Study Guide. Writing Expressions and Equations

Transcription

1 Writing Epressions and Equations If the cost of one CD is $5 and ou want to bu three CDs, ou know that our total cost will be $5, or $45. The epression 5 is called a numerical epression. You could use a letter such as n to represent the number of CDs ou might bu. Then the epression 5 n, or 5n, would represent the cost of buing n CDs. The epression 5n is called an algebraic epression because it contains a variable. A variable such as n is a letter used to represent a number. You can use variables to write verbal epressions as algebraic epressions. Verbal Epressions plus c c more than k minus 5 k decreased b 5 two times the product of and q divided b the sum of 6 and b the quotient of q and the sum of 6 and b Epressions do not contain equal signs, but tell onl which operations to perform. Equations alwas contain an equal sign. Algebraic Epression c k 5 q 6 b Eample: Write an equation for each sentence. a. Eight multiplied b equals b. Three less than 7 times a number n is 4. 7n 4 Write an algebraic epression for each verbal epression.. the difference of r and 0. increase the product of and a b Write a verbal epression for each algebraic epression. p (4 w) Write an equation for each sentence. 6. Seven minus a number z is the same as Ten more than twelve times a number h equals 5. Glencoe/McGraw-Hill Algebra: Concepts and Applications

2 rder of perations Read this sentence: Jason said Leona is smart. You need punctuation to tell ou whether the sentence means Jason said, Leona is smart. or Jason, said Leona, is smart. The meaning of a mathematical epression such as 0 can also be confusing unless ou know which numbers and operations should be grouped together. The order of operations at the right tells ou that 0 means 0 ( ) or 4. rder of perations. Find the values of the epressions inside grouping smbols, such as parentheses ( ) and brackets [ ], and as indicated b fraction bars.. Do all multiplications and divisions from left to right.. Do all additions and subtractions from left to right. Eample : Find the value of Multipl and divide from left to right. 0 Eample : Find the value of 7(0 ). 7(0 ) 7 7 Simplif within parentheses first. 49 Eample : Find the value of. 0 ( ) ( ) 5 5 Simplif within parentheses first. Evaluate the numerator and the denominator separatel. Find the value of each epression (7 6) ( 8) ( 7) Glencoe/McGraw-Hill 6 Algebra: Concepts and Applications

3 Commutative and Associative Properties Carlos makes salad dressings with olive oil and balsamic vinegar. Sometimes he adds the olive oil first and other times he adds the vinegar first. The salad dressing is alwas the same, so the order doesn t matter. The order in which an two numbers are either added or multiplied doesn t change the sum or product. Addition and multiplication are said to be commutative. Eample : Eample : You can also regroup numbers when ou are adding or multipling without changing the sum or product. Addition and multiplication are said to be associative. Eample : (0 5) 8 0 (5 8) Eample 4: (00 4) 5 00 (4 5) 000 Commutative Propert For an two numbers of Addition a and b, a b b a. Commutative Propert For an two numbers of Multiplication a and b, a b b a. Commutative () Commutative () Associative Propert For an numbers a, b, and c, of Addition (a b) c a (b c). Associative Propert For an numbers a, b, and c, of Multiplication (a b) c a (b c). Associative () Associative () Simplif each epression.. (d 7) k 5 4. (4a b) (a b) (8 m) 4 Name the propert shown b each statement b b 9 8. (4 6) ( 4)6 9. ( ) 7 ( 7) Glencoe/McGraw-Hill Algebra: Concepts and Applications

4 4 Distributive Propert Judi bus a cup of juice and a bagel for Hans and herself at the cafeteria. Juice costs $ and a bagel costs $0.50. To find the total, Judi finds the total for herself and doubles it. ($ $0.50) $.50 $ Judi and Hans find the same total. ($ $0.50) ($) ($0.50). This is an eample of the Distributive Propert. To find the total, Hans finds the cost of bagels and juices and adds them. ($) ($0.50) $ $ $ Distributive Propert For all numbers a, b, and c, a(b c) ab ac and a(b c) ab ac You can use the Distributive Propert to simplif epressions. Eample : Simplif ( ) 4. ( ) 4 4 Distributive Propert 4 Commutative Propert 7 Substitution Propert Eample : Simplif 7(m p) (m p). 7(m p) (m p) 7m 7p m p Distributive Propert 7m m 7p p Commutative Propert 9m 5p Substitution Propert Simplif each epression.. (u v). 5(k ). 4( 5s) 4. 7( h) 5. (a j k) 6. 7(c d) 7. 5(ab c) 8. 4(w ) 9. (a b) (a b) 0. 7( ). (e 4f ef ). (m ) 4m Glencoe/McGraw-Hill 6 Algebra: Concepts and Applications

5 5 A Plan for Problem Solving A problem-solving plan can help ou identif and organize the information in a problem, then plan and eecute a solution. A problem-solving plan should include these steps.. Eplore Read the problem carefull. Identif the information that is given and determine what ou need to find.. Plan Select a strateg for solving the problem. If possible, estimate what ou think the answer should be before solving the problem.. Solve Use our strateg to solve the problem. You ma have to choose a variable for the unknown, and then write an epression. 4. Eamine Check our answer. Does it make sense? Is it reasonabl close to our estimate? Eample: A tree in our ard grows 7 inches a ear and is now 9 inches tall. In about how man ears will the tree be inches tall? Eplore: Plan: The tree is alread 9 inches tall and it grows 7 inches a ear. You need to find how man ears it will take the tree to grow to inches. Since the tree needs to grow about 0 more inches and 0 7 4, ou can estimate that it will take more than 4 ears for the tree to reach inches. Solve the problem. Solve: Number of Years From Present Height of Tree in Inches Eamine: Your table shows ou that the tree will be inches in a little over 4 ears. Since the answer matches our estimate, the answer is reasonable. Janine is selling subscriptions to an Internet service. She began b selling one subscription the first da. n the second da she sold two more subscriptions, and on the third da she sold more. If she continues to sell subscriptions according to this pattern, how man will she have sold at the end of one week? Glencoe/McGraw-Hill Algebra: Concepts and Applications

6 6 Collecting Data Alarge ski area surveed 50 skiers to find out how long the waited in a lift line during a bus period. Their responses are in the chart below. Time in Lift Line (minutes) A frequenc table is one wa to organize data so ou can draw conclusions more easil from the data. In a frequenc table, ou use tall marks to record how frequentl events occur. Eample: Make a frequenc table to organize the surve data. Waiting times var from minutes to 6 minutes. If ou group the waiting times in sets of three, our table will not be too long. The groups are called intervals. This table has intervals of three minutes. Each result from the surve is recorded in the tall chart. The total number of tallies is recorded in the Frequenc column. Lift Line Wait Time (min) Tall Frequenc Make a frequenc table to organize the data in the table. Use intervals of $4. Profits ($) Glencoe/McGraw-Hill 6 Algebra: Concepts and Applications

7 7 Displaing and Interpreting Data Data can be easier to analze if the are presented in the form of a graph. There are man was to graph data, such as line graphs, histograms, and stem-and-leaf plots. A histogram is a graph of the data in a frequenc table. Eample: The frequenc table shows the amount of time skiers waited in a lift line. Construct a histogram for the data. Lift Line Wait Time (min) Tall Frequenc The horizontal ais displas the time intervals from the table. The vertical ais displas equal intervals of. For each time interval, draw a bar. The height of the bar is equal to its frequenc. Label the two aes and title the histogram. 0 5 Frequenc Lift Line Wait Time (min) Make a histogram of the data in the frequenc table. Students Visiting Museum Ages Tall Frequenc Glencoe/McGraw-Hill Algebra: Concepts and Applications

8 Graphing Integers on a Number Line The numbers displaed on the number line below belong to the set of integers. The arrows at both ends of the number line indicate that the numbers continue indefinitel in both directions. Notice that the integers are equall spaced. negative integers positive integers Use dots to graph numbers on a number line. You can label the dots with capital letters. A B C D E F integers The coordinate of B is and the coordinate of D is 0. Because is to the right of on the number line,. And because 5 is to the left of, 5. Because and are the same distance from 0, the have the same absolute value,. Use two vertical lines to represent absolute value. units units The absolute value of is The absolute value of is. Eample: Evaluate and 0 0 Name the coordinate of each point.. B. D. G 5 4 B D E F G Graph each set of numbers on a number line. 4. {,, 4} 5. {, 0, } Write or in each blank to make a true sentence Evaluate each epression Glencoe/McGraw-Hill 5 Algebra: Concepts and Applications

9 The Coordinate Plane The two intersecting lines and the grid at the right form a coordinate sstem. The horizontal number line is called the -ais, and the vertical number line is called the -ais. The - and -aes divide the coordinate plane into four quadrants. Point S in Quadrant I is the graph of the ordered pair (, ). The -coordinate of point S is, and the -coordinate of point S is. Quadrant II Quadrant III Quadrant I S 4 Quadrant IV The point at which the aes meet has coordinates (0, 0) and is called the origin. Eample : What is the ordered pair for point J? In what quadrant is point J located? You move 4 units to the left of the origin and then unit up to get to J. So the ordered pair for J is (4, ). Point J is located in Quadrant II. Eample : Graph M(, 4) on the coordinate plane. Start at the origin. Move left on the -ais to and then down 4 units. Draw a dot here and label it M. Quadrant II J 4 4 M 4 Quadrant III Quadrant I 4 Quadrant IV Write the ordered pair that names each point.. P. Q. R 4. T 4 P R T 4 4 Q 4 Graph each point on the coordinate plane. Name the quadrant, if an, in which each point is located. 5. A(5, ) 6. B(, 0) 7. C(, ) 8. D(0, ) 9. E(, ) 0. F(, ) Glencoe/McGraw-Hill 56 Algebra: Concepts and Applications

10 Adding Integers You can use a number line to add integers. Start at 0. Then move to the right for positive integers and move to the left for negative integers () Both integers are positive. Both integers are negative. First move units right from 0. First move units left from 0. Then move more unit right. Then move more unit left. When ou add one positive integer and one negative integer on the number line, ou change directions, which results in one move being subtracted from the other move () 4 Move units left, then unit right. Move units right, then unit left. Use the following rules to add two integers and to simplif epressions. Rule Eamples To add integers with the same sign, add 7 4 their absolute values. Give the result the 8 () 0 same sign as the integers. 5 () 8 To add integers with different signs, 9 (6) subtract their absolute values. Give the (5) 4 result the same sign as the integer with 9 7 the greater absolute value. (4) Find each sum (9). (8) (8) (5) 6. 8 (8) () Simplif each epression. 8. (6) 9. 5 (7) 0. m (4m) (m) Glencoe/McGraw-Hill 6 Algebra: Concepts and Applications

11 4 Subtracting Integers If the sum of two integers is 0, the numbers are opposites or additive inverses. Eample : a. is the opposite of because 0 b. 7 is the opposite of 7 because 7 (7) 0 Use this rule to subtract integers. To subtract an integer, add its opposite or additive inverse. Eample : Eample : Find each difference. a () Subtracting is the same as adding its opposite,. b. 7 () 7 () 7 Subtracting is the same as adding its 6 opposite,. Evaluate c d e if c, d 7, and e. c d e 7 () Replace c with, d with 7, and e with. 7 Write 7 () as Find each difference (9) (5) (0) (8) Simplif each epression (6). m 8m (m) Evaluate each epression if,, and z 4... z 5 4. z () z z 7. 0 Glencoe/McGraw-Hill 66 Algebra: Concepts and Applications

12 5 Multipling Integers Use these rules to multipl integers and to simplif epressions. The product of two positive integers is positive. The product of two negative integers is positive. The product of a positive integer and a negative integer is negative. Eample : Find each product. a. 7() 7() 84 Both factors are positive, so the product is positive. b. 5(9) 5(9) 45 c. 4(8) 4(8) Both factors are negative, so the product is positive. The factors have different signs, so the product is negative. Eample : Evaluate ab if a and b 5. ab ()(5) Replace a with and b with 5. 9(5) 9 45 Both factors are negative. Eample : Simplif (4). (4) ( 4)() 48 Associative Propert 4 48 Find each product.. (8). (7)(9). () 4. 6(5) 5. 4()(5) 6. (8)(8)() 7. (5)(0) Evaluate each epression if a, b, and c. 8. 5c 9. ab 0. abc. b c Simplif each epression.. (6). 5(7) 4. (p)(4q) Glencoe/McGraw-Hill 7 Algebra: Concepts and Applications

13 6 Dividing Integers Eample : Use the multiplication problems at the right to find each quotient. a. 5 5 Since 5 5, 5 5. b. 5 (5) Since (5) 5, 5 (5). c. 5 5 Since 5 5, 5 5. d. 5 (5) Since (5) 5, 5 (5). 5 5 (5) (5) 5 Use these rules to divide integers. The quotient of two positive integers is positive. The quotient of two negative integers is positive. The quotient of a positive integer and a negative integer is negative. r s Eample : Evaluate if r 8 and s. r s 8 Replace r with 8 and s with () Find each quotient (7). 5 () Evaluate each epression if k, m, and n. n k 9. m 0.. m k. m 5 n Glencoe/McGraw-Hill 76 Algebra: Concepts and Applications

14 Rational Numbers The chart below shows the fraction and decimal forms of some rational numbers. Rational Number Fraction Form Decimal Form You can compare numbers using a number line or using cross products Eample: Write,, or in each blank to make a true sentence. a. 7 (7) (5) Cross multipl So. b Since 0.4 is to the right of, 0.4. Write,, or in each blank to make a true statement Write the numbers in each set from least to greatest. 8.,, 4., 0.4, ,, ,, Glencoe/McGraw-Hill 95 Algebra: Concepts and Applications

15 Adding and Subtracting Rational Numbers During track practice, Sheila recorded the time it took her to run two consecutive miles. She ran the first mile in 7.6 seconds. She ran the second mile in 7.0 seconds. The net change in times from the first mile to the second mile is or 0.5 seconds. Sheila ran the second mile 0.5 seconds faster. To add and subtract rational numbers, ou use the same rules that ou learned for adding and subtracting integers. Adding Rational Numbers Subtracting Rational Numbers To add rational numbers with the same sign, add their absolute values. Give the result the same sign as the rational numbers. To add rational numbers with different signs, subtract their absolute values. Give the result the same sign as the number with the greater absolute value. To subtract a rational number, add its opposite. Eamples: Find each sum or difference. a. 7.4 (0.) b c (9.) ( ) 5. (9.) [8. 5.] (9.) (7.4 0.) 4. (9.) Find each sum or difference (0.) 6..5 (0.) (5.66) (.6) Evaluate each epression if a, b, c 9.5, and d a b 7. c d 8. d c Glencoe/McGraw-Hill 00 Algebra: Concepts and Applications

16 Mean, Median, Mode, and Range For his Social Studies class, Carlos surveed five gas stations and recorded these prices for gallon of gasoline. $. $.8 $.4 $. $.0 The mean, the median, the mode, and the range of these prices can all be used to describe the prices. mean median mode range Eample: The mean, or average, of a set of data is the sum of the data divided b the number of pieces of data. The median of a set of data is the middle number when the data in the set are arranged in numerical order. If there are two middle numbers, the median is the mean of those two numbers. The mode of a set of data is the number that occurs most often in the set. A set can have no mode or more than one mode. The range of a set of data is the difference between the greatest and the least values of the set. Find the mean, the median, the mode, and the range of the gasoline prices that Carlos recorded. Find the mean of the data The mean is $... Find the mode of the data. Since $. occurs most often, $. is the mode. Find the median of the data. Find the range of the data. First arrange the data in order. Subtract the greatest and least values. Then identif the middle number. $.4 $.8 $0.5.8,.0,.,.,.4 The range of the data is $0.5. The median is $.. Find the mean, median, mode, and range of each set of data.. 48, 5, 9, 4, 6, 6. 5.,.7,.7,.7. 0,, , 0.4, 8.,.6, 0.4 Glencoe/McGraw-Hill 05 Algebra: Concepts and Applications

17 4 Equations Jason has won gold medals in swim meets this ear. Net Saturda he will swim in events. How man events will he win if he has 5 gold medals at the end of the meet? Let m the number of gold medals Jason wins net Saturda. Then the equation m 5 models the number of gold medals Jason will have at the end of the meet. The replacement set for m is {,, }. Since is the onl number from the replacement set that makes the equation true, is the solution of the equation. Eample : Find the solution of 0 if the replacement set is {0,, }. Find the value in the replacement set that makes the equation true false true false Since makes the equation 0 true, the solution is. Sometimes ou can solve equations b appling the order of operations. Eample : Solve each equation. a. b 7 () Multipl. b 7 6 Subtract. b The solution is. 5 8() 4 b. 8 h Evaluate the fraction. 6 8 h Divide. 8 h Subtract. 5 h The solution is 5. Find the solution of each equation if the replacement sets are {,, }, {5, 4, }, and z {, 0, } z. 6 z z () z 5 z 9 Solve each equation g. 6 9 p. n (5). d 4. w 5. s 4 5 4() Glencoe/McGraw-Hill 0 Algebra: Concepts and Applications

18 5 Solving Equations b Using Models You can use algebra tiles to model and solve equations. Eample: Use algebra tiles to solve. Model b placing -tile and one-tiles on one side of the mat to represent. Place negative one-tile on the other side of the mat to represent. To get the -tile b itself, add negative one-tiles to each side. Then remove the zero pairs. The -tile is matched with 4 negative one-tiles. Therefore, 4. () () 4 Solve each equation. Use algebra tiles if necessar b. 6 4 a 4. r () 5 5. h m 6 7. n (4) 8. 5 p 8 9. c k 4. 7 (). Yolanda scored points higher on her math test than she scored on her previous test. If her grade was 8 on this test, what was her score on the previous test? a. Write an equation that can be used to find Yolanda s score on the previous test. b. What was Yolanda s score on the previous test? Glencoe/McGraw-Hill 5 Algebra: Concepts and Applications

19 6 Solving Addition and Subtraction Equations You can use the addition and subtraction properties of equalit to solve equations. Addition Propert of Equalit Subtraction Propert of Equalit If ou add the same number to each side of an equation, the two sides remain equal. Eample: If 5, then 5. If ou subtract the same number from each side of an equation, the two sides remain equal. Eample: If, then 7 7. Eample: Solve each equation. Check our solution. a. b 5 b 5 b 5 Add to each side. b 0 Check: b 5 5 Replace b with. 5 5 The solution is. b Subtract 4 from each side Check: 4 4 Replace with. The solution is. Solve each equation. Check our solution n 6 9. p (5) 4. 7 k 0 5. d (6) 6. s w 8. 8 v c (6.) a 5 5. m a Glencoe/McGraw-Hill 0 Algebra: Concepts and Applications

20 7 Solving Equations Involving Absolute Value Some equations involve absolute value. Eample : Eample : Eample : Solve 5. Check our solution. 5 means 5 or 5. Solve both equations Add 5 to each side Check: 7 5 Check: 5 The solution set is {, 7}. Solve 8. Check our solution. 8 Simplif b subtracting from each side. 5 5 or 5 Check: 5 8 Check: The solution set is {5, 5}. Solve 6. Check our solution. 6 Simplif b adding to each side. 4 This sentence can never be true. The solution is the empt set or. Solve each equation. Check our solution.. 7. m z 5. d 0 6. r 0 7. p 5 8. c 9. r () 6 0. Ashle said it was too cold to go snowboarding because the temperature was onl F awa from 0 F. What are the two possible temperatures? Glencoe/McGraw-Hill 5 Algebra: Concepts and Applications

21 4 Multipling Rational Numbers Use the following rules to multipl two rational numbers and to simplif epressions. Rules Eamples Rational Numbers The product of two rational 5(.5).5 with Different Signs numbers having different 6(4.8) 8.8 signs is negative. 0.4(.5) 0.6 Rational Numbers The product of two rational (4.)(8).6 with the Same Sign numbers having the same (.) 8.4 sign is positive. (w)(.) 6.w Fractions To multipl fractions, 5 4 multipl the numerators 7 and multipl the denominators or 6 Find each product...8(4). 6(.). 6(6.) (5.) 5. (8.4) (0.5) Simplif each epression.. (.4a) () 5. b 4 Glencoe/McGraw-Hill 45 Algebra: Concepts and Applications

22 4 Counting utcomes Jerem has three sweatshirts and two pairs of jeans. ne wa to find the number of different outfits is to draw a tree diagram, as shown below. Sweatshirts blue red hooded Jeans blue black Sweatshirt blue red hooded Jeans blue black blue black blue black utcome blue, blue blue, black red, blue red, black hooded, blue hooded, black Since there are si outcomes, there are si outfits possible. The Fundamental Counting Principle can also be used to count outcomes. It states that if an event M can occur in m was and is followed b event N that can occur in n was, then the event M followed b event N can occur in m n was. sweatshirts jeans or 6 outfits Find the number of possible outcomes b drawing a tree diagram.. sandwich with one condiment. computer with one peripheral Menu Item Condiment Computer Peripheral hamburger ketchup desk model DVD drive cheeseburger mustard laptop CD-Rom drive hot dog pickle flopp disk drive onion Find the number of possible outcomes b using the Fundamental Counting Principle.. A certain game includes cards with different pictures, 4 colors, and 6 numbers. Find the number of cards in the game. 4. Find the number of possible was of answering a true-false question followed b a multiple choice question with 4 choices. 5. Suppose ou can order sweaters from a catalog in 8 sizes, colors, and deliver methods. Find the number of possible sweater orders. Glencoe/McGraw-Hill 50 Algebra: Concepts and Applications

23 4 Dividing Rational Numbers Use the following rules to divide rational numbers or fractions. Rule or Propert Eamples Dividing Rational The quotient of two numbers Numbers having different signs is negative 4.8 (6) 0.8 The quotient of two numbers.8 (0.4) 7 having the same sign is positive..8 (0.4) 7 Multiplicative For ever number, where and b 4 are Inverse Propert a, b 0, there is eactl one multiplicative inverses. number such that. a c b a a d a Dividing, where b, c, d 0 b d b c or Fractions (6) a b b a or 8 Find each quotient () (5.5) (0.5) (4) Glencoe/McGraw-Hill 55 Algebra: Concepts and Applications

24 4 4 Solving Multiplication and Division Equations Use the following rules to solve algebraic equations. Rule or Propert Eamples Division Propert For an numbers a, b, and c, with 4 4 of Equalit a b 4 c 0, if a b, then. 6 Multiplication For an numbers a, b, and c, 4 Propert of if a b, then ac bc. Equalit c c Solve each equation n 4..5 w n n n Glencoe/McGraw-Hill 60 Algebra: Concepts and Applications

25 4 5 Solving Multi-Step Equations Some equations require more than one step to solve. To solve a problem that has more than one step, the best strateg is to undo the operations in reverse order. Alwas check our solution. Eample : Solve Check: 8 () Eample : Solve Check: Solve each equation. Check our solution n 9 4. m w Glencoe/McGraw-Hill 65 Algebra: Concepts and Applications

26 4 6 Variables on Both Sides Some equations contain variables on both sides and require more than one step to solve. To solve these equations, first use the Addition or Subtraction Propert of Equalit to write an equivalent equation that has all of the variables on one side. Then solve and check. Eample : Solve Check: 6 4 (0) Eample : Solve () Check: 4 (4) (4) 8 8 Solve each equation. Check our solution.. 6m 40 m n n w.8w m.5m Glencoe/McGraw-Hill 70 Algebra: Concepts and Applications

27 4 7 Grouping Smbols The first step in solving an equation that contains grouping smbols is to remove the parentheses b using the Distributive Propert. Then solve and check. Eample : Solve 6( ) 4. 6( ) Check: 6( ) 4 6(() ) 4 6(6 ) 4 6(7) Eample : Solve ( 8) 4 4 ( 8) Check: 4 4 ( 8) ( () 8) 4 (6 8) Solve each equation. Check our solution.. (n ). 4 4( ). (4 ) 4 (8) 4. 6 ( 5) 4 5. ( 9) 6. ( 9) ( ) w.5(w 6) 9..6.n.8(n 5) 0. ( 9) 8. (8 4) 4. ( 8) 4 4 Glencoe/McGraw-Hill 75 Algebra: Concepts and Applications

28 5 Solving Proportions An equation stating that two ratios are equal is called a proportion. 8 6 For eample, 6 is a proportion. Use the Propert of Proportions and other algebraic properties ou know to solve proportions. Propert of Proportions The cross products of a proportion are equal. a b c d If, then ad bc. If ad bc, then. a b c d Eample : Solve. Eample : Solve. Solve each proportion. g n 8 n 6n 5(8) Find the cross products. 6p (p 4) 6n 90 Simplif. 6p p 48 n 5 6p 48 p Check: Check: 6(5) 5(8) 8(6) (8 4) m p a 60 p p 4 6 p b c 8 4 d 8 k Glencoe/McGraw-Hill 95 Algebra: Concepts and Applications

29 5 Scale Drawings and Models A scale drawing or scale model is used to represent an object that is too large or too small to be drawn or built at its actual size. Eample : Eample : The Statue of Libert is about 50 feet tall. A model is 5 inches tall. Find the scale used. The scale used is the ratio of the length of the model to the actual length, or 5 in. 50 feet. Reduce the ratio to get a scale of in. 0 feet. Another method is to convert 50 feet to inches, or 50 ft 50() in. 800 in. The scale is 5 in. 800 in. Reduce the ratio to get :0. The units do not need to be included if the are the same. The scale of a map of Florida is inch 0 miles. Find the actual distance between Miami and St. Petersburg if the distance between them on the map is inches. inch 0 miles inches miles 0() 60 Use a proportion. The distance from Miami to St. Petersburg is 60 miles. Find each scale or distance.. n the blueprint of a house, the kitchen is 5 inches long. If the actual kitchen is 0 feet long, find the scale of the blueprint.. n a map, the scale is inch 5 miles. Find the actual distance for each map distance. From To Map Distance Portsmouth, H Springfield, H 5. inches Chicago, IL Lawrenceville, IL 0 inches Santa Fe, NM Clovis, NM 8 inches. The Sears Tower is 450 feet tall. If a model is 5 inches tall, find the scale. 4. In an H scale model of a train, the length of the engine is 6 inches. If the H scale is : 87, find the actual length. Write the answer in feet. 5. Las Vegas, Nevada, is 445 miles from Reno, Nevada. If the distance on the map is inches, find the scale used for the map. 8 5 Glencoe/McGraw-Hill 00 Algebra: Concepts and Applications

30 5 The Percent Proportion Percent is a ratio that compares a number to 00. For eample, 6 6 out of 00 can be epressed as 00, 6 : 00, or 6%. Use the percent proportion to find percents. If P is the percentage, B is the base, and r is the percent, the percent proportion is. P B r 00 Eample : 80% of what number is 6? Eample : 75 is what percent of 00? P B r 00 P B r 00 6 B r 00 6(00) 80B Cross multipl. 75(00) 00r B Simplif r, B B 5 r So, 80% of So, 75 is 5% of 00. Use the percent proportion to find each number.. 40 is what percent of 80?. 8 is what percent of 5? 00r 00. Find 0% of is % of what number? 5. What number is 50% of? 6. is 400% of what number? 7. 0 is what percent of 50? 8. Find 84% of % of what number is 8.? 0. What number is 5% of 4?. 50% of 0.75 is what number?. Find 40% of % of what number is.? 4. Find 0.5% of is what percent of 40? 6. is what percent of 50? Glencoe/McGraw-Hill 05 Algebra: Concepts and Applications

31 5 4 The Percent Equation P B r 00 The percent proportion can also be written as the equation P RB, where P is the percentage, B is the base, and R is the rate epressed as a decimal. Eample : Find 60% of 5. Eample : 8 is 40% of what number? P RB P RB R 0.6 Write R as decimal. R 0.4 P 0.6(5) 8 0.4(B) P 5 Divide b 0.4. So, 60% of B 0.4B 0.4 So, 8 is 40% of 45. Use the percent equation to find each number.. 7 is what percent of 90?. Find 0% of is what percent of 60? 4. Find 8% of Find % of % of what number is 9.8? 7. What number is 4% of 90? 8. 50% of 0.6 is what number? is what percent of 50? is 00% of what number?. is what percent of 500?. Find 0.5% of.. Find 60% of What number is 65% of 4? is what percent of 5? is 5% of what number? 7. Marcia bought $80 in groceries. What is her final cost after sales ta of 6.5% is added? 8. Mr. Perez paid 4% of his income in taes. If his taes were $7686, what was his income? Glencoe/McGraw-Hill 0 Algebra: Concepts and Applications

32 5 5 Percent of Change The median price of a home increased from $90,600 in 988 to $4,600 in 998. The change in the price can be written as a percent. Since the price increased, it is called the percent of increase. Likewise, the percent that an amount decreases is known as the percent of decrease. To find the percent of increase or decrease, write a ratio that compares the amount of increase or decrease to the original amount. Eample: Find the percent of increase in the median price of a home. original price: $90,600 new price: $4,600 The amount of increase 4,600 90,600 44,000. Substitute the amount of increase for P. You are comparing it to the original price, so substitute $90,600 for B. Method Method P r B P RB 44,000 90,600 44,000(00) 90,600r 4,400,000 90,600r 4,400,000 90, r 00 r 00 90,600r 90,600 So the percent of increase was about 49%. 44,000 R 90,600 44,000 90,600 R R 90,600 90,600 Find the percent of change. Round to the nearest percent.. original: 5. original: $0,000 new: 6 new: $7,500. original: original: 5 new: 54 new: 5. In Januar, gasoline cost $.05 per gallon. In Jul, gasoline cost $.0 per gallon. Find the percent of increase. 6. The original price of a CD plaer is $05. A discount of $ is given. Find the percent of decrease. 7. The original cost of a computer is $900. If the price is decreased b 5%, what is the sale price? Divide each side b 90,600. Glencoe/McGraw-Hill 5 Algebra: Concepts and Applications

33 5 6 Probabilit and dds You can measure the chances of an event happening with probabilit, a number that is alwas between 0 and, inclusive. You can epress the probabilit of an event as a fraction, as a decimal, or as a percent, using the following relationship. The probabilit of an event is a ratio that compares the number of favorable outcomes to the number of possible outcomes. P(event) number of favorable outcomes number of possible outcomes Eample : The spinner at the right is spun. Find the probabilit of spinning a number greater than 5. There are favorable outcomes, spinning a 6, 7, or an 8. There are 8 possible outcomes Therefore P(a number greater than 5). 8 Another wa to measure the chance of an event is with odds. The odds of an event is a ratio that compares the number of favorable outcomes to the number of unfavorable outcomes. dds number of favorable outcomes number of unfavorable outcomes Eample : Refer to the spinner above. Find the odds of spinning an even number. There are 4 favorable outcomes:, 4, 6, or 8. The number of unfavorable outcomes is Therefore the odds of spinning an even number are 4 : 4 or :. Refer to the spinner above. Find the probabilit of each outcome.. a. an even number. a number less than 4. a number greater than 4 Refer to the spinner above. Find the odds of each outcome. 5. an odd number 6. a multiple of 7. not a 6 8. a number greater than 4 Glencoe/McGraw-Hill 0 Algebra: Concepts and Applications

34 5 7 Compound Events To win a game, ou have to roll either an even number or a with one die. What is the probabilit that ou will win? The probabilit of tossing an even number P(even number), and the probabilit of tossing a is P() 6. These events cannot occur at the same time and so the are mutuall eclusive. This is a compound event because there is more than event occurring. You find the probabilit of mutuall eclusive events b adding the probabilities of each event. P(even number or a ) P(even number) P() or Compound events are called independent if the outcome of one event does not affect the outcome of the other event. For eample, if two dice are rolled, rolling an odd number with the first die does not affect rolling a 4 on the second die. You find the probabilit of two independent events b multipling the probabilities of each event. P(odd number and a 4) P(odd number) P(4) 6 6 or Probabilit of Mutuall Eclusive Events A and B Probabilit of Independent Events A and B P(A or B) P(A) P(B) P(A and B) P(A) P(B) A die is rolled. Find the probabilit of each compound event.. P(a 4 or a ). P(an odd number or a 6). P(a or a ) 4. P(a or a 6) 5. P(an even number or a 5) 6. P(a or a multiple of ) 7. P(a or a number greater than 4) 8. P(an odd number or an even number) Two dice are rolled. Find the probabilit of each compound event. 9. P(an odd number and a ) 0. P(an odd number and an even number). P(two odd numbers). P(a 6 and a number greater than 4). P(two numbers less than ) 4. P(an even number and a 5) Glencoe/McGraw-Hill 5 Algebra: Concepts and Applications

35 6 Relations A set of ordered pairs is called a relation. The set of all first coordinates of the ordered pairs is the domain of the relation. The set of all second coordinates is the range. You can use a table or a graph to represent a relation. Eample : Epress the relation {(, ), (0, ), (, ), (, )} as a table and as a graph. Then determine the domain and range. The domain is {, 0,, } and the range is {,,, }. 0 (0, ) (, ) (, ) (, ) Eample : Epress the relation shown on the graph as a set of ordered pairs. Then find the domain and the range. The set of ordered pairs for the relation is {(, 0), (, ), (, ), (, ), (4, )}. The domain is {,,,, 4} and the range is {0,,,, }. (, ) (4, ) (, 0) (, ) (, ) Identif the domain and the range of each function.. {(6, 0), (, ), (4, )}. 4. Epress the relation 4. Epress the relation shown on the graph as {(4, ), (, ), (0, 0)} as a table as a set of ordered pairs and in a table. and as a graph. Then determine the Then determine the domain and the range. domain and the range. (4, ) 4 6 (, ) (, ) (, ) Glencoe/McGraw-Hill 45 Algebra: Concepts and Applications

36 6 Equations as Relations Since 5 9 is true when 4, we sa that 4 is a solution of the equation. Equations with two variables have solutions that ma include one or more ordered pairs. The ordered pair (, 0) is a solution of the equation because 0. But (4, ) is not solution because 4. Eample: Solve if the domain is {,, 0, }. Graph the solution. Substitute each value of into the equation to find the corresponding -value. Then graph the ordered pairs. The solution set is {(, 4), (, ), (0, ), (, 0)}. Which ordered pairs are solutions of each equation?. 4 a. (, 5) b. (, ) c. (0, 4) d. (5, 9). b a 9 a. (7, ) b. (, 5) c. (, ) d. (6, ). 4g h 6 a. (4, 0) b. (0, 4) c. (0, 6) d. (, 8) 4. 5 a. (, ) b. (, ) c. (4, 7) d. (0, 5) Solve each equation if the domain is {,, 0,, }. Graph the solution set (, ) () 4 (, 4) () (, ) 0 0 (0, ) 0 (, 0) (, 0) (, ) (0, ) (, 4) Glencoe/McGraw-Hill 50 Algebra: Concepts and Applications

37 6 Graphing Linear Relations The solution set of the equation contains an infinite number of ordered pairs. A few of the solutions are shown in the table at the right. (, ) () (, ) 0 (0) (0, ) () (, ) () (, ) (, ) (, ) = (0, ) (, ) Graphing the ordered pairs indicates that the graph of the equation is a straight line. An equation whose graph is a straight line is a linear equation. Eample: Graph. Select at least three values for. Determine the values for. Graph the ordered pairs and use them to draw a line. (, ) () 4 (, 4) 0 0 (0, ) (, ) (, ) = Graph each equation Glencoe/McGraw-Hill 55 Algebra: Concepts and Applications

38 6 4 Functions A function is a special kind of relation in which each member of the domain is paired with eactl one member of the range. Stud these eamples. {(4, 7), (0, ), (5, )} This relation is a function because ever first coordinate is matched with eactl one second coordinate This relation is not a function because 7 is paired with two -values, 0 and 8. This relation is a function because ever -value on the graph is paired with eactl one -value. Equations that are functions can be written in functional notation. Notice that f() replaces in this eample. Equation 5 We read f() as f of. If, then f() 5() or. Functional Notation f() 5 Eample: If f(), find f(). f() f() () Replace with. f() 7 Simplif. Determine whether each relation is a function.. {(, ), (, 0), (, 5), (5, 6)}. {(0, ), (, ), (, )}. {(8, 4), (, ), (8, ), (7, 0)} 4. {(9, 0), (9, 0), (4, 5), (5, 4)} If f() and g() 5, find each value. 8. f() 9. f(6) 0. g(). g(5) Glencoe/McGraw-Hill 60 Algebra: Concepts and Applications

39 6 5 Direct Variation A linear function that can be written in the form k, where k 0, is called a direct variation. In a direct variation, varies directl as. Because a direct variation is a linear function whose solution contains (0, 0), the graph of a direct variation equation is a straight line that passes through the origin. Eample : Determine whether each function is a direct variation. a. 0 b. The function is a linear function. Since 0 0(0), (0, 0) is a solution of 0. Therefore, the function is a direct variation and the graph will pass through the origin. The function is a linear function. But since 0 (0), (0, 0) is not a solution of. Therefore, the function is not a direct variation and the graph will not pass through the origin. Eample : Assume that varies directl as and 0 when 5. Find when. Step Find the constant of variation, k. k 0 k(5) When 0, 5. 6 k Solve for k. When,. Step Use k 6 to find when. k 6 Substitute 6 for k. 6 Substitute for. Solve for. Determine whether each equation is a direct variation Solve. Assume that varies directl as. 9. Find when 4 if 8 when Find when 6 if 8 when 4. Glencoe/McGraw-Hill 65 Algebra: Concepts and Applications

40 6 6 Inverse Variation Shauna wants to improve her running speed. As her speed increases, her time decreases. rate time distance r t d r t d 9 8 mile/minute 9 minutes mile mile/minute 7 minutes mile mile/minute 8 minutes mile mile/minute 6 minutes mile 7 6 The equation rt d is an eample of an inverse variation. An inverse variation is described b an equation of the form k, where k 0. We sa that varies inversel as. Eample: Suppose varies inversel as and when 4. Find when. Step Find the constant of variation, k. k (4) k When, 4. k Solve for k. When,. Step Use k to find when. k Substitute for k. Substitute for. Solve for. Determine if each equation is an inverse variation or a direct variation.. 5. ab 5. Solve. Assume that varies inversel as. 4. Find when 9 if when Suppose 5 when 6. Find when If 4.5 when 6, find when. 7. Find when 0.5 if.5 when Find when 0.9, if.5 when Suppose 8 when 6. Find when If when 8, find when. 5 Glencoe/McGraw-Hill 70 Algebra: Concepts and Applications

41 7 Slope Slope is the ratio of the rise, or the vertical change, to the run, or the horizontal change. A greater ratio indicates a steeper slope. A tpical ski mountain has a slope of about 4, while a car windshield ma have a slope of. For an two points (, ) and (, ), slope m Eamples: Find the slope of the line containing each pair of points. a. (4, ) and (, 7) b. (, 6) and (, ) c. (4, 5) and (, 5) m m m 7 () 4 m m m 9 change in change in 6 () m m m or 0 Since ou cannot divide b 0, the slope is undefined. Determine the slope of the line passing through the points whose coordinates are listed.. (, ) and (4, ). (0, ) and (4, ) (, 5), (5, 7) 4. (4, ) and (4, ) 5. (8, ) and (, ) 6. (5, ) and (, 5) 7. (7, ) and (6, 6) 8. (5, ) and (5, ) 9. (7, 7) and (6, 6) 0. (4, 4) and (0, ). (, 4) and (, 9). (0, 8) and (, 8) Glencoe/McGraw-Hill 89 Algebra: Concepts and Applications

42 7 Writing Equations in Point-Slope Form You can write the equation of a line if ou know its slope and the coordinates of one point or if ou know the coordinates of two points on the line. Use the point-slope form. Point-Slope Form For a nonvertical line through the point at (, ) with slope m, the point-slope form of a linear equation is Eamples: Write the point-slope form of an equation for each line. a. the line passing through the point at b. the line passing through points at (4, ) and having a slope of (4, 4) and (, ) m( ) m Find m. ( (4)) ( 4) m (4) or (4) ( 4) Substitute m ( 4) Write the point-slope form of an equation for each line, given either the coordinates of a point and the slope or the coordinates of two points.. (, ), m. (4, ), m. (, 5), m (4, ), m 5. (8, ), m 0 6. (5, ), m 7. (7, ) and (6, 6) 8. (5, ) and (5, ) 9. (7, 7) and (6, 6) 0. (4, 4) and (0, ). (, 4) and (, 9). (0, 8) and (, 8) Glencoe/McGraw-Hill 94 Algebra: Concepts and Applications

43 7 Writing Equations in Slope-Intercept Form Given the slope m and the -intercept b of a line, the slope-intercept form of an equation of the line is m b. Sometimes it is more convenient to epress the equation of a line in slope-intercept form instead of point-slope form. This form is especiall useful when graphing lines. Eamples: Write the point-intercept form of an equation for each line. a. the line with m and b 4 m b 4 b. the line passing through points at (4, ) and (6, ) m Find m. () m ( 6) Multipl. Add to each side. 0 Simplif. Write the slope-intercept form of an equation for each line, given either the slope and the -intercept or the coordinates of two points.. m, b. m, b m, (0, ) 4. m, b m, b 6. m 0, b 5 7. (5, ) and (4, ) 8. (5, ) and (, 5) 9. (0, ) and (6, 5) 0. (, ) and (4, ). (, 4) and (, 6). (0, 8) and (, 4). (0, 8) and (, 7) 4. (0, ) and (6, 5) Glencoe/McGraw-Hill 99 Algebra: Concepts and Applications

44 7 4 Scatter Plots The scatter plot below is a graph of the number of games plaed in the highest-scoring World Cup finals compared to the number of goals scored. The data are listed in the table. Number of Goals Scored Number of Games Plaed Year Number of Number of Games Plaed Goals Scored You can use the scatter plot to draw conclusions about the data. The data points fall roughl along a line with a positive slope. Therefore, we sa there is a positive relationship between the number of games plaed and the number of goals scored. A line with a negative slope would indicate a negative relationship, and no line would indicate no relationship between the variables. A valid conclusion from this scatter plot is that as the number of games plaed increases, more goals are scored. Determine whether each scatter plot has a positive relationship, negative relationship, or no relationship. If there is a relationship, describe it... Number of Cameras Sold Price of Camera Weight 0 (pounds) Height (inches) Length 00 of Swim (meters) Practice per Da (hours) Number of Seeds Sprouted Temperature (C) Glencoe/McGraw-Hill 04 Algebra: Concepts and Applications

45 7 5 Graphing Linear Equations You ma use the slope-intercept form to graph linear equations, as shown in the eample below. Eample: Graph 6. Rewrite in slope-intercept form. 6 6 Subtract from each side. 6 6 Divide each side b. To graph, plot a point at the -intercept,. Then use the slope,. From (0, ), go down units. Then go right units. Graph a point at (, 0). Draw the line through the points. (, 0) Graph each equation Glencoe/McGraw-Hill 09 Algebra: Concepts and Applications

46 7 6 Families of Linear Graphs Graphs of linear equations that have at least one characteristic in common are called families of graphs. Graphs are families if the have the same slope, the same -intercept, or the same -intercept. An eample of each is shown below. same slope same -intercept same -intercept Eample: Graph the pair of equations. Eplain wh the are a famil of graphs. Since both graphs have the same slope, this is a famil of graphs. Graph each pair of equations. Eplain wh the are a famil of graphs Glencoe/McGraw-Hill 4 Algebra: Concepts and Applications

47 7 7 Parallel and Perpendicular Lines Lines that have the same slope are parallel. The graphs of the equations of two lines are a famil of graphs because the have the same slope. For eample, the graphs of and are parallel because their corresponding equations have the same slope,. Lines whose slopes are negative reciprocals are perpendicular. That is, the product of the slopes is. For eample, the graphs of and are perpendicular because the product of the slopes, and, is. Parallel Lines same slope, 4 different -intercept Perpendicular Product of slopes Lines is. 9 5 Determine whether the graphs of the equations are parallel, perpendicular, or neither Glencoe/McGraw-Hill 9 Algebra: Concepts and Applications

48 8 Powers and Eponents The product of a number with itself is called a perfect square. For eample, 64 is a perfect square because The eponent can be used to write 8 8 as 8. Likewise, the eponent 4 can be used to write as 8 4. Numbers that are epressed using eponents are called powers. base 84 eponent Eample : Eample : Eample : Write using eponents. The base is. There are 5 factors of, so the eponent is 5. 5 Write using eponents. The base is. There are 4 factors of, so the eponent is 4. 4 Write 5 as a multiplication epression. There is factor of 5, factors of, and factors of. 5 Write each epression using eponents ()()()(). 4. () a a a a b Write each power as a multiplication epression d 4 e 0. 9ab. w. () 5 Evaluate each epression if,, and z.. z 4. ( z) Glencoe/McGraw-Hill 9 Algebra: Concepts and Applications

49 8 Multipling and Dividing Powers You can multipl powers b using the following rules. Product of Powers To multipl powers with the same base, add the eponents. a m a n a m n Eamples 4 4 or 6 or 5 (ab ) (a b 4 ) (a a )(b b 4 ) or a 4 b 6 You can divide powers b using the following rules. Quotient of Powers To divide powers with the same base, subtract the eponents. a m a n a m n b 8 c 4 b c Eamples 0 4 or 0 5 or b 8 c 4 or b 6 c Simplif each epression () (5m )(m n 4 ) 5. (ab )(a b ) 6. (0 4 )( ) w 6 w a 7 b a6 b j 5 j m 6 n 6m n Glencoe/McGraw-Hill 44 Algebra: Concepts and Applications

50 8 Negative Eponents In science and technolog, negative eponents are sometimes used to represent ver small numbers. For eample, the diameter of an atom is epressed as 0 0 meter. This is the decimal This number epressed as a fraction is. 0,000,000,000 When simplifing an epression with a negative eponent, ou ma need to use the Quotient of Powers rule. Negative Eponents Eamples a n a n or 9 b b b () or b 5 Remember that a negative eponent is used to write a reciprocal, not to represent a negative number. Simplif each epression m 5. (a )( b ) k k a b c 4m 6 n.. m n z 9 Glencoe/McGraw-Hill 49 Algebra: Concepts and Applications

51 8 4 Scientific Notation In science and in other applications, scientific notation is often used to represent ver small or ver large numbers. For eample, the speed of light is about 0 8 meters per second. This represents the number 00,000,000 meters per second. In scientific notation, a number is epressed in the form a 0 n, where a is a number greater than and less than 0 and n is an integer. Scientific Notation Eamples. Place the decimal point after 6,00, the first non-zero digit in the given number.. Find the power of 0 b counting decimal places. When the given number is one or greater, the power of 0 is positive. When the given number is between zero and one, the eponent of 0 is negative. Epress each number in standard form Epress each number in scientific notation ,000, ,000,000, Glencoe/McGraw-Hill 54 Algebra: Concepts and Applications