition Subtraction the sum of a and 8 a+8 the difference of 23 and P 23 – P 4 plus c 16 added to m 4 more than t 20 greater than F T increased by r exceeds y by 35 4+c m + 16 t+4 F + 20 T+r y + 35 550 minus h 18 less than w 7 decreased by j M reduced by x 12 subtracted from L 5 less f 550 – h w – 18 7–j M–x L – 12 5–f compare “18 less than w” and “5 less f” 4 3.1 Algebraic Expressions 1. Translate word phrases to algebraic expressions Multiplication the product of 4 and x 20 times B twice r double the amount a triple the profit P three-fourth of m Division 4x 20B 2r 2a 3P 3 4 m the quotient of R and 19 R 19 s divided by d s d k split into 4 parts k 4 the ratio of c to d c d 5 3.1 Algebraic Expressions 1. Translate word phrases to algebraic expressions e.g.1 Write each phrase as an algebraic expression Ans. p – 15 a. 15 less than the population p b. Twice the weight w Ans. 2w c. The product of the volume V and 3, increased by 60 Ans. 3V + 60 6 3.1 Algebraic Expressions 2. Write algebraic expressions to represent unknown quantities e.g.2 PACKAGING A regular-size package of Oreos cookies weights x ounces. A family-size package of Oreos weights 16 ounces more. Write an algebraic expression that represents the weight (in ounces) of the family-size pack. Ans. x + 16 e.g.3 JEWELRY The value of a wedding ring is three times that of an engagement ring. Choose a variable to represent the value of one of the rings. Then write an algebraic expression that represent the value of the other ring. Ans. x = value of the engagement ring Then, 3x = value of the wedding ring 7 3.1 Algebraic Expressions 2. Write algebraic expressions to represent unknown quantities e.g.4 CAMPING A rope is to be cut into six equally long pieces. Write an algebraic expression that represents the length of each piece. let r = length of the rope in feet. Then, length of each piece (in ft) = r 6 e.g.5 VEHICLE WEIGHTS The weight of a Cadillac Escalade is 900 lb less than four times a Smart Car. Choose a variable to represent the weight of one of he cars. Write an algebraic expression that represents the weight of the other car. Let x (in lb) = weight of a Smart Car, then weight of a Cadillac (in lb) = 4x – 900 8 3.1 Algebraic Expressions 2. Write algebraic expressions to represent unknown quantities e.g.6 POLICE FORCE Part of a 500-officer police force works in the field. The others have desk jobs. Choose a variable to represent the number of officers that work in the field. Then write an algebraic expression that represents the number of that work force at a desk. Let x = number of officers that work in the field, then number of officers that work at a desk = 500 – x e.g.7 LEADERS Hillary Clinton was born 7 years before Oprah Winfrey. Michelle Obama was born 10 years after Oprah. Write algebraic expressions to represent the ages of each of these famous women. Let x = the age of Oprah, then the age of Hillary = x + 7, the age of Michelle = x – 10 9 3.1 Algebraic Expressions 2. Write algebraic expressions to represent unknown quantities e.g.8 Write an algebraic expression that represents the number of hours in d days. Ans. 24d hours Comments: introducing tables, and inductive method 10 3.2 Evaluating Algebraic Expressions and Formulas 1. Evaluating algebraic expressions 2. Use formulas from business to solve application problems 3. Use formulas from science to solve application problems 4. Find the mean (average) of a set of values 11 3.2 Evaluating Algebraic Expressions and Formulas 1. Evaluating algebraic expressions e.g.2 Evaluate each expression for x = 4: Ans. 22 a. 6x – 2 − x −8 Ans. –4 b. 3 e.g.3 Evaluate each expression for x = –4: Ans. 44 a. 2x2 – 3x Ans. –17 b. –x + 7(1 + x) Ans. –89 c. x3 – 25 12 3.2 Evaluating Algebraic Expressions and Formulas 1. Evaluating algebraic expressions e.g.4 Evaluate each expression for c = –3 and d = 9: a. ( 2cd + 5d )2 Ans. 81 b. | –8d – 3c | Ans. 63 13 3.2 Evaluating Algebraic Expressions and Formulas 2. Use formulas from business to solve application problems e.g.5 FILMS It cost DreamWorks SKG about $528 million to make and distribute the film Shrek. If the studio has received approximately $916 million to date in worldwide revenue from the film, find the profit the studio has made on this movie. Ans. $388 millions Formula relating profit with revenue and cost. 14 3.2 Evaluating Algebraic Expressions and Formulas 3. Use formulas from science to solve application problems e.g.6 SPEED LIMITS Texas’s daytime speed limit for cars on rural interstate highways is 75 mph. How far would a car travel in 4 hours at this speed? Ans. 300 mi Distance-Speed-Time formula: d = r t 15 3.2 Evaluating Algebraic Expressions and Formulas 3. Use formulas from science to solve application problems e.g.7 LAND of ENCHANTMENT The record high temperature for New Mexico is 122°F, on June 27, 1994. Convert this temperature to degree Celsius. Ans. 50°C Celsius-to-Fahrenheit: Fahrenheit-to-Celsius: F = 95 C + 32 C= 5 ( F − 32 ) 9 16 3.2 Evaluating Algebraic Expressions and Formulas 4. Find the mean (average) of a set of values e.g.9 STUDENT PARKING As part of a traffic survey, the number of cars in a college parking lot at noon were counted. The results are listed below. Mon: 121, Tues: 204, Wed: 166, Thurs: 152, Fri: 87. Find the mean (average) number of cars in the parking lot at that time. Ans. 146 cars 17 3.3 Simplifying Algebraic Expressions and the Distribution Property 1. Simplify products 2. Use the distribution property 3. Distribute a factor of –1 18 3.3 Simplifying Algebraic Expressions and the Distribution Property 1. Simplify products e.g.1 Simplify: a. 6·9m b. –11a(–10) e.g.2 Simplify: a. –6x ·3y b. 3(–5b)(2a) Ans. 54m Ans. Ans. Ans. 110a –18xy –30ab 19 3.3 Simplifying Algebraic Expressions and the Distribution Property 2. Use the distribution property The Distributive Property For any numbers a, b and c a(b + c) = ab + ac and e.g.3 Multiply: a. 10(d + 7) b. 4(3y – 8) a(b – c) = ab – ac Ans. 10d + 70 Ans. 12y – 32 20 3.3 Simplifying Algebraic Expressions and the Distribution Property 2. Use the distribution property e.g.4 a. b. c. d. Multiply: –2(21x + 3) –8(4 – 10n) –4(–3t – 16) –1(a – 7) Ans. −42x − 6 Ans. −32 + 80n Ans. 12t + 64 Ans. −a + 7 21 3.3 Simplifying Algebraic Expressions and the Distribution Property 2. Use the distribution property Distributive property can be extended, for example, (b + c)a = ba + ca, (b – c)a = ba – ca It can also be extended to more then two terms: a(b + c + d) = ab + ac + ad a(b – c – d) = ab – ac – ad 22 3.3 Simplifying Algebraic Expressions and the Distribution Property 2. Use the distribution property e.g.5 a. b. c. d. Multiply: (14x + 21)3 (32 – n)4 3(x – 6y)2 –7(–2a – 3b +1) Ans. 42x + 63 Ans. 128 − 4n Ans. 6x − 36y Ans. 14a + 21b − 7 23 3.3 Simplifying Algebraic Expressions and the Distribution Property 3. Distribution a factor of –1 The opposite of a Sum The opposite of a sum is the sum of opposite For any numbers a and b, –(a + b) = –a + (–b) e.g.6 Simplify: –(16r – 8) Ans. −16r + 8 24 3.4 Combining Like Terms (long) 1. Identifying terms and coefficients of terms 2. Identify like terms 3. Combine like terms 4. Find perimeter of a rectangle and square 25 3.4 Combining Like Terms 1. Identifying terms and coefficients of terms Addition symbols separates expressions into parts called Terms. For example, x + 8 has two terms: x + 8 1st term 2nd term Since subtraction can be written as addition of opposite, the expression a2 – 3a – 9 has three terms: a2 – 3a – 9 = a2 + (–3a) + (–9) 1st term 2nd term 3rd term 26 3.4 Combining Like Terms 1. Identifying terms and coefficients of terms In general, a term is a product or quotient of numbers and/or variables. A single number or variable is also a term. Examples of terms are: 3 3 5 2 4, y, 6r, –w , 7x , , –15ab n e.g.1 Identifying terms of each expression: Ans. three terms: 2y2, 7y, 15 a. 2y2 + 7y + 15 b. –6ab Ans. one term: –6ab Ans. four terms: t, –4, –9t, 8 c. t – 4 – 9t + 8 27 3.4 Combining Like Terms 1. Identifying terms and coefficients of terms It is important to distinguish between terms and factors. e.g.2 Is x used as a factor or a term in each expression? Ans. term a. x+2 b. 2x Ans. factor 28 3.4 Combining Like Terms 1. Identifying terms and coefficients of terms Coefficient: the numerical factor of a term is called the coefficient of the term. e.g.3 Identify the coefficients of each term in the Expression: p3 – 12p2 + 3p – 4 Ans. 1, –12, 3, –4 29 3.4 Combining Like Terms 2. Identify like terms Like terms Like terms are terms containing exactly the same variables raised to the same powers. Any two constants are like terms. For example, in expression x – 4 – 9x + 8, –4 and 8 are like terms. 30 3.4 Combining Like Terms 2. Identify like terms e.g.4 Identify the like terms in each expressions: Ans. 6x and 2x are like terms a. 6x + 7 + 2x Ans. no like terms b. 8a4 – 8a3 – 8a2 c. –4t3 + 1 – 9 + t3 Ans. –4t3 and t3 are like terms, 1 and –9 are like terms 31 3.4 Combining Like Terms 3. Combine like terms 3x + 4x can be simplified (why) 3x + 4y cannot be simplified Combining Like Terms Like terms can be combined by adding or subtracting the coefficients of the terms and keeping the same variables with the same exponents 32 3.4 Combining Like Terms 3. Combine like terms e.g.5 Simplify by combining like terms, if possible: Ans. 8x a. 6x + 2x Ans. –11m b. –7m + (–6m) + 3m Ans. 3t3 c. 7t3 – 4t3 Ans. does not simplify d. 9r + 11 Ans. 12t e. 6t + 9t – 3t 33 3.4 Combining Like Terms 3. Combine like terms e.g.6 Simplify by combining like terms: Ans. y a. 12y – 11y Ans. 11y b. 12y – y Ans. −y c. 11y – 12y Ans. 13y d. 12y + y 34 3.4 Combining Like Terms 3. Combine like terms e.g.7 Simplify: 5x2 + 13x – 3x – 4 Ans. 5x2 + 10x – 4 e.g.8 Simplify: 7(b + 6) – 1 – (3b – 9) Ans. 4b + 50 35 3.4 Combining Like Terms 4. Finding the perimeter of a rectangle and square l Rectangle w w l The Formula for the Perimeter of a Rectangle The perimeter P of a rectangle with length l and width w is given by P = 2l + 2w Why? 36 3.4 Combining Like Terms 4. Finding the perimeter of a rectangle and square s Square s s s The Formula for the Perimeter of a Rectangle The perimeter P of a rectangle with sides of length s is given by P = 4s Why? 37 3.5 Simplifying Expressions to Solve Equations 1. Determine whether a number is a solution 2. Combine like terms to solve equations 3. Solve equations that have variable terms on both sides 4. Use the distributive property to solve equations 5. Applying strategies to solve equations 38 3.5 Simplifying Expressions to Solve Equations 1. Determine whether a number is a solution e.g.1 Is 8 a solution of 2x – 4 = x + 3? Ans. no ---------------------------------------------------------------------2. Combine like terms to solve equations Ans. m = 6 e.g.2 Solve: 10m – 6m = 24 Ans. a = 18 e.g.3 Solve: 100 + 27 = 8a – 35 + a 39 3.5 Simplifying Expressions to Solve Equations 3. Solve equations that have variable terms on both sides e.g.4 Solve: e.g.5 Solve: Ans. x = –26 7x – 22 = 8x + 4 –38 – 2b – b = 4b + 11 Ans. x = –7 ---------------------------------------------------------------------4. Use the distributive property to solve equations e.g.6 Solve: 4(8n + 7) =28 Ans. n = 0 40 3.5 Simplifying Expressions to Solve Equations 5. Apply a strategy to solve equations Strategy for Solving Equations (for detail read the book) 1) simplify each side of equation 2) isolate the variable term on one side 3) isolate the variable 4) Check the result e.g.7 Solve: 3(5x – 40) + 9x = 6(x + 70) Ans. x = 30 41 3.6 Using Equations to Solve Application Problems 1. Solve application problems to find one unknown 2. Solve application problems to find two unknowns 3. Solve number-valued problems 42 3.6 Using Equations to Solve Application Problems 1. Solve application problems to find one unknown e.g.1 COLLECTORS A poster collector purchases 6 classic film posters each year. If he now owns 36 posters, in how many years will he have a collection of 120 posters? Ans. In 14 years he will have a collection of 120 posters. Five-step problem solving strategy 43 3.6 Using Equations to Solve Application Problems 1. Solve application problems to find one unknown e.g.2 TAX REFUNDS After receiving tax refund check, a husband and wife split the refunded money equally. The wife then give $110 of her share to her son, leaving her with $344. What was the amount of the tax refund check? Ans. The amount of the tax refund check is $908. Five-step problem solving strategy 44 3.6 Using Equations to Solve Application Problems 2. Solve application problems to find two unknowns e.g.3 MARCHING BAND At a field competition, a marching band scored 6 points more on the music portion of the judging than they did on the visual portion. If their combined score was 84, what were their scores on music portion and on the visual portion? Ans. The amount of the tax refund check is $908. • • Five-step problem solving strategy Two unknown, one variable 45 3.6 Using Equations to Solve Application Problems 2. Solve application problems to find two unknowns e.g.4 GEOMETRY The perimeter of a rectangle is 100 inches. Find the length and width if the length is four times the width. Ans. The width is 10 inches, the length is 40 inches. • • Five-step problem solving strategy Two unknown, one variable 46 3.6 Using Equations to Solve Application Problems 3. Solve number-valued problems Number-Value Formula (Number)(Value) = Total Value e.g.5 Bananas sells for 99 cents a pound. Find the cost of: a. 6 lb of bananas Ans. the cost is 994 cents or $9.94 b. p lb of bananas Ans. the cost is 99p cents. c. (p + 3) lb of bananas Ans. the cost is 99(p + 3) cents.47 3.6 Using Equations to Solve Application Problems 3. Solve number-valued problems e.g.6 TICKET SALES Nine hundred fifty tickets were sold for a concert. Ticket price were $55 for general admission and $105 for reserved. Write algebraic expressions that represent the income received from the sale of general admission and the sale of reserved tickets. Use a table to present the result. Ans. let g be the number of tickets sold for general admission, then, the income from the sale of general admission = 55g dollars, The income from the sale of reserved tickets = 105(950 – g) dollars 48