Study Guide and Intervention

Transcription

1 - Stud Guide and Intervention Variables and Epressions Write Mathematical Epressions In the algebraic epression, w, the letters and w are called variables. In algebra, a variable is used to represent unspecified numbers or values. An letter can be used as a variable. The letters and w are used above because the are the first letters of the words length and width. In the epression w, and w are called factors, and the result is called the product. Eample Write an algebraic epression for each verbal epression. a. four more than a number n The words more than impl addition. four more than a number n 4 n The algebraic epression is 4 n. b. the difference of a number squared and 8 The epression difference of implies subtraction. the difference of a number squared and 8 n 8 The algebraic epression is n 8. Lesson - Eample a. 4 Evaluate each epression. 4 Use as a factor 4 times. 8 Multipl. b. five cubed Cubed means raised to the third power Use 5 as a factor times. 5 Multipl. Write an algebraic epression for each verbal epression.. a number decreased b 8. a number divided b 8. a number squared 4. four times a number 5. a number divided b 6 6. a number multiplied b 7 7. the sum of 9 and a number 8. less than 5 times a number 9. twice the sum of 5 and a number 0. one-half the square of b. 7 more than the product of 6 and a number. 0 increased b times the square of a number Evaluate each epression Glencoe/McGraw-Hill Glencoe Algebra

2 - Write Verbal Epressions Translating algebraic epressions into verbal epressions is important in algebra. Eample a. 6n Stud Guide and Intervention (continued) Variables and Epressions Write a verbal epression for each algebraic epression. the product of 6 and n squared b. n m the difference of n cubed and twelve times m Write a verbal epression for each algebraic epression.. w. a c n 4 8. a b k 5. b. 7n k 5 5. b a 6. 4(n ) n Glencoe/McGraw-Hill Glencoe Algebra

3 - Stud Guide and Intervention rder of perations Evaluate Rational Epressions Numerical epressions often contain more than one operation. To evaluate them, use the rules for order of operations shown below. rder of perations Step Evaluate epressions inside grouping smbols. Step Evaluate all powers. Step Do all multiplication and/or division from left to right. Step 4 Do all addition and/or subtraction from left to right. Eample Eample Evaluate each epression. Evaluate each epression. a Multipl and Add 7 and 8. Subtract 4 from 5. b. () 4( 6) () 4( 6) () 4(8) Add and 6. 6 Multipl left to right. 8 Add 6 and. a. [ ( ) ] [ ( ) ] ( 4 ) Divide b. ( 6) Find 4 squared. (8) Add and Multipl and 8. b Evaluate power in numerator. Add and 8 in the numerator. Evaluate power in denominator. Lesson - 48 Multipl. Evaluate each epression.. (8 4). ( 4) (0 7) ( 8) () [5( 7 4)] (5 ) 5 4(5 6. ) (4 5 ) 0() () ( 8) 4 Glencoe/McGraw-Hill 7 Glencoe Algebra

4 - Stud Guide and Intervention (continued) rder of perations Evaluate Algebraic Epressions Algebraic epressions ma contain more than one operation. Algebraic epressions can be evaluated if the values of the variables are known. First, replace the variables b their values. Then use the order of operations to calculate the value of the resulting numerical epression. Eample Evaluate 5( ) if and. 5( ) 5( ) Replace with and with. 8 5( ) Evaluate. 8 5(9) Subtract from Multipl 5 and 9. 5 Add 8 and 45. The solution is 5. 4 Evaluate each epression if,, z 4, a, and b z 5. 6a 8b 6. (a b) z 5 9. ( z) 0. (0) 4 00a.. a b 7 z. (z ) 4. 6z ab 5a b 8. (z ) a z 9. z z 0.. z z z z Glencoe/McGraw-Hill 8 Glencoe Algebra

5 - Stud Guide and Intervention pen Sentences Solve Equations A mathematical sentence with one or more variables is called an open sentence. pen sentences are solved b finding replacements for the variables that result in true sentences. The set of numbers from which replacements for a variable ma be chosen is called the replacement set. The set of all replacements for the variable that result in true statements is called the solution set for the variable. A sentence that contains an equal sign,, is called an equation. Eample Eample Find the solution set of a 9 if the replacement set is {6, 7, 8, 9, 0}. Replace a in a 9 with each value in the replacement set. (6) false (7) 9 9 false (8) false (9) true (0) false Since a 9 makes the equation a 9 true, the solution is 9. The solution set is {9}. ( ) (7 4) (4) () 8 9 b b b ( ) Solve b. (7 4) riginal equation Add in the numerator; subtract in the denominator. Simplif. 8 The solution is. 9 Find the solution of each equation if the replacement sets are X, 4 and Y {, 4, 6, 8} ,,, Lesson - 7. ( ) 7 8. ( ) Solve each equation. 0. a. n 6 4. w k 4. p 5. s m 7. k c 4 Glencoe/McGraw-Hill Glencoe Algebra

6 - Solve Inequalities An open sentence that contains the smbol,,, or is called an inequalit. Inequalities can be solved the same wa that equations are solved. Eample Find the solution set for a 8 0 if the replacement set is {4, 5, 6, 7, 8}. Replace a in a 8 0 with each value in the replacement set. (4) 8? false (5) 8? false (6) 8? false (7) 8? 0 0 true (8) 8? true Since replacing a with 7 or 8 makes the inequalit a 8 0 true, the solution set is {7, 8}. Stud Guide and Intervention (continued) pen Sentences Find the solution set for each inequalit if the replacement set is X {0,,,, 4, 5, 6, 7} (8 ) ( ) 0 Find the solution set for each inequalit if the replacement sets are X,,,,, 5, 8 and Y {, 4, 6, 8, 0} ( ) (6 ) 4 Glencoe/McGraw-Hill 4 Glencoe Algebra

7 -4 Stud Guide and Intervention Identit and Equalit Properties Identit and Equalit Properties The identit and equalit properties in the chart below can help ou solve algebraic equations and evaluate mathematical epressions. Additive Identit For an number a, a 0 a. Multiplicative Identit For an number a, a a. Multiplicative Propert of 0 For an number a, a 0 0. Multiplicative Inverse Propert a b a b For ever number, a, b 0, there is eactl one number such that. b a b a Refleive Propert For an number a, a a. Smmetric Propert For an numbers a and b, if a b, then b a. Transitive Propert For an numbers a, b, and c, if a b and b c, then a c. Substitution Propert If a b, then a ma be replaced b b in an epression. Eample Eample Name the propert used in each equation. Then find the value of n. Name the propert used to justif each statement. a. 8n 8 Multiplicative Identit Propert n, since 8 8 b. n Multiplicative Inverse Propert n, since a Refleive Propert b. If n, then 4n 4. Substitution Propert Name the propert used in each equation. Then find the value of n.. 6n 6. n 8. 6 n n 9 5. n 0 6. n 8 4 Name the propert used in each equation. 7. If 4 5 9, then Lesson (5) 0 0. () If 6 and 6, then (4 6) 8 Glencoe/McGraw-Hill 9 Glencoe Algebra

8 -4 Use Identit and Equalit Properties The properties of identit and equalit can be used to justif each step when evaluating an epression. Eample Evaluate 4 8 5(9 ). Name the propert used in each step (9 ) 4 8 5( ) Substitution; (0) Substitution; (0) Multiplicative Identit; Multiplicative Propert of Zero; 5(0) Substitution; Additive Identit; Evaluate each epression. Name the propert used in each step. 4 Stud Guide and Intervention (continued) Identit and Equalit Properties (5 5). ( 5 4) (6 ) (5 5 ) 7 Glencoe/McGraw-Hill 0 Glencoe Algebra

9 -5 Stud Guide and Intervention The Distributive Propert Evaluate Epressions The Distributive Propert can be used to help evaluate epressions. Distributive Propert For an numbers a, b, and c, a(b c) ab ac and (b c)a ba ca and a(b c) ab ac and (b c)a ba ca. Eample Rewrite 6(8 0) using the Distributive Propert. Then evaluate. 6(8 0) Distributive Propert Multipl. 08 Add. Eample Rewrite ( 5 ) using the Distributive Propert. Then simplif. ( 5 ) ( ) ( )(5) ( )() 6 ( 0) ( ) 6 0 Distributive Propert Multipl. Simplif. Rewrite each epression using the Distributive Propert. Then simplif.. (0 5). 6( t). ( ) 4. 6( 5) 5. ( 4) 6. ( ) 7. 5(4 9) 8. (8 ) ( 4t). ( ) 4. ( z) 4. ( ) 5. (a b c) 6. (6 4z) 7. ( ) 8. ( ) 4 Lesson -5 Glencoe/McGraw-Hill 5 Glencoe Algebra

10 -5 Stud Guide and Intervention (continued) The Distributive Propert Simplif Epressions A term is a number, a variable, or a product or quotient of numbers and variables. Like terms are terms that contain the same variables, with corresponding variables having the same powers. The Distributive Propert and properties of equalities can be used to simplif epressions. An epression is in simplest form if it is replaced b an equivalent epression with no like terms or parentheses. Eample Simplif 4(a ab) ab. 4(a ab) ab 4(a ab) ab Multiplicative Identit 4a ab ab Distributive Propert 4a ( )ab Distributive Propert 4a ab Substitution Simplif each epression. If not possible, write simplified.. a a g 0g a a p q. 0 4( ). c 8c b b a b c 6. 4 (6 0) Glencoe/McGraw-Hill 6 Glencoe Algebra

11 -6 Stud Guide and Intervention Commutative and Associative Properties Commutative and Associative Properties The Commutative and Associative Properties can be used to simplif epressions. The Commutative Properties state that the order in which ou add or multipl numbers does not change their sum or product. The Associative Properties state that the wa ou group three or more numbers when adding or multipling does not change their sum or product. Commutative Properties For an numbers a and b, a b b a and a b b a. Associative Properties For an numbers a, b, and c, (a b) c a (b c ) and (ab)c a(bc). Lesson -6 Eample Eample Evaluate Commutative Propert (6 )( 5) Associative Propert 8 0 Multipl. 80 Multipl. The product is 80. Evaluate (8..8) (.5.5) 0 5 Add. 5 Add. The sum is 5. Commutative Prop. Associative Prop. Evaluate each epression Glencoe/McGraw-Hill Glencoe Algebra

12 -6 Stud Guide and Intervention (continued) Commutative and Associative Properties Simplif Epressions The Commutative and Associative Properties can be used along with other properties when evaluating and simplifing epressions. Eample Simplif 8( ) 7. 8( ) Distributive Propert Commutative ( ) (8 7) 6 Distributive Propert 5 6 Substitution The simplified epression is 5 6. Simplif each epression.. 4. a 4b a. 8rs rs 7rs 4. a 4b 0a 5. 6( ) ( ) 6. 6n (4n 5) 7. 6(a b) a b 8. 5( ) 6( ) 9. 5(0. 0.) ( 0). z 9 4 z. 6( 4) ( 9) Write an algebraic epression for each verbal epression. Then simplif.. twice the sum of and z is increased b 4. four times the product of and decreased b 5. the product of five and the square of a, increased b the sum of eight, a, and 4 6. three times the sum of and increased b twice the sum of and Glencoe/McGraw-Hill Glencoe Algebra

13 -7 Stud Guide and Intervention Logical Reasoning Conditional Statements A conditional statement is a statement of the form If A, then B. Statements in this form are called if-then statements. The part of the statement immediatel following the word if is called the hpothesis. The part of the statement immediatel following the word then is called the conclusion. Eample Eample Identif the hpothesis and conclusion of each statement. Identif the hpothesis and conclusion of each statement. Then write the statement in if-then form. a. If it is Wednesda, then Jerri has aerobics class. Hpothesis: it is Wednesda Conclusion: Jerri has aerobics class b. If 4 0, then 7. Hpothesis: 4 0 Conclusion: 7 a. You and Marlnn can watch a movie on Thursda. Hpothesis: it is Thursda Conclusion: ou and Marlnn can watch a movie If it is Thursda, then ou and Marlnn can watch a movie. b. For a number a such that a, a. Hpothesis: a Conclusion: a If a, then a. Lesson -7 Identif the hpothesis and conclusion of each statement.. If it is April, then it might rain.. If ou are a sprinter, then ou can run fast.. If 4 4, then. 4. If it is Monda, then ou are in school. 5. If the area of a square is 49, then the square has side length 7. Identif the hpothesis and conclusion of each statement. Then write the statement in if-then form. 6. A quadrilateral with equal sides is a rhombus. 7. A number that is divisible b 8 is also divisible b Karln goes to the movies when she does not have homework. Glencoe/McGraw-Hill 7 Glencoe Algebra

14 -7 Stud Guide and Intervention (continued) Logical Reasoning Deductive Reasoning and Countereamples Deductive reasoning is the process of using facts, rules, definitions, or properties to reach a valid conclusion. To show that a conditional statement is false, use a countereample, one eample for which the conditional statement is false. You need to find onl one countereample for the statement to be false. Eample Determine a valid conclusion from the statement If two numbers are even, then their sum is even for the given conditions. If a valid conclusion does not follow, write no valid conclusion and eplain wh. a. The two numbers are 4 and 8. 4 and 8 are even, and 4 8. Conclusion: The sum of 4 and 8 is even. b. The sum of two numbers is 0. Consider and However, 8, 9, and 8 all equal 0. There is no wa to determine the two numbers. Therefore there is no valid conclusion. Eample Provide a countereample to this conditional statement. If ou use a calculator for a math problem, then ou will get the answer correct. Countereample: If the problem is and ou press 475 5, ou will not get the correct answer. Determine a valid conclusion that follows from the statement If the last digit of a number is 0 or 5, then the number is divisible b 5 for the given conditions. If a valid conclusion does not follow, write no valid conclusion and eplain wh.. The number is 0.. The number is a multiple of 4.. The number is 0. Find a countereample for each statement. 4. If Susan is in school, then she is in math class. 5. If a number is a square, then it is divisible b. 6. If a quadrilateral has 4 right angles, then the quadrilateral is a square. 7. If ou were born in New York, then ou live in New York. 8. If three times a number is greater than 5, then the number must be greater than si. 9. If 0, then 4. Glencoe/McGraw-Hill 8 Glencoe Algebra

15 -8 Stud Guide and Intervention Graphs and Functions Interpret Graphs A function is a relationship between input and output values. In a function, there is eactl one output for each input. The input values are associated with the independent variable, and the output values are associated with the dependent variable. Functions can be graphed without using a scale to show the general shape of the graph that represents the function. Eample Eample The graph below represents the height of a football after it is kicked downfield. Identif the independent and the dependent variable. Then describe what is happening in the graph. The graph below represents the price of stock over time. Identif the independent and dependent variable. Then describe what is happening in the graph. Height Time The independent variable is time, and the dependent variable is height. The football starts on the ground when it is kicked. It gains altitude until it reaches a maimum height, then it loses altitude until it falls to the ground. Price Time The independent variable is time and the dependent variable is price. The price increases steadil, then it falls, then increases, then falls again. Lesson -8. The graph represents the speed of a car as it travels to the grocer store. Identif the independent and dependent variable. Then describe what is happening in the graph. Speed Time. The graph represents the balance of a savings account over time. Identif the independent and the dependent variable. Then describe what is happening in the graph. Account Balance (dollars) Time. The graph represents the height of a baseball after it is hit. Identif the independent and the dependent variable. Then describe what is happening in the graph. Height Time Glencoe/McGraw-Hill 4 Glencoe Algebra

16 -8 NAME DATE PERID Stud Guide and Intervention (continued) Graphs and Functions Draw Graphs You can represent the graph of a function using a coordinate sstem. Input and output values are represented on the graph using ordered pairs of the form (, ). The -value, called the -coordinate, corresponds to the -ais, and the -value, or -coordinate corresponds to the -ais. Graphs can be used to represent man real-world situations. Eample A music store advertises that if ou bu CDs at the regular price of $6, then ou will receive one CD of the same or lesser value free. a. Make a table showing the cost of buing to 5 CDs. Number of CDs 4 5 Total Cost ($) b. Write the data as a set of ordered pairs. (, 6), (, ), (, 48), (4, 48), (5, 64) c. Draw a graph that shows the relationship between the number of CDs and the total cost. Cost ($) CD Cost Number of CDs. The table below represents the length of a bab versus its age in months. Age (months) 0 4 Length (inches) 0 4 a. Identif the independent and dependent variables. ind: age; dep: length b. Write a set of ordered pairs representing the data in the table. (0, 0), (, ), (, ), (, ), (4, 4) c. Draw a graph showing the relationship between age and length. Length (inches) Age (months). The table below represents the value of a car versus its age. Age (ears) Value ($) 0 4 0,000 8,000 6,000 4,000,000 a. Identif the independent and dependent variables. ind: age; dep: value b. Write a set of ordered pairs representing the data in the table. (0, 0,000), (, 8,000), (, 6,000), (, 4,000), (4,,000) c. Draw a graph showing the relationship between age and value. Value (thousands of $) Age (ears) Glencoe/McGraw-Hill 44 Glencoe Algebra

17 -9 NAME DATE PERID Stud Guide and Intervention Statistics: Analzing Data b Using Tables and Graphs Analze Data Graphs or tables can be used to displa data. A bar graph compares different categories of data, while a circle graph compares parts of a set of data as a percent of the whole set. A line graph is useful to show how a data set changes over time. Eample The circle graph at the right shows the number of international visitors to the United States in 000, b countr. International Visitors to the U.S., 000 a. If there were a total of 50,89,000 visitors, how man were from Meico? 50,89,000 0% 0,78,00 b. If the percentage of visitors from each countr remains the same each ear, how man visitors from Canada would ou epect in the ear 00 if the total is 59,000,000 visitors? 59,000,000 9% 7,0,000 thers % United Kingdom 9% Japan 0% Source: TInet Canada 9% Meico 0%. The graph shows the use of imported steel b U. S. companies over a 0-ear period. a. Describe the general trend in the graph. The general trend is an increase in the use of imported steel over the 0-ear period, with slight decreases in 996 and 000. b. What would be a reasonable prediction for the percentage of imported steel used in 00? about 0% Percent Imported Steel as Percent of Total Used Year Source: Chicago Tribune Lesson -9. The table shows the percentage of change in worker productivit at the beginning of each ear for a 5-ear period. a. Which ear shows the greatest percentage increase in productivit? 998 b. What does the negative percent in the first quarter of 00 indicate? Worker productivit decreased in this period, as compared to the productivit one ear earlier. Worker Productivit Inde Year (st Qtr.) % of Change Source: Chicago Tribune Glencoe/McGraw-Hill 49 Glencoe Algebra

18 -9 NAME DATE PERID Stud Guide and Intervention (continued) Statistics: Analzing Data b Using Tables and Graphs Misleading Graphs Graphs are ver useful for displaing data. However, some graphs can be confusing, easil misunderstood, and lead to false assumptions. These graphs ma be mislabeled or contain incorrect data. r the ma be constructed to make one set of data appear greater than another set. Eample The graph at the right shows the number of students per computer in the U.S. public schools for the school ears from 995 to 999. Eplain how the graph misrepresents the data. The values are difficult to read because the vertical scale is too condensed. It would be more appropriate to let each unit on the vertical scale represent student rather than 5 students and have the scale go from 0 to. Students per Computer, U.S. Public Schools 0 Students Years since 994 Source: The World Almanac Eplain how each graph misrepresents the data.. The graph below shows the U.S.. The graph below shows the amount of greenhouse gases emissions for 999. mone spent on tourism for Nitrous ide 6% Methane 9% HCFs, PFCs, and Sulfur Heafluoride % U.S. Greenhouse Gas Emissions 999 Carbon Dioide 8% World Tourism Receipts 460 Billions of $ Year Source: The World Almanac Source: Department of Energ The graph is misleading because The graph is misleading because the sum of the percentages is not the vertical ais starts at %. Another section needs to billion. This gives the impression be added to account for the that $400 billion is a minimum missing %, or.6. amount spent on tourism. Glencoe/McGraw-Hill 50 Glencoe Algebra

19 - Stud Guide and Intervention Rational Numbers on the Number Line Graph Rational Numbers The figure at the right is part of a number line. A number line can be used to show the sets of natural numbers, whole numbers, and integers. Positive numbers, are located to the right of 0, and negative numbers are located to the left of 0. Another set of numbers that ou can displa on a number line is the set of rational numbers. A rational number can a be written as, where a and b are integers and b 0. Some b 7 eamples of rational numbers are,,, and Eample Eample Name the coordinates of the points graphed on each number line. a The dots indicate each point on the graph. The coordinates are {,,,, 5} b The bold arrow to the right means the graph continues indefinitel in that direction. The coordinates are {,.5,,.5, 4, 4.5, 5, }. numbers. Natural Numbers 4 0 Negative Numbers Integers Whole Numbers 4 Positive Numbers Graph each set of a. {,,,, 0,, } b.,0,, Lesson - Name the coordinates of the points graphed on each number line Graph each set of numbers. 5. {,,, } 6. { 5,,, } 7. {integers less than 0} {,,, 0, } 9.,,, 0. {, 4,, 0,, } Glencoe/McGraw-Hill 75 Glencoe Algebra

20 - Stud Guide and Intervention (continued) Rational Numbers on the Number Line Absolute Value n a number line, is three units from zero in the negative direction, and is three units from zero in the positive direction. The number line at the right illustrates the meaning of absolute value. The absolute value of a number n is the distance from zero on a number line and is represented n. For this eample, and. units units direction direction 4 5 Eample Eample Find each absolute value. a. 6 6 is si units from zero in the negative direction. 6 6 b. is three halves units from zero in the positive direction. Evaluate 4 if Replace with Simplif. Find each absolute value Evaluate each epression if a 5, b, 8, and a 5. 6 b 6. b 7. b a 8. b 4 Glencoe/McGraw-Hill 76 Glencoe Algebra

21 - Stud Guide and Intervention Adding and Subtracting Rational Numbers Add Rational Numbers Adding Rational Numbers, Same Sign Adding Rational Numbers, Different Signs Add the numbers. If both are positive, the sum is positive; if both are negative, the sum is negative. Subtract the number with the lesser absolute value from the number with the greater absolute value. The sign of the sum is the same as the sign of the number with the greater absolute value. Eample Eample Use a number line to find the sum ( ). Step Draw an arrow from 0 to. Step From the tip of the first arrow, draw a second arrow units to the left to represent adding. Step The second arrow ends at the sum 5. So ( ) Find each sum. a ( 8 5 ) (8 5) b Lesson - 4 Find each sum ( 5) (.5) 6..5 ( 5.) ( 05) ( 6) ( 90) (.8) ( 0.5) 40 0 Glencoe/McGraw-Hill 8 Glencoe Algebra

22 - Stud Guide and Intervention (continued) Adding and Subtracting Rational Numbers Subtract Rational Numbers Ever positive rational number can be paired with a negative rational number so that their sum is 0. The numbers, called opposites, are additive inverses of each other. Additive Inverse Propert For ever number a, a ( a) 0. To subtract a rational number, add its inverse and use the rules for addition given on page 8. Subtraction of Rational Numbers For an numbers a and b, a b a ( b). Eample Find ( 0.) To subtract 0., add its inverse. ( ).7 Simplif. Find each difference. 0. is greater, so the result is negative ( ). ( 7) 4. 8 ( ) (.5) (.5) ( 5) (6.8) ( 7) (.) ( 5.6) Sanelle was plaing a video game. Her scores were 50, 75, 8, and. What was her final score? 0. The football team offense began a drive from their 0-ard line. The gained 8 ards, lost ards and lost ards before having to kick the ball. What ard line were the on when the had to kick the ball? Glencoe/McGraw-Hill 8 Glencoe Algebra

23 - Stud Guide and Intervention Multipling Rational Numbers Multipl Integers You can use the rules below when multipling integers and rational numbers. Multipling Numbers with the Same Sign Multipling Numbers with Different Signs The product of two numbers having the same sign is positive. The product of two numbers having different signs is negative. Eample Eample Find each product. a. 7(6) The signs are different, so the product is negative. 7(6) 4 b. 8( 0) The signs are the same, so the product is positive. 8( 0) 80 Find each product. Simplif the epression ( )5. ( )5 ( )(5) ( 5) Associative Propert 0 Simplif.. (4). 5( ). ( 4)( ) Commutative Propert ( ) 4. (60)( ) 5. ( )( )( 4) 6. 8( 5) 7. 5() 8. ()( 0) 9. ( )( )() Lesson - 0. (5)( 5)(0)(4). ( 5)(45). ( )( ) Simplif each epression.. 4( ) ( n ) 0n 5. 6( ) 6. (d d) 7. () () 8. 4m( n) d( 4e) 9. 5( ) ( ) 0. ()( 4 ). ( )( 8n 6m) Glencoe/McGraw-Hill 87 Glencoe Algebra

24 - Stud Guide and Intervention (continued) Multipling Rational Numbers Multipl Rational Numbers Multipling a rational number b gives ou the additive inverse of the number. Multiplicative Propert of The product of an number and is its additive inverse. ( )(5) 5( ) 5 Eample Eample Evaluate a b if a and b 5. a b ( ) ( 5) Substitution ( 8)(5) ( ) 8 and ( 5) 5 00 different signs negative product n Substitution Evaluate n if n or 4 different signs negative product Find each product.. ( ) (6.0)( 0.) ( 5) ( 4) 8. ( 0) 9. ( ) ( )(0) Evaluate each epression if a.5, b 4., c 5.5, and d 0... a 4. 5( b) 5. 6(cd) 6. (d c) 7. ad c 8. b (c d) 9. 5bcd 0. d 4. ( )( 8a b) Glencoe/McGraw-Hill 88 Glencoe Algebra

25 -4 Stud Guide and Intervention Dividing Rational Numbers Divide Integers The rules for finding the sign of a quotient are similar to the rules for finding the sign of a product. Dividing Two Numbers with the Same Sign Dividing Two Numbers with Different Signs The quotient of two numbers having the same sign is positive. The quotient of two numbers having different signs is negative. Find each quotient. a. 88 ( 4) 88 ( 4) b. Eample Eample Find each quotient. same signs positive quotient different signs negative quotient 4( 0 ) Simplif. ( ) 4( 0 ) 4( 8). ( ) ( ) ( ) ( 0) ( ) 5. ( ) 6. 8 ( ) 7. 5 ( ) 8. ( ) ( 8). 5 ( 5) Simplif. ( 4) 5( 0 ( )) 6( 6 ) ( ) ( ) 0 ( ) Lesson -4 ( ( )) 4( 8 ( 4)) 4( 4) ( ) (8) Glencoe/McGraw-Hill 9 Glencoe Algebra

26 -4 Stud Guide and Intervention (continued) Dividing Rational Numbers Divide Rational Numbers The rules for division with integers also appl to division c with rational numbers. To divide b an nonzero number,, multipl b the reciprocal of d d that number,. c a c a Division of Rational Numbers d b d b c Eample Eample 0a 5 Simplif. 5 a. Find a 5 ( 0a 5) ( 0a 5) a or 4a b. Find Find each quotient ( ) 5..9 ( 0.) (.5)..5 (.5) ( 4.) Simplif each epression. 44a a 6b 6a a 6 57 Evaluate each epression if a 6, b.5, c., and d 4.8. ab a d a b d b c d Glencoe/McGraw-Hill 94 Glencoe Algebra

27 -5 Stud Guide and Intervention Statistics: Displaing and Analzing Data Create Line Plots and Stem-and-Leaf Plots ne wa to displa data graphicall is with a line plot. A line plot is a number line labeled with a scale that includes all the data and s placed above a data point each time it occurs in the data list. The s represent the frequenc of the data. A stem-and-leaf plot can also be used to organize data. The greatest common place value is called the stem, and the numbers in the net greatest place value form the leaves. Eample Eample Draw a line plot for the data Step The value of the data ranges from to 0, so construct a number line containing those points Step Then place an above the number each time it occurs Use the data below to create a stem-and-leaf plot The greatest common place value is tens, so the digits in the tens place are the stems. Thus 6 would have a stem of 6 and 04 would have a stem of ten. The stem-and-leaf plot is shown below. Stem Leaf Use the table at the right for.. Make a line plot representing the weights of the wrestlers shown in the table at the right. Weights of Junior Varsit Wrestlers (pounds) How man wrestlers weigh over 40 lb?. What is the greatest weight? Use each set of data to make a stem-and-leaf plot Lesson -5 Glencoe/McGraw-Hill 99 Glencoe Algebra

28 -5 Analze Data Numbers that represent the centralized, or middle, value of a set of data are called measures of central tendenc. Three measures of central tendenc are the mean, median, and mode. Mean Definition Sum of the data values divided b the number of values in the data set. Eample Data: 4, 6,, 0,, 0; 7 6 The middle number in a data set when the numbers are arranged in numerical Median order. If there is an even number of 5 0 Data:,, 5, 0,, 4; 7.5 values, the median is halfwa between the two middle values. Mode Stud Guide and Intervention (continued) Statistics: Displaing and Analzing Data The number or numbers that occur most often in the set of data. Data:,, 4, 0, 0, 6; and 0 are modes Eample Which measure of central tendenc best represents the data? Stem Leaf Find the mean, median, and mode. Mean 05 Median 0 Modes 99 and The median best represents the center of the data since the mean is too high. Find the mean, median, and mode for each data set. Then tell which best represents the data.... Stem Leaf Stem Leaf Stem Leaf Month Das above Ma 4 June 7 Jul 4 August September Glencoe/McGraw-Hill 00 Glencoe Algebra

29 -6 Stud Guide and Intervention Probabilit: Simple Probabilit and dds Probabilit The probabilit of a simple event is a ratio that tells how likel it is that the event will take place. It is the ratio of the number of favorable outcomes of the event to the number of possible outcomes of the event. You can epress the probabilit either as a fraction, as a decimal, or as a percent. number of favorable outcomes Probabilit of a Simple Event For an event a, P(a). number of possible outcomes Lesson -6 Eample Eample Mr. Babcock chooses 5 out of 5 students in his algebra class at random for a special project. What is the probabilit of being chosen? number of students chosen P(being chosen) total number of students 5 The probabilit of being chosen is or. 5 5 A bowl contains pears, 4 bananas, and apples. If ou take a piece of fruit at random, what is the probabilit that it is not a banana? There are 4 or 9 pieces of fruit. There are or 5 pieces of fruit that are not bananas. P(not banana) number of other pieces of fruit total number of pieces of fruit The probabilit of not choosing a banana is. 9 A card is selected at random from a standard deck of 5 cards. Determine each probabilit.. P(0). P(red ). P(king or queen) 4. P(black card) 5. P(ace of spades) 6. P(spade) Two dice are rolled and their sum is recorded. Find each probabilit. 7. P(sum is ) 8. P(sum is 6) 9. P(sum is less than 4) 0. P(sum is greater than ). P(sum is less than 5). P(sum is greater than 8) A bowl contains 4 red chips, blue chips, and 8 green chips. You choose one chip at random. Find each probabilit.. P(not a red chip) 4. P(red or blue chip) 5. P(not a green chip) A number is selected at random from the list {,,,, 0}. Find each probabilit. 6. P(even number) 7. P(multiple of ) 8. P(less than 4) 9. A computer randoml chooses a letter from the word CMPUTER. Find the probabilit that the letter is a vowel. Glencoe/McGraw-Hill 05 Glencoe Algebra

30 -6 Stud Guide and Intervention (continued) Probabilit: Simple Probabilit and dds dds The odds of an event occurring is the ratio of the number of was an event can occur (successes) to the number of was the event cannot occur (failures). dds number of successes number of failures Eample A die is rolled. Find the odds of rolling a number greater than 4. The sample space is {,,, 4, 5, 6}. Therefore, there are si possible outcomes. Since 5 and 6 are the onl numbers greater than 4, two outcomes are successes and four are failures. So the odds of rolling a number greater than 4 is, or :. 4 Find the odds of each outcome if the spinner at the right is spun once.. multiple of 4. odd number. even or a 5 4. less than even number greater than 5 Find the odds of each outcome if a computer randoml chooses a number between and the number is less than 0 7. the number is a multiple of 4 8. the number is even 9. the number is a one-digit number A bowl of mone at a carnival contains 50 quarters, 75 dimes, 00 nickels, and 5 pennies. ne coin is randoml selected. 0. Find the odds that a dime will not be chosen.. What are the odds of choosing a quarter if all the dimes are removed?. What are the odds of choosing a penn? Suppose ou drop a chip onto the grid at the right. Find the odds of each outcome.. land on a shaded square 4. land on a square on the diagonal land on square number 6 6. land on a number greater than 7. land on a multiple of 5 Glencoe/McGraw-Hill 06 Glencoe Algebra

31 -7 Stud Guide and Intervention Square Roots and Real Numbers Square Roots A square root is one of two equal factors of a number. For eample, the square roots of 6 are 6 and 6, since 6 6 or 6 is 6 and ( 6)( 6) or ( 6) is also 6. A rational number like 6, whose square root is a rational number, is called a perfect square. The smbol is a radical sign. It indicates the nonnegative, or principal, square root of the number under the radical sign. So 6 6 and 6 6. The smbol 6 represents both square roots. 5 Find represents the negative square root of Eample Eample Find represents the positive and negative square roots of and 0.6 ( 0.4) Lesson -7 Find each square root Glencoe/McGraw-Hill Glencoe Algebra

32 -7 NAME DATE PERID Stud Guide and Intervention (continued) Square Roots and Real Numbers Classif and rder Numbers Numbers such as and are not perfect squares. Notice what happens when ou find these square roots with our calculator. The numbers continue indefinitel without an pattern of repeating digits. Numbers that cannot be written as a terminating or repeating decimal are called irrational numbers. The set of real numbers consists of the set of irrational numbers and the set of rational numbers together. The chart below illustrates the various kinds of real numbers. Natural Numbers {,,, 4, } Whole Numbers {0,,,, 4, } Integers {,,,, 0,,,, } Rational Numbers a {all numbers that can be epressed in the form, where a and b are integers and b 0} b Irrational Numbers a {all numbers that cannot be epressed in the form, where a and b are integers and b 0} b Eample Name the set or sets of numbers to which each real number belongs. a. 4 Because 4 and are integers, this number is a rational number. b. 8 Because 8 9, this number is a natural number, a whole number, an integer, and a rational number. c. Because , which is not a repeating or terminating decimal, this number is irrational. Name the set or sets of numbers to which each real number belongs natural, whole, rational rational irrational integer, rational rational natural, whole, rational irrational integer, rational Write each set of numbers in order from least to greatest. 7 9., 5, 5, , 0.,.. 5, 0.05,, ,,, 5 0., 0.09,. 5,, 0.05, ,, 4,. 4 5.,,, , ,,, , 0.5,,0. 5, 5, 5 5 Glencoe/McGraw-Hill Glencoe Algebra

33 - Stud Guide and Intervention Writing Equations Write Equations Writing equations is one strateg for solving problems. You can use a variable to represent an unspecified number or measure referred to in a problem. Then ou can write a verbal epression as an algebraic epression. Eample Eample Translate each sentence into an equation or a formula. a. Ten times a number is equal to.8 times the difference minus z. 0.8 ( z) The equation is 0.8( z). b. A number m minus 8 is the same as a number n divided b. m 8 n n The equation is m 8. c. The area of a rectangle equals the length times the width. Translate this sentence into a formula. Let A area, length, and w width. Formula: Area equals length times width. A w The formula for the area of a rectangle is A w. Use the Four-Step Problem-Solving Plan. The population of the United States in 00 was about 84,000,000, and the land area of the United States is about,500,000 square miles. Find the average number of people per square mile in the United States. Source: Step Eplore You know that there are 84,000,000 people. You want to know the number of people per square mile. Step Plan Write an equation to represent the situation. Let p represent the number of people per square mile.,500,000 p 84,000,000 Step Solve,500,000 p 84,000,000.,500,000p 84,000,000 Divide each side b p 8.4,500,000. There about 8 people per square mile. Step 4 Eamine If there are 8 people per square mile and there are,500,000 square miles, 8,500,000 8,500,000, or about 84,000,000 people. The answer makes sense. Lesson - Translate each sentence into an equation or formula.. Three times a number t minus twelve equals fort.. ne-half of the difference of a and b is 54.. Three times the sum of d and 4 is. 4. The area A of a circle is the product of and the radius r squared. WEIGHT LSS For 5 6, use the following information. Lou wants to lose weight to audition for a part in a pla. He weighs 60 pounds now. He wants to weigh 50 pounds. 5. If p represents the number of pounds he wants to lose, write an equation to represent this situation. 6. How man pounds does he need to lose to reach his goal? Glencoe/McGraw-Hill 7 Glencoe Algebra

34 b. a b c a b c - Write Verbal Sentences You can translate equations into verbal sentences. Eample a. 4n 8. Stud Guide and Intervention (continued) Writing Equations Translate each equation into a verbal sentence. 4n 8 Four times n minus eight equals twelve. The sum of the squares of a and b is equal to the square of c. Translate each equation into a verbal sentence.. 4a 5. 0 k 4k p p 6. b (h ) (g h) 9. p 5 p 9 0. C (F ) 9. V Bh. A hb Glencoe/McGraw-Hill 8 Glencoe Algebra

35 - Stud Guide and Intervention Solving Equations b Using Addition and Subtraction Solve Using Addition If the same number is added to each side of an equation, the resulting equation is equivalent to the original one. In general if the original equation involves subtraction, this propert will help ou solve the equation. Addition Propert of Equalit For an numbers a, b, and c, if a b, then a c b c. Eample Eample Solve m 8. m 8 m 8 m 50 Simplif. The solution is 50. riginal equation Add to each side. Solve 8 p. 8 p 8 p p 6 Simplif. The solution is 6. riginal equation Add to each side. Solve each equation. Then check our solution.. h. m 8. p k w 8 Lesson - 7. h k 9. j b m ( ) 0. w 4 Write an equation for each problem. Then solve the equation and check the solution.. Twelve subtracted from a number equals 5. Find the number. 4. What number decreased b 5 equals? 5. Fift subtracted from a number equals eight. Find the number. 6. What number minus one-half is equal to negative one-half? 7. The difference of a number and eight is equal to 4. What is the number? 8. A number decreased b fourteen is equal to eighteen. What is the number? Glencoe/McGraw-Hill 4 Glencoe Algebra

36 - Stud Guide and Intervention (continued) Solving Equations b Using Addition and Subtraction Solve Using Subtraction If the same number is subtracted from each side of an equation, the resulting equation is equivalent to the original one. In general if the original equation involves addition, this propert will help ou solve the equation. Subtraction Propert of Equalit For an numbers a, b, and c, if a b, then a c b c. Eample Solve p. p riginal equation p Subtract from each side. p 4 Simplif. The solution is 4. Solve each equation. Then check our solution.. 6. z. 7 b s ( 9) ( 0.) h 4 8. k 4 9. j b m ( 8). w 8 Write an equation for each problem. Then solve the equation and check the solution.. Twelve added to a number equals 8. Find the number. 4. What number increased b 0 equals 0? 5. The sum of a number and fift equals eight. Find the number. 6. What number plus one-half is equal to four? 7. The sum of a number and is equal to 5. What is the number? Glencoe/McGraw-Hill 44 Glencoe Algebra

37 - Stud Guide and Intervention Solving Equations b Using Multiplication and Division Solve Using Multiplication If each side of an equation is multiplied b the same number, the resulting equation is equivalent to the given one. You can use the propert to solve equations involving multiplication and division. Multiplication Propert of Equalit For an numbers a, b, and c, if a b, then ac bc. 7 Eample Eample Solve p. Solve n 6. 4 p riginal equation n 6 riginal equation 4 7 p Rewrite each mied number as an improper fraction. 7 p Multipl each side b. 7 7 p 7 The solution is. 7 Simplif. 4 n 64 The solution is n 4(6) Multipl each side b 4. Simplif. Solve each equation. Then check our solution. h.. m 6. p m k j 7. h 4 8. k 9. 5 Lesson - 7 p 0. b 5. m Write an equation for each problem. Then solve the equation.. ne-fifth of a number equals 5. Find the number. 4. What number divided b equals 8? 5. A number divided b eight equals. Find the number. 6. ne and a half times a number equals 6. Find the number. Glencoe/McGraw-Hill 49 Glencoe Algebra

38 - Stud Guide and Intervention (continued) Solving Equations b Using Multiplication and Division Solve Using Division To solve equations with multiplication and division, ou can also use the Division Propert of Equalit. If each side of an equation is divided b the same number, the resulting equation is true. a b Division Propert of Equalit For an numbers a, b, and c, with c 0, if a b, then. c c Eample Eample 8n 64 8n n 8 Solve 8n 64. riginal equation Divide each side b 8. Simplif. The solution is 8. 5n 60 5n n Solve 5n 60. riginal equation Divide each side b 5. Simplif. The solution is. Solve each equation. Then check our solution.. h 4. 8m 6. t 5 4. r k m 6 7. h p j m. 6m 5..5p 75 Write an equation for each problem. Then solve the equation.. Four times a number equals 64. Find the number. 4. What number multiplied b 4 equals 6? 5. A number times eight equals 6. Find the number. Glencoe/McGraw-Hill 50 Glencoe Algebra

39 -4 Stud Guide and Intervention Solving Multi-Step Equations Work Backward Working backward is one of man problem-solving strategies that ou can use to solve problems. To work backward, start with the result given at the end of a problem and undo each step to arrive at the beginning number. Eample Eample A number is divided b, and then 8 is subtracted from the quotient. The result is 6. What is the number? Solve the problem b working backward. The final number is 6. Undo subtracting 8 b adding 8 to get 4. To undo dividing 4 b, multipl 4 b to get 48. The original number is 48. Solve each problem b working backward. A bacteria culture doubles each half hour. After hours, there are 6400 bacteria. How man bacteria were there to begin with? Solve the problem b working backward. The bacteria have grown for hours. Since there are one-half hour periods in one hour, in hours there are 6 one-half hour periods. Since the bacteria culture has grown for 6 time periods, it has doubled 6 times. Undo the doubling b halving the number of bacteria 6 times. 6,400 6, There were 00 bacteria to begin with.. A number is divided b, and then 4 is added to the quotient. The result is 8. Find the number.. A number is multiplied b 5, and then is subtracted from the product. The result is. Find the number.. Eight is subtracted from a number, and then the difference is multiplied b. The result is 4. Find the number. 4. Three times a number plus is 4. Find the number. 5. CAR RENTAL Angela rented a car for $9.99 a da plus a one-time insurance cost of $5.00. Her bill was $4.96. For how man das did she rent the car? Lesson MNEY Mike withdrew an amount of mone from his bank account. He spent one fourth for gasoline and had $90 left. How much mone did he withdraw? 7. TELEVISINS In 999, 68% of households with TV s subscribed to cable TV. If 8,000 more subscribers are added to the number of households with cable, the total number of households with cable TV would be 67,600,000. How man households were there with TV in 999? Source: World Almanac Glencoe/McGraw-Hill 55 Glencoe Algebra

40 -4 Solve Multi-Step Equations To solve equations with more than one operation, often called multi-step equations, undo operations b working backward. Reverse the usual order of operations as ou work. Eample Stud Guide and Intervention (continued) Solving Multi-Step Equations Solve 5. riginal equation. Subtract from each side. Simplif. Divide each side b 5. Simplif. Solve each equation. Then check our solution n p d n g b ( ) k Write an equation and solve each problem. 6. Find three consecutive integers whose sum is Find two consecutive odd integers whose sum is Find three consecutive integers whose sum is 9. Glencoe/McGraw-Hill 56 Glencoe Algebra

41 -5 Stud Guide and Intervention Solving Equations with the Variable on Each Side Variables on Each Side To solve an equation with the same variable on each side, first use the Addition or the Subtraction Propert of Equalit to write an equivalent equation that has the variable on just one side of the equation. Then solve the equation. Eample Eample Solve The solution is 0. Solve each equation. Then check our solution. Solve The solution is.. 6 b 5b n 8 n s s.8 7. b 4 b k 5 k p 4p b 8 0 b k 0 k 5. 0 a 0a 6. n 8 n r 5 5 4r k 0 4k. 0 Lesson -5 Glencoe/McGraw-Hill 6 Glencoe Algebra

42 -5 Stud Guide and Intervention (continued) Solving Equations with the Variable on Each Side Grouping Smbols When solving equations that contain grouping smbols, first use the Distributive Propert to eliminate grouping smbols. Then solve. Eample Solve 4(a ) 0(a 5). 4(a ) 0(a 5) riginal equation 8a 4 0a 50 8a 4 0a 0a 50 0a 8a 4 50 Simplif. 8a a 54 Simplif. 8a Distributive Propert Add 0a to each side. Add 4 to each side. Divide each side b 8. a The solution is. Simplif. Solve each equation. Then check our solution.. ( 5) ( ). (7 t) t. (a ) 5 a g 5( 4 g) 5. 5(f ) ( f ) 6. 4(p ) (c ) 8. (d 8) d 9. 5(p ) 9 (p ) (b ) (5 b)..( ). 4 8 a 8 a (4 k) 0 k 5. (w ) 4 4(w ) 6. 6(n ) (n 4) 7. [ ( )] 8. 4(r ) 4( 4r) 9. ( 8) (4 4k) 0 6k. 6( ) 5( ) Glencoe/McGraw-Hill 6 Glencoe Algebra

43 -6 Stud Guide and Intervention Ratios and Proportions Ratios and Proportions A ratio is a comparison of two numbers b division. The ratio of to can be epressed as to, : or. Ratios are usuall epressed in simplest form. An equation stating that two ratios are equal is called a proportion.to determine whether two ratios form a proportion, epress both ratios in simplest form or check cross products. Determine whether the 4 ratios and form a proportion Eample Eample when epressed in simplest form. when epressed in simplest form. 8 4 The ratios and form a proportion 6 8 because the are equal when epressed in simplest form. Use cross products to 0 5 determine whether and form a 8 45 proportion. 0 5 Write the proportion (45) 8(5) Cross products Simplif. 0 5 The cross products are equal, so Since the ratios are equal, the form a proportion. Lesson -6 Use cross products to determine whether each pair of ratios forms a proportion ,.,., , 5., 6., , 8., 9., :, 0:0. 5 to 9, 5 to 45., :5, 0: to 4, 50 to :75, 44: , 7., 8., Glencoe/McGraw-Hill 67 Glencoe Algebra

44 -6 Stud Guide and Intervention (continued) Ratios and Proportions Solve Proportions If a proportion involves a variable, ou can use cross products to solve 0 the proportion. In the proportion, and are called etremes and 5 and 0 are 5 called means. In a proportion, the product of the etremes is equal to the product of the means. a c Means-Etremes Propert of Proportions For an numbers a, b, c, and d, if, then ad bc. b d Eample The solution is. 0 Solve riginal proportion () 5(0) Cross products 50 Simplif. 50 Divide each side b. Simplif. Solve each proportion t p d b 4 8 a w k k 6 9 Use a proportion to solve each problem. 6. MDELS To make a model of the Guadeloupe River bed, Hermie used inch of cla for 5 miles of the river s actual length. His model river was 50 inches long. How long is the Guadeloupe River? 7. EDUCATIN Josh finished 4 math problems in one hour. At that rate, how man hours will it take him to complete 7 problems? Glencoe/McGraw-Hill 68 Glencoe Algebra

45 -7 Stud Guide and Intervention Percent of Change Percent of Change When an increase or decrease in an amount is epressed as a percent, the percent is called the percent of change. If the new number is greater than the original number, the percent of change is a percent of increase. If the new number is less than the original number, the percent of change is the percent of decrease. Eample Eample Find the percent of increase. original: 48 new: 60 First, subtract to find the amount of increase. The amount of increase is Then find the percent of increase b using the original number, 48, as the base. r Percent proportion (00) 48(r) Cross products 00 48r Simplif r Divide each side b r Simplif. The percent of increase is 5%. Find the percent of decrease. original: 0 new: First, subtract to find the amount of decrease. The amount of decrease is 0 8. Then find the percent of decrease b using the original number, 0, as the base. 8 r Percent proportion (00) 0(r) Cross products 800 0r Simplif r Divide each side b r Simplif. The percent of decrease is 6 %, or about 7%. Lesson -7 State whether each percent of change is a percent of increase or a percent of decrease. Then find each percent of change. Round to the nearest whole percent.. original: 50. original: 90. original: 45 new: 80 new: 00 new: 0 4. original: original: original: 5 new: 6 new: 50 new: original: 0 8. original: original: 7.5 new: 80 new: 70 new: 5 0. original: 84. original:.5. original: 50 new: 98 new: 0 new: 500 Glencoe/McGraw-Hill 7 Glencoe Algebra

46 -7 Solve Problems Discounted prices and prices including ta are applications of percent of change. Discount is the amount b which the regular price of an item is reduced. Thus, the discounted price is an eample of percent of decrease. Sales ta is amount that is added to the cost of an item, so the price including ta is an eample of percent of increase. Eample Stud Guide and Intervention (continued) Percent of Change A coat is on sale for 5% off the original price. If the original price of the coat is $75, what is the discounted price? The discount is 5% of the original price. 5% of $ % Use a calculator. Subtract $8.75 from the original price. $75 $8.75 $56.5 The discounted price of the coat is $56.5. Find the final price of each item. When a discount and a sales ta are listed, compute the discount price before computing the ta.. Compact disc: $6. Two concert tickets: $8. Airline ticket: $48.00 Discount: 5% Student discount: 8% Superair discount: % 4. Shirt: $ CD plaer: $ Celebrit calendar: $0.95 Sales ta: 4% Sales ta: 5.5% Sales ta: 7.5% 7. Class ring: $ Software: $ Video recorder: $0.95 Group discount: 7% Discount: % Discount: 0% Sales ta: 5% Sales ta: 6% Sales ta: 5% 0. VIDES The original selling price of a new sports video was $ Due to the demand the price was increased to $ What was the percent of increase over the original price?. SCHL A high school paper increased its sales b 75% when it ran an issue featuring a contest to win a class part. Before the contest issue, 0% of the school s 800 students bought the paper. How man students bought the contest issue?. BASEBALL Baseball tickets cost $5 for general admission or $0 for bo seats. The sales ta on each ticket is 8%, and the municipal ta on each ticket is an additional 0% of the base price. What is the final cost of each tpe of ticket? Glencoe/McGraw-Hill 74 Glencoe Algebra

47 -8 Stud Guide and Intervention Solving Equations and Formulas Solve for Variables Sometimes ou ma want to solve an equation such as V wh for one of its variables. For eample, if ou know the values of V, w, and h, then the equation V is more useful for finding the value of. If an equation that contains more than one wh variable is to be solved for a specific variable, use the properties of equalit to isolate the specified variable on one side of the equation. Eample Eample Solve 4 8 for or The value of is. 4 Solve m n km 8 for m. m n km 8 m n km km 8 km m n km 8 m n km n 8 n m km 8 n m( k) 8 n m( k) k m 8 n k 8 n k, or n 8 k n 8 The value of m is. Since division b 0 is k undefined, k 0, or k. Lesson -8 Solve each equation or formula for the variable specified.. a b c for. 5 for. ( f) j for 4. z 9 for 5. (4 k) p for k 6. 7 m for 7. 4(c ) t for c 8. b c for 9. ( ) z for h(a b) 0. 6z 4 for. d rt for r. A for h 5. C (F ) for F 4. P w for w 5. A w for 9 Glencoe/McGraw-Hill 79 Glencoe Algebra

48 -8 Use Formulas Man real-world problems require the use of formulas. Sometimes solving a formula for a specified variable will help solve the problem. Eample The formula C d represents the circumference of a circle, or the distance around the circle, where d is the diameter. If an airplane could fl around Earth at the equator without stopping, it would have traveled about 4,900 miles. Find the diameter of Earth. C d Given formula C d Solve for d. 4,900 d.4 Use.4. d 790 Simplif. The diameter of Earth is about 790 miles. Stud Guide and Intervention (continued) Solving Equations and Formulas. GEMETRY The volume of a clinder V is given b the formula V r h, where r is the radius and h is the height. a. Solve the formula for h. b. Find the height of a clinder with volume 500 feet and radius 0 feet.. WATER PRESSURE The water pressure on a submerged object is given b P 64d, where P is the pressure in pounds per square foot, and d is the depth of the object in feet. a. Solve the formula for d. b. Find the depth of a submerged object if the pressure is 67 pounds per square foot.. GRAPHS The equation of a line containing the points (a, 0) and (0, b) is given b the formula. a b a. Solve the equation for. b. Suppose the line contains the points (4, 0), and (0, ). If, find. 4. GEMETRY The surface area of a rectangular solid is given b the formula S w h wh, where length, w width, and h height. a. Solve the formula for h. b. The surface area of a rectangular solid with length 6 centimeters and width centimeters is 7 square centimeters. Find the height. Glencoe/McGraw-Hill 80 Glencoe Algebra

49 -9 Stud Guide and Intervention Weighted Averages Miture Problems Weighted Average The weighted average M of a set of data is the sum of the product of each number in the set and its weight divided b the sum of all the weights. Miture Problems are problems where two or more parts are combined into a whole. The involve weighted averages. In a miture problem, the weight is usuall a price or a percent of something. Eample Delectable Cookie Compan sells chocolate chip cookies for $6.95 per pound and white chocolate cookies for $5.95 per pound. How man pounds of chocolate chip cookies should be mied with 4 pounds of white chocolate cookies to obtain a miture that sells for $6.75 per pound. Let w the number of pounds of chocolate chip cookies Number of Pounds Price per Pound Total Price Chocolate Chip w w White Chocolate (5.95) Miture w (w 4) Equation: 6.95w 4(5.95) 6.75(w 4) Solve the equation. 6.95w 4(5.95) 6.75(w 4) riginal equation 6.95w w 7 Simplif. 6.95w w 6.75w w Subtract 6.75w from each side. 0.w.80 7 Simplif. 0.w Subtract.80 from each side. 0.w. Simplif. w 6 Simplif. 6 pounds of chocolate chip cookies should be mied with 4 pounds of white chocolate cookies. Lesson -9. SLUTINS How man grams of sugar must be added to 60 grams of a solution that is % sugar to obtain a solution that is 50% sugar?. NUTS The Quik Mart has two kinds of nuts. Pecans sell for $.55 per pound and walnuts sell for $.95 per pound. How man pounds of walnuts must be added to 5 pounds of pecans to make a miture that sells for $.75 per pound?. INVESTMENTS Alice Gleason invested a portion of $,000 at 9% interest and the balance at % interest. How much did she invest at each rate if her total income from both investments was $, MILK Whole milk is 4% butterfat. How much skim milk with 0% butterfat should be added to ounces of whole milk to obtain a miture that is.5% butterfat? Glencoe/McGraw-Hill 85 Glencoe Algebra

50 -9 Uniform Motion Problems Motion problems are another application of weighted averages. Uniform motion problems are problems where an object moves at a certain speed, or rate. Use the formula d rt to solve these problems, where d is the distance, r is the rate, and t is the time. Eample Stud Guide and Intervention (continued) Weighted Averages Bill Gutierrez drove at a speed of 65 miles per hour on an epresswa for hours. He then drove for.5 hours at a speed of 45 miles per hour on a state highwa. What was his average speed? M.5 Definition of weighted average 56.4 Simplif. Bill drove at an average speed of about 56.4 miles per hour.. TRAVEL Mr. Anders and Ms. Rich each drove home from a business meeting. Mr. Anders traveled east at 00 kilometers per hour and Ms. Rich traveled west at 80 kilometers per hours. In how man hours were the 00 kilometers apart.. AIRPLANES An airplane flies 750 miles due west in hours and 750 miles due south in hours. What is the average speed of the airplane?. TRACK Sprinter A runs 00 meters in 5 seconds, while sprinter B starts.5 seconds later and runs 00 meters in 4 seconds. If each of them runs at a constant rate, who is further in 0 seconds after the start of the race? Eplain. 4. TRAINS An epress train travels 90 kilometers per hour from Smallville to Megatown. A local train takes.5 hours longer to travel the same distance at 50 kilometers per hour. How far apart are Smallville and Megatown? 5. CYCLING Two cclists begin traveling in the same direction on the same bike path. ne travels at 5 miles per hour, and the other travels at miles per hour. When will the cclists be 0 miles apart? 6. TRAINS Two trains leave Chicago, one traveling east at 0 miles per hour and one traveling west at 40 miles per hour. When will the trains be 0 miles apart? Glencoe/McGraw-Hill 86 Glencoe Algebra

51 4- Stud Guide and Intervention The Coordinate Plane Identif Points In the diagram at the right, points are located in reference to two perpendicular number lines called aes. The horizontal number line is the -ais, and the vertical number line is the -ais. The plane containing the - and -aes is called the coordinate plane. Points in the coordinate plane are named b ordered pairs of the form (, ). The first number, or -coordinate corresponds to a number on the -ais. The second number, or -coordinate, corresponds to a number on the -ais. The aes divide the coordinate plane into Quadrants I, II, III, and IV, as shown. The point where the aes intersect is called the origin. The origin has coordinates (0, 0). Eample Eample Write an ordered pair for point R above. The -coordinate is 0 and the -coordinate is 4. Thus the ordered pair for R is (0, 4). Quadrant II P Quadrant III R Quadrant I Q Quadrant IV Write ordered pairs for points P and Q above. Then name the quadrant in which each point is located. The -coordinate of P is and the -coordinate is. Thus the ordered pair for P is (, ). P is in Quadrant III. The -coordinate of Q is 4 and the -coordinate is. Thus the ordered pair for Q is (4, ). Q is in Quadrant IV. Lesson 4- Write the ordered pair for each point shown at the right. Name the quadrant in which the point is located. R Q V. N. P. Q 4. R T W Z U N 5. S 6. T P B A S 7. U 8. V 9. W 0. Z. A. B. Write the ordered pair that describes a point 4 units down from and units to the right of the origin. 4. Write the ordered pair that is 8 units to the left of the origin and lies on the -ais. Glencoe/McGraw-Hill Glencoe Algebra

52 4- Stud Guide and Intervention (continued) The Coordinate Plane Graph Points To graph an ordered pair means to draw a dot at the point on the coordinate plane that corresponds to the ordered pair. To graph an ordered pair (, ), begin at the origin. Move left or right units. From there, move up or down units. Draw a dot at that point. Eample Plot each point on a coordinate plane. a. R(, ) Start at the origin. Move left units since the -coordinate is. Move up units since the -coordinate is. Draw a dot and label it R. R b. S(0, ) Start at the origin. Since the -coordinate is 0, the point will be located on the -ais. Move down units since the -coordinate is. Draw a dot and label it S. S Plot each point on the coordinate plane at the right.. A(, 4). B(0, ). C( 4, 4) 4. D(, 0) 5. E(, 4) 6. F(0, 0) H Q D J C A M F L K B E P G N I 7. G(5, 0) 8. H(, 4) 9. I(4, 5) 0. J(, ). K(, ). L(, ). M(0, ) 4. N(5, ) 5. P(4, 5) 6. Q( 5, ) Glencoe/McGraw-Hill 4 Glencoe Algebra

53 4- Stud Guide and Intervention Transformations on the Coordinate Plane Transform Figures Transformations are movements of geometric figures. The preimage is the position of the figure before the transformation, and the image is the position of the figure after the transformation. Reflection Translation Dilation Rotation A figure is flipped over a line. A figure is slid horizontall, verticall, or both. A figure is enlarged or reduced. A figure is turned around a point. Eample Determine whether each transformation is a reflection, translation, dilation, or rotation. a. The figure has been flipped over a line, so this is a reflection. b. The figure has been turned around a point, so this is a rotation. c. The figure has been reduced in size, so this is a dilation. Lesson 4- d. The figure has been shifted horizontall to the right, so this is a translation. Determine whether each transformation is a reflection, translation, dilation, or rotation Glencoe/McGraw-Hill 9 Glencoe Algebra

54 4- Stud Guide and Intervention (continued) Transformations on the Coordinate Plane Transform Figures on the Coordinate Plane You can perform transformations on a coordinate plane b changing the coordinates of each verte. The vertices of the image of the transformed figure are indicated b the smbol, which is read prime. Reflection over -ais (, ) (, ) Reflection over -ais (, ) (, ) Translation (, ) ( a, b) Dilation (, ) (k, k) Rotation 90 counterclockwise (, ) (, ) Rotation 80 (, ) (, ) Eample A triangle has vertices A(, ), B(, 4), and C(, 0). Find the coordinates of the vertices of each image below. a. reflection over the -ais To reflect a point over the -ais, multipl the -coordinate b. A(, ) A (, ) B(, 4) B (, 4) C(, 0) C (, 0) The coordinates of the image vertices are A (, ), B (, 4), and C (, 0). b. dilation with a scale factor of Find the coordinates of the dilated figure b multipling the coordinates b. A(, ) A (, ) B(, 4) B (4, 8) C(, 0) C (6, 0) The coordinates of the image vertices are A (, ), B (4, 8), and C (6, 0). Find the coordinates of the vertices of each figure after the given transformation is performed.. triangle RST with R(, 4), S(, 0), T(, ) reflected over the -ais. triangle ABC with A(0, 0), B(, 4), C(, 0) rotated about the origin 80. parallelogram ABCD with A(, 0), B(, ), C(, ), D(, 0) translated units down 4. quadrilateral RSTU with R(, ), S(, 4), T(4, 4), U(4, 0) dilated b a factor of 5. triangle ABC with A( 4, 0), B(, ), C(0, 0) rotated counterclockwise heagon ABCDEF with A(0, 0), B(, ), C(0, 4), D(, 4), E(4, ), F(, 0) translated units up and unit to the left Glencoe/McGraw-Hill 0 Glencoe Algebra

55 4- Stud Guide and Intervention Relations Represent Relations A relation is a set of ordered pairs. A relation can be represented b a set of ordered pairs, a table, a graph, or a mapping. A mapping illustrates how each element of the domain is paired with an element in the range. Eample Eample Epress the relation {(, ), (0, ), (, )} as a table, a graph, and a mapping. State the domain and range of the relation. X 0 0 Y The domain for this relation is {0,, }. The range for this relation is {,, }. A person plaing racquetball uses 4 calories per hour for ever pound he or she weighs. a. Make a table to show the relation between weight and calories burned in one hour for people weighing , 0, 0, and 0 pounds Source: The Math Teacher s Book of Lists b. Give the domain and range. domain: {00, 0, 0, 0} range: {400, 440, 480, 50} 0 50 c. Graph the relation. Calories Weight (pounds). Epress the relation {(, ), (, ), (4, )} as a table, a graph, and a mapping. Then determine the domain and range. X 4 Y Lesson 4-. The temperature in a house drops for ever hour the air conditioner is on between the hours of 6 A.M. and A.M. Make a graph to show the relationship between time and temperature if the temperature at 6 A.M. was 8 F. Temperature ( F) Time (A.M.) Glencoe/McGraw-Hill 5 Glencoe Algebra

56 4- Inverse Relations The inverse of an relation is obtained b switching the coordinates in each ordered pair. Eample Epress the relation shown in the mapping as a set of ordered pairs. Then write the inverse of the relation. X Stud Guide and Intervention (continued) Relations Y Relation: {(6, 5), (, ), (, 4), (0, )} Inverse: {(5, 6), (, ), (4, ), (, 0)} Epress the relation shown in each table, mapping, or graph as a set of ordered pairs. Then write the inverse of each relation X 4 5 Y X 0 4 Y Glencoe/McGraw-Hill 6 Glencoe Algebra

57 4-4 Stud Guide and Intervention Equations as Relations Solve Equations The equation 4 is an eample of an equation in two variables because it contains two variables, and. The solution of an equation in two variables is an ordered pair of replacements for the variables that results in a true statement when substituted into the equation. Eample Eample Find the solution set for, given the replacement set {(, ), (0, ), (, ), (, )}. Make a table. Substitute the and -values of each ordered pair into the equation. True or False 0 ( ) (0) () () 7 True True False False The ordered pairs (, ), and (0, ) result in true statements. The solution set is {(, ), (0, )}. Solve b a if the domain is {,, 0,, 4}. Make a table. The values of a come from the domain. Substitute each value of a into the equation to determine the corresponding values of b in the range. a a b (a, b) ( ) 5 (, 5) ( ) (, ) 0 (0) (0, ) () (, ) 4 (4) 7 (4, 7) The solution set is {(, 5), (, ), (0, ), (, ), (4, 7)}. Find the solution set of each equation, given the replacement set.. ; {(0, ),,,,,(, )}. 6; {(, ), (0, ), (0, ), (, 0)}. 5 ; {(, ), (, ), (, ), (4, )} Solve each equation if the domain is ( 4,, 0,, 4} a b 0 Lesson Glencoe/McGraw-Hill Glencoe Algebra

58 4-4 Graph Solution Sets You can graph the ordered pairs in the solution set of an equation in two variables. The domain contains values represented b the independent variable. The range contains the corresponding values represented b the dependent variable, which are determined b the given equation. Eample Stud Guide and Intervention (continued) Equations as Relations Solve 4 if the domain is (, 0,, 4}. Graph the solution set. First solve the equation for in terms of. 4 riginal equation Subtract 4 from each side. 4 Simplif. 4 Divide each side b. 6 Simplif. Substitute each value of from the domain to determine the corresponding value of in the range. 6 (, ) 6 ( ) 8 (, 8) 0 6 (0) 6 (0, 6) 6 () (, ) 4 6 (4) (4, ) Graph the solution set. Solve each equation for the given domain. Graph the solution set.. 4 for {, 0,, 4}. for {,, 0, }. 6 for {, 0,, 6} for { 4,, 0, } Glencoe/McGraw-Hill Glencoe Algebra

59 4-5 Stud Guide and Intervention Graphing Linear Equations Identif Linear Equations A linear equation is an equation that can be written in the form A B C. This is called the standard form of a linear equation. Standard Form of a Linear Equation A B C, where A 0, A and B are not both zero, and A, B, and C are integers whose GCF is. Eample Eample Determine whether 6 is a linear equation. If so, write the equation in standard form. First rewrite the equation so both variables are on the same side of the equation. 6 riginal equation 6 Add to each side. 6 Simplif. The equation is now in standard form, with A, B and C 6. This is a linear equation. Determine whether 4 is a linear equation. If so, write the equation in standard form. Since the term has two variables, the equation cannot be written in the form A B C. Therefore, this is not a linear equation. Determine whether each equation is a linear equation. If so, write the equation in standard form a b 8 b Lesson 4-5 Glencoe/McGraw-Hill 7 Glencoe Algebra

60 4-5 Stud Guide and Intervention (continued) Graphing Linear Equations Graph Linear Equations The graph of a linear equation is a line. The line represents all solutions to the linear equation. Also, ever ordered pair on this line satisfies the equation. Eample Graph the equation. Solve the equation for. riginal equation Add to each side. Simplif. Select five values for the domain and make a table. Then graph the ordered pairs and draw a line through the points. (, ) ( ) (, ) ( ) (, ) 0 (0) (0, ) () (, ) () 5 (, 5) Graph each equation Glencoe/McGraw-Hill 8 Glencoe Algebra

61 4-6 Stud Guide and Intervention Functions Identif Functions Relations in which each element of the domain is paired with eactl one element of the range are called functions. Eample Eample Determine whether the relation {(6, ), (4, ), (7, ), (, )} is a function. Eplain. Since each element of the domain is paired with eactl one element of the range, this relation is a function. Determine whether 6 is a function. Since the equation is in the form A B C, the graph of the equation will be a line, as shown at the right. If ou draw a vertical line through each value of, the vertical line passes through just one point of the graph. Thus, the line represents a function. Lesson 4-6 Determine whether each relation is a function.... X Y {(4, ), (, ), (6, )} 8. {(, ), (, 4), (, 4)} 9. {(, 0), (, 0)} Glencoe/McGraw-Hill 4 Glencoe Algebra

62 4-6 Stud Guide and Intervention (continued) Functions Function Values Equations that are functions can be written in a form called function notation. For eample, can be written as f(). In the function, represents the elements of the domain, and f() represents the elements of the range. Suppose ou want to find the value in the range that corresponds to the element in the domain. This is written f() and is read f of. The value of f() is found b substituting for in the equation. Eample If f() 4, find each value. a. f() f() () 4 Replace with b. f( ) f( ) ( ) Multipl. Simplif. Replace with. Multipl. Simplif. If f() 4 and g() 4, find each value.. f(4). g(). f( 5) 4. g( ) 5. f(0) 6. g(0) 7. f() 8. f 9. g f(a ). f(k ). g(c). f() 4. f() 5. g( 4) Glencoe/McGraw-Hill 44 Glencoe Algebra

63 4-7 Stud Guide and Intervention Arithmetic Sequences Recognize Arithmetic Sequences A sequence is a set of numbers in a specific order. If the difference between successive terms is constant, then the sequence is called an arithmetic sequence. Arithmetic Sequence a numerical pattern that increases or decreases at a constant rate or value called the common difference Eample Eample Determine whether the sequence,, 5, 7, 9,, is an arithmetic sequence. Justif our answer. If possible, find the common difference between the terms. Since, 5, and so on, the common difference is. Since the difference between the terms of,, 5, 7, 9,, is constant, this is an arithmetic sequence. Determine whether the sequence,, 4, 8, 6,, is an arithmetic sequence. Justif our answer. If possible, find the common difference between the terms. Since and 4, there is no common difference. Since the difference between the terms of,, 4, 8, 6,, is not constant, this is not an arithmetic sequence. Lesson 4-7 Determine whether each sequence is an arithmetic sequence. If it is, state the common difference.., 5, 9,, 7,. 8, 4, 0, 4, 8,.,, 9, 7, 8, 4. 0, 5, 5, 40, 60, 5. 0, 5, 0, 5, 0, 6. 8, 6, 4,, 0,, 7. 4, 8,, 6, 8. 5,, 0, 9, 9..,.,., 4., 5., 0. 8, 7, 6, 5, 4,. 0.5,.5,.5,.5, 4.5,., 4, 6, 64,. 0, 4, 8,, 4., 6, 9,, 5. 7, 0, 7, 4, Glencoe/McGraw-Hill 49 Glencoe Algebra

64 4-7 Stud Guide and Intervention (continued) Arithmetic Sequences Write Arithmetic Sequences You can use the common difference of an arithmetic sequence to find the net term of the sequence. Each term after the first term is found b adding the preceding term and the common difference. Terms of an Arithmetic Sequence nth Term of an Arithmetic Sequence a n a (n )d If a is the first term of an arithmetic sequence with common difference d, then the sequence is a, a d, a d, a d,. Find the net three terms of the arithmetic sequence 8,, 6, 40,. Find the common difference b subtracting successive terms. 8 The common difference is 4. Add 4 to the last given term, 40, to get the net term. Continue adding 4 until the net three terms are found. 40 Eample Eample The net three terms are 44, 48, 5. Write an equation for the nth term of the sequence, 5, 8,,. In this sequence, a is. Find the common difference. 5 8 Find the net three terms of each arithmetic sequence. The common difference is. Use the formula for the nth term to write an equation. a n a (n )d Formula for the nth term a n (n ) a, d a n n Distributive Propert a n n 9 Simplif. The equation for the nth term is a n n 9.. 9,, 7,, 5,. 4, 0, 4, 8,,. 9, 5, 4, 47, 4. 0, 5, 0, 5, 5..5, 5, 7.5, 0, 6.., 4., 5., 6., Find the nth term of each arithmetic sequence described. 7. a 6, d, n 0 8. a, d, n 8 9. a, d 5, n 0 0. a, d, n 50. a, d 4, n 0. a, d, n Write an equation for the nth term of the arithmetic sequence..,, 5, 7, 4., 4, 7, 0, 5. 4, 9, 4, 9, Glencoe/McGraw-Hill 50 Glencoe Algebra

65 4-8 Stud Guide and Intervention Writing Equations from Patterns Look for Patterns A ver common problem-solving strateg is to look for a pattern. Arithmetic sequences follow a pattern, and other sequences can follow a pattern. Find the net three terms in the sequence, 9, 7, 8,. Stud the pattern in the sequence. Eample Eample Find the net three terms in the sequence 0, 6,, 7,, 8,. Stud the pattern in the sequence Successive terms are found b multipling the last given term b The net three terms are 4, 79, 87. Assume that the pattern continues The net three terms are, 9, 4. Lesson 4-8. Give the net two items for the pattern below. Give the net three numbers in each sequence..,, 7, 4,. 7, 4, 8, 56, 4. 0, 0, 5, 5, 0, 5. 0,,, 6, 0, 6.,,, 7.,,,, 4 Glencoe/McGraw-Hill 55 Glencoe Algebra

66 4-8 Write Equations Sometimes a pattern can lead to a general rule that can be written as an equation. Eample Stud Guide and Intervention (continued) Writing Equations from Patterns Suppose ou purchased a number of packages of blank compact disks. If each package contains compact disks, ou could make a chart to show the relationship between the number of packages of compact disks and the number of disks purchased. Use for the number of packages and for the number of compact disks. Make a table of ordered pairs for several points of the graph. Number of Packages 4 5 Number of CDs The difference in the values is, and the difference in the values is. This pattern shows that is alwas three times. This suggests the relation. Since the relation is also a function, we can write the equation in functional notation as f().. Write an equation for the function in. Write an equation for the function in functional notation. Then complete functional notation. Then complete the table. the table Write an equation in functional notation. 4. Write an equation in functional notation. Glencoe/McGraw-Hill 56 Glencoe Algebra

67 5- Find Slope Stud Guide and Intervention Slope Slope of a Line rise m or m, where (, ) and (, ) are the coordinates run of an two points on a nonvertical line Eample Eample Find the slope of the line that passes through (, 5) and (4, ). Let (, 5) (, ) and (4, ) (, ). m 5 4 ( ) 7 7 Slope formula, 5, 4, Simplif. Find the value of r so that the line through (0, r) and (, 4) has a slope of. 7 m Slope formula 4 r 7 0 m 7, 4, r,, 0 4 r Simplif. 7 7 ( 7) 7(4 r) Cross multipl r Distributive Propert 4 7r r Divide each side b 7. Subtract 8 from each side. Lesson 5- Find the slope of the line that passes through each pair of points.. (4, 9), (, 6). ( 4, ), (, 5). ( 4, ), ( 4, 5) 4. (, ), (8, 9) 5. (4, 8), (7, 6) 6. (4, ), (8, ) 7. (, ), (6, ) 8. (, 5), (6, ) 9. (4,.5), ( 4,.5) Determine the value of r so the line that passes through each pair of points has the given slope. 0. (6, 8), (r, ), m. (, ), (7, r), m 4. (, 8), (r, 4) m. (7, 5), (6, r), m 0 4. (r, 4), (7, ), m 4 5. (7, 5), (r, 9), m 6 6. (0, r), (, 4), m 7. (0, 4), (, r), m (r, ), (7, r), m 7 5 Glencoe/McGraw-Hill 8 Glencoe Algebra

68 5- Stud Guide and Intervention (continued) Slope Rate of Change The rate of change tells, on average, how a quantit is changing over time. Slope describes a rate of change. Eample PPULATIN The graph shows the population growth in China. a. Find the rates of change for and for change in population : change in time or change in population : change in time or Population Growth in China People (billions) * Year *Estimated Source: United Nations Population Division b. Eplain the meaning of the slope in each case. From , the growth was 0.05 billion per ear, or 5. million per ear. From , the growth was 0.04 billion per ear, or.4 million per ear. c. How are the different rates of change shown on the graph? There is a greater vertical change for than for Therefore, the section of the graph for has a steeper slope. LNGEVITY The graph shows the predicted life epectanc for men and women born in a given ear.. Find the rates of change for women from and Find the rates of change for men from and Eplain the meaning of our results in and. 4. What pattern do ou see in the increase with each 5-ear period? Predicting Life Epectanc Age * 050* Year Born Women Men *Estimated Source: USA TDAY 5. Make a prediction for the life epectanc for Eplain how ou arrived at our prediction. Glencoe/McGraw-Hill 8 Glencoe Algebra

69 5- Stud Guide and Intervention Slope and Direct Variation Direct Variation A direct variation is described b an equation of the form k, where k 0. We sa that varies directl as. In the equation k, k is the constant of variation. Eample Eample Name the constant of variation for the equation. Then find the slope of the line that passes through the pair of points. For, the constant of variation is. m Slope formula 0 0 (, ) (0, 0), (, ) (, ) The slope is. Simplif. (, ) (0, 0) Suppose varies directl as, and 0 when 5. a. Write a direct variation equation that relates and. Find the value of k. k Direct variation equation 0 k(5) Replace with 0 and with 5. 6 k Divide each side b 5. Therefore, the equation is 6. b. Use the direct variation equation to find when 8. 6 Direct variation equation 8 6 Replace with 8. Divide each side b 6. Therefore, when 8. Lesson 5- Name the constant of variation for each equation. Then determine the slope of the line that passes through each pair of points.... (, ) (0, 0) (, ) (0, 0) (, ) (0, 0) Write a direct variation equation that relates to. Assume that varies directl as. Then solve. 4. If 4 when, find when If 9 when, find when If 4.8 when.6, find when If when, find when Glencoe/McGraw-Hill 87 Glencoe Algebra

70 5- Solve Problems The distance formula d rt is a direct variation equation. In the formula, distance d varies directl as time t, and the rate r is the constant of variation. Eample TRAVEL A famil drove their car 5 miles in 5 hours. a. Write a direct variation equation to find the distance traveled for an number of hours. Use given values for d and t to find r. d rt riginal equation 5 r(5) d 5 and t 5 45 r Divide each side b 5. Therefore, the direct variation equation is d 45t. b. Graph the equation. The graph of d 45t passes through the origin with slope rise m run CHECK (5, 5) lies on the graph. c. Estimate how man hours it would take the famil to drive 60 miles. d 45t riginal equation 60 45t Replace d with 60. t 8 Divide each side b 45. Therefore, it will take 8 hours to drive 60 miles. Stud Guide and Intervention (continued) Slope and Direct Variation Distance (miles) d Automobile Trips d 45t (5, 5) 90 (, 45) t Time (hours) RETAIL The total cost C of bulk jell beans is $4.49 times the number of pounds p.. Write a direct variation equation that relates the variables.. Graph the equation on the grid at the right.. Find the cost of 4 pound of jell beans. Cost (dollars) Cost of Jell Beans C 0 4 w Weight (pounds) CHEMISTRY Charles s Law states that, at a constant pressure, volume of a gas V varies directl as its temperature T. A volume of 4 cubic feet of a certain gas has a temperature of 00 (absolute temperature). 4. Write a direct variation equation that relates the variables. 5. Graph the equation on the grid at the right. 6. Find the volume of the same gas at 50 (absolute temperature). Volume (cubic feet) V 4 Charles s Law T Temperature ( K) Glencoe/McGraw-Hill 88 Glencoe Algebra

71 5- Stud Guide and Intervention Slope-Intercept Form Slope-Intercept Form Slope-Intercept Form m b, where m is the given slope and b is the -intercept Eample Write an equation of the line whose slope is 4 and whose -intercept is. m b Slope-intercept form 4 Replace m with 4 and b with. Eample Graph riginal equation 4 8 Subtract from each side. 4 8 (0, ) Divide each side b Simplif. (4, ) 4 8 The -intercept of is and the slope is 4 4. So graph the point (0, ). From this point, move up units and right 4 units. Draw a line passing through both points. Write an equation of the line with the given slope and -intercept.. slope: 8, -intercept. slope:, -intercept. slope:, -intercept 7 Write an equation of the line shown in each graph (0, ) (, 0) (, 0) (0, ) (4, ) (0, 5) Lesson 5- Graph each equation Glencoe/McGraw-Hill 9 Glencoe Algebra

72 5- Model Real-World Data Eample Stud Guide and Intervention (continued) Slope-Intercept Form MEDIA Since 997, the number of cable TV sstems has decreased b an average rate of sstems per ear. There were 0,94 sstems in 997. a. Write a linear equation to find the average number of cable sstems in an ear after 997. The rate of change is sstems per ear. In the first ear, the number of sstems was 0,94. Let N the number of cable TV sstems. Let the number of ears after 997. An equation is N 0,94. b. Graph the equation. The graph of N 0,94 is a line that passes through the point at (0, 0,94) and has a slope of. c. Find the approimate number of cable TV sstems in 000. N 0,94 riginal equation N () 0,94 Replace with. N 0,580 Simplif. There were about 0,580 cable TV sstems in 000. Number of Cable TV Sstems Cable TV Sstems N 0,900 0,800 0,700 0,600 0, Years Since 997 Source: The World Almanac ENTERTAINMENT In 995, 65.7% of all households with TV s in the U.S. subscribed to cable TV. Between 995 and 999, the percent increased b about 0.6% each ear.. Write an equation to find the percent P of households that subscribed to cable TV for an ear between 995 and Graph the equation on the grid at the right.. Find the percent that subscribed to cable TV in 999. PPULATIN The population of the United States is projected to be 00 million b the ear 00. Between 00 and 050, the population is epected to increase b about.5 million per ear. 4. Write an equation to find the population P in an ear between 00 and Graph the equation on the grid at the right. 6. Find the population in 050. Percent of Households with TV Having Cable Percent Years Since 995 Source: The World Almanac Population (millions) P Projected United States Population P Years Since 00 Source: The World Almanac Glencoe/McGraw-Hill 94 Glencoe Algebra

73 5-4 Stud Guide and Intervention Writing Equations in Slope-Intercept Form Write an Equation Given the Slope and ne Point Eample Eample Write an equation of a line that passes through ( 4, ) with slope. The line has slope. To find the -intercept, replace m with and (, ) with ( 4, ) in the slope-intercept form. Then solve for b. m b Slope-intercept form ( 4) b m,, and 4 b Multipl. 4 b Add to each side. Therefore, the equation is 4. Write an equation of the line that passes through (, ) with slope. 4 The line has slope. Replace m with and (, ) 4 4 with (, ) in the slope-intercept form. m b Slope-intercept form ( ) b m,, and 4 4 b Multipl. b Add to each side. Therefore, the equation is. 4 Write an equation of the line that passes through each point with the given slope.... (, 5) m (, 4) m m (0, 0) 4. (8, ), m 5. (, ), m 5 6. (4, 5), m 4 7. ( 5, 4), m 0 8. (, ), m 9. (, 4), m 6 Lesson Write an equation of a line that passes through the -intercept with slope.. Write an equation of a line that passes through the -intercept 4 with slope.. Write an equation of a line that passes through the point (0, 50) with slope. 5 Glencoe/McGraw-Hill 99 Glencoe Algebra

74 5-4 Write an Equation Given Two Points Eample Stud Guide and Intervention (continued) Writing Equations in Slope-Intercept Form Write an equation of the line that passes through (, ) and (, ). Find the slope m. To find the -intercept, replace m with its computed value and (, ) with (, ) in the slope-intercept form. Then solve for b. m Slope formula m,,, m Simplif. m b Slope-intercept form () b Replace m with, with, and with. b Multipl. 4 b Add to each side. Therefore, the equation is 4. Write an equation of the line that passes through each pair of points.... (, ) (0, 4) (0, ) (, 0) (0, ) (4, 0) 4. (, 6), (7, 0) 5. (0, ), (, 7) 6. (6, 5), (, ) 7. (, ), (, ) 8. (0, ), (4, ) 9. ( 4, ), (7, 7) 0. Write an equation of a line that passes through the -intercept 4 and -intercept.. Write an equation of a line that passes through the -intercept and -intercept 5.. Write an equation of a line that passes through (0, 6) and ( 0, 0). Glencoe/McGraw-Hill 00 Glencoe Algebra

75 5-5 Stud Guide and Intervention Writing Equations in Point-Slope Form Point-Slope Form Point-Slope Form m( ), where (, ) is a given point on a nonvertical line and m is the slope of the line Eample Eample Write the point-slope form of an equation for a line that passes 5 through (6, ) and has a slope of. m( ) Point-slope form 5 5 ( 6) m ; (, ) (6, ) 5 Therefore, the equation is ( 6). Write the point-slope form of an equation for a horizontal line that passes through (4, ). m( ) Point-slope form ( ) 0( 4) 0 Simplif. Therefore, the equation is 0. m 0; (, ) (4, ) Write the point-slope form of an equation for a line that passes through each point with the given slope.... (4, ) m 0 m (, ) m (, ) 4. (, ), m 4 5. ( 7, ), m 6 6. (8, ), m 7. ( 6, 7), m 0 8. (4, 9), m 9. ( 4, 5), m 4 0. Write the point-slope form of an equation for the horizontal line that passes through (4, ).. Write the point-slope form of an equation for the horizontal line that passes through ( 5, 6).. Write the point-slope form of an equation for the horizontal line that passes through (5, 0). Lesson 5-5 Glencoe/McGraw-Hill 05 Glencoe Algebra

76 5-5 Stud Guide and Intervention (continued) Writing Equations in Point-Slope Form Forms of Linear Equations Slope-Intercept Form Point-Slope Form m b m( ) m slope; b -intercept m slope; (, ) is a given point. Standard A B C A and B are not both zero. Usuall A is nonnegative and A, B, and C Form are integers whose greatest common factor is. Eample Eample Write 5 ( 6) in Write ( 8) 4 standard form. in slope-intercept form. 5 ( 6) riginal equation ( 8) riginal equation 4 ( 5) ( 6) Multipl each side b. Distributive Propert 4 5 ( 6) Distributive Propert 4 Add to each side. 5 Distributive Propert 4 7 Subtract 5 from each side. Therefore, the slope-intercept form of the 7 Add to each side. 7 Multipl each side b. equation is 4. 4 Therefore, the standard form of the equation is 7. Write each equation in standard form.. ( ). ( 6). ( 9) 5 4. ( 5) 5. 4 ( ) 6. 4 ( ) 5 Write each equation in slope-intercept form ( ) 8. 5 ( 6) 9. 8 ( 8) ( 5). ( ) 5 Glencoe/McGraw-Hill 06 Glencoe Algebra

77 5-6 Stud Guide and Intervention Geometr: Parallel and Perpendicular Lines Parallel Lines Two nonvertical lines are parallel if the have the same slope. All vertical lines are parallel. Eample Write the slope-intercept form for an equation of the line that passes through (, 6) and is parallel to the graph of. A line parallel to has the same slope,. Replace m with and (, ) with (, 6) in the point-slope form. m( ) Point-slope form 6 ( ( )) m ; (, ) (, 6) 6 ( ) Simplif. 6 Distributive Propert 8 Slope-intercept form Therefore, the equation is 8. Lesson 5-6 Write the slope-intercept form for an equation of the line that passes through the given point and is parallel to the graph of each equation.... (5, ) ( 8, 7) 8 4 (, ) 4 4. (, ), 4 5. (6, 4), 6. (4, ), 7. (, 4), 0 8. (, 6), 9. (4, 6), 5 0. Find an equation of the line that has a -intercept of that is parallel to the graph of the line Find an equation of the line that has a -intercept of that is parallel to the graph of the line 6.. Find an equation of the line that has a -intercept of 4 that is parallel to the graph of the line 6. Glencoe/McGraw-Hill Glencoe Algebra

78 5-6 Stud Guide and Intervention (continued) Geometr: Parallel and Perpendicular Lines Perpendicular Lines Two lines are perpendicular if their slopes are negative reciprocals of each other. Vertical and horizontal lines are perpendicular. Eample Write the slope-intercept form for an equation that passes through ( 4, ) and is perpendicular to the graph of 9. Find the slope of 9. 9 riginal equation 9 Divide each side b. Subtract from each side. The slope of is. So, the slope of the line passing through ( 4, ) that is perpendicular to this line is the negative reciprocal of, or. Use the point-slope form to find the equation. m( ) Point-slope form ( ( 4)) m ; (, ) ( 4, ) ( 4) Simplif. 6 Distributive Propert 4 Slope-intercept form Write the slope-intercept form for an equation of the line that passes through the given point and is perpendicular to the graph of each equation.. (4, ),. (, ), 4. (6, 4), ( 8, 7), 8 5. (6, ), 6 6. ( 5, ), 7. ( 9, 5), 8. (, ), 4 9. (6, 6), 6 0. Find an equation of the line that has a -intercept of and is perpendicular to the graph of the line 5.. Find an equation of the line that has a -intercept of 5 and is perpendicular to the graph of the line 4 8. Glencoe/McGraw-Hill Glencoe Algebra