Answers may vary. Sample: The set of all positive numbers is one example.

Transcription

1 6-1 Solving Sstems b Graphing Vocabular Review Write I if the amount described is infinite. Write F if the amount is finite. I 1. the rational numbers greater than 6 F. the number of seats in a movie theater F 3. the number of grams in one kilogram I. the set of odd numbers 5. Give one eample of an infinite amount. Eplain wh the amount is infinite. Answers ma var. Sample: The set of all positive numbers is one eample. It is infinite because there is no end to the set of positive numbers. Vocabular Builder sstem (noun) SIS tum Other Word Forms: sstematic (adjective), sstematize (verb) Main Idea: A sstem of linear equations has two or more linear equations. A solution to this sstem is an ordered pair that makes all of the equations true. Use Your Vocabular Complete each statement with the appropriate word from the list. sstematic sstem sstematize 6. The librarian planned to 9 the donated magazines. 7. The American 9 of government is based on the Constitution. 8. Sam was 9 in his approach to studing for his final eam. 9. A 9 of linear equations might consist of two equations. sstematize sstem sstematic sstem Chapter 6 170

2 Problem 1 Solving a Sstem of Equations b Graphing Got It? What is the solution of the 5 1 sstem? Use a graph. Check our answer Graph both lines on the coordinate plane at the right. 11. Circle the point of intersection of the two lines. (, 0) (0, ) (, 0) (, ) 1. Check that our answer to Eercise 11 makes both equations true O 0 0? ( ) The solution of the sstem is (, 0 ). Problem Writing a Sstem of Equations Got It? One satellite radio service charges $10 per month plus an activation fee of $0. A second service charges $11 per month plus an activation fee of $15. In what month was the cost of the service the same? 1. Complete the model below. Relate Define Write Let c 5 Service 1: Service : total cost total cost c c is 5 5 monthl charge Let n 5 month number.. times number of months plus activation fee $10? n 1 $0 $ Which variable will ou graph on the horizontal ais? Which variable will ou graph on the vertical ais?? n What will the intersection of the graphs of the two linear equations tell ou? $ 15 Answers ma var. Sample: The intersection will show when (in what month) the cost of the services is the same. c n 171 Lesson 6-1

3 17. Graph the equations ou wrote in Eercise 1 on the coordinate grid at the right. 18. Multiple Choice Which ordered pair gives the coordinates of the point of intersection of the two lines? (0, 0) (5, 70) (1, 30) (70, 5) 19. So, in month 5 the total cost of either service is the same. Total Cost ($) c O Number of Months n A sstem of equations that has at least one solution is consistent. A consistent sstem can be either independent or dependent. A consistent sstem that is independent has eactl one solution. A consistent sstem that is dependent has infinitel man solutions. A sstem of equations that has no solution is inconsistent. Underline the correct word, words, or number to complete each sentence. 0. If two lines intersect at one point (have different slopes), the sstem of equations is independent and consistent / inconsistent. The sstem of equations has 0 / 1 / infinitel man solution(s). 1. If two lines are the same (have the same slope and -intercept), the sstem of equations is dependent and consistent / inconsistent. The sstem of equations has 0 / 1 / infinitel man solution(s).. If two lines are parallel (have the same slope and different -intercepts), the sstem of equations is consistent / inconsistent. The sstem of equations has 0 / 1 / infinitel man solution(s). Problem 3 6 Sstems With Infinitel Man Solutions or No Solution Got It? What is the solution of the sstem? Use a graph. 5 3 Describe the number of solutions Multiple Choice Which shows the solution of the two equations? 6 O O O 6 6. The lines are parallel / perpendicular. The sstem has no solution / one solution. Chapter 6 17

4 Lesson Check Do ou know HOW? Solve the sstem b graphing The slope of is 1. The -intercept of is The slope of 5 is 1. The -intercept of 5 is. 7. Graph each line in the sstem on the coordinate grid at the right. 8. The solution of the sstem O is ( 16, 1 ) Lesson Check Do ou UNDERSTAND? Vocabular Draw a line from each tpe of sstem in Column A to the number of solutions the sstem has in Column B. Math Success Check off the vocabular words that ou understand. sstem of linear equations Need to review Column A solution of a sstem of linear equations consistent independent dependent inconsistent Rate how well ou can solve a sstem of linear equations. Column B 9. inconsistent eactl one solution 30. consistent and dependent infinitel man solutions 31. consistent and independent no solution Now I get it! 173 Lesson 6-1

5 6- Solving Sstems Using Substitution Vocabular Review 1. Cross out the epression that does NOT include a variable. 1 9 a b Circle the equation in which the variable is isolated. 8k 5 16 m a z 1 Vocabular Builder substitution (noun) sub stuh TOO shun Related Words: substitute (verb or adjective) Definition: A substitution is something taking the place of something else. Eample: A substitution of for and 8 for in 1 gives 1 8, or 1. Use Your Vocabular Complete each statement with the appropriate form of the word substitution. 3. ADJECTIVE We had a 9 teacher in social studies class toda.. NOUN The coach made a 9 of one plaer for another. 5. VERB To evaluate the epression 1 6, ou can 9 a number for. 6. Write a combination of coins that ou could substitute for each dollar amount. $1.00 $.00 $5.00 Answers ma var. Sample: 10 dimes Answers ma var. Sample: 0 nickels substitute substitution substitute Answers ma var. Sample: 0 quarters You can solve linear sstems b solving one of the equations for one of the variables. Then substitute the epression for the variable into the other equation. This is called the substitution method. Chapter 6 17

6 Problem 1 Using Substitution Got It? What is the solution of the sstem? Use substitution Check our answer Circle the equation that shows a substitution from one equation into the other Now find the value of. 9. Use the value of to find the value of ? (8) The solution is ( 8, 9 ). 11. Check our answer b substituting the values for and in both equations ? ( 8 ) Problem Solving for a Variable and Using Substitution Got It? What is the solution of the sstem? Use substitution Complete the reasoning model below. Think I need to solve one of the equations for one of the variables. Solving the second equation for is quickest. Net I substitute into the other equation. Then I solve the equation for. Now I substitute the -value into either original equation and solve for. Write ( 7 3 ) < 1 3(.8 )5 7 < The solution is about ( 7.,.8). 175 Lesson 6-

7 Problem 3 Using Sstems of Equations Got It? You pa $ to rent 6 video games. The store charges $ for new games and $ for older games. How man new games did ou rent? 1. Define the variables. Let 5 the number of $ games. Let 5 the number of $ games. 15. Complete the models below. Relate total games is number of $ games plus number of $ games Write Relate total cost is cost of $ games plus cost of $ games Write 5? 1? 16. Solve the first equation for Substitute our answer from Eercise 16 to find the -value (6 ) The student rented 5 new ($) games. If ou get an identit, such as 5, when ou solve a sstem of equations, then the sstem has infinitel man solutions. If ou get a false statement, such as 8 5, then the sstem has no solution. Chapter 6 176

8 Problem Sstems with Infinitel Man Solutions or No Solution Got It? How man solutions does the sstem have? Use substitution to solve the sstem of equations. Check students' work. Sample: ( ) I obtained an identit / a false statement, so this sstem of equations has infinitel man / no solutions. Lesson Check Do ou UNDERSTAND? For the sstem, tell which equation ou would first use to solve for a variable in 1 51 the first step of the substitution method. Eplain our choice Each of the equations has been solved for a variable. Eplain which variable ou would choose to solve for and wh. Equation 1 Equation Math Success Check off the vocabular words that ou understand. substitution sstem of equations solution of a sstem Rate how well ou can solve sstems using substitution. Need to review solved for : solved for : solved for : Now I get it! solved for : 5 6 Answers ma var. Sample: Solving the first equation for is eas because its coefficient is Lesson 6-

9 6-3 Solving Sstems Using Elimination Vocabular Review 1. Multiple Choice Which of the following does NOT describe a formula? a rule or principle a fied method for doing something a recipe or prescription a constant. Circle the formula ou can use to find the volume of a rectangular prism. (/ 1 w) πr /hw s 3. Cross out the pairs of equations that are NOT equivalent ; a 1 b 5 17; b 5 17 a m 1 6n 5 10; m 1 3n 5 5 j 1 k 5 11; j 5 k 1 11 Vocabular Builder elimination (noun) ee LIM in a shun Other Word Forms: eliminate (verb), eliminated (verb), eliminating (verb) Definition: Elimination is the act of removing something. Eample: A gardener works toward the elimination of weeds from a garden. Math Usage: In elimination, ou use properties of equalit to add or subtract equations to eliminate a variable in a sstem. Use Your Vocabular Complete each sentence with the word elimination or one of its other word forms.. The cit council will not 9 funding for the public librar. 5. The 9 of the dance portion of the contest was disappointing. 6. The new floor cleaner is great for 9 dirt. eliminate elimination eliminating Chapter 6 178

10 Problem 1 Solving a Sstem b Adding Equations Got It? What is the solution of the sstem? Use elimination Reasoning Wh should our first step be to eliminate? Place a in the bo if the response is correct. Place an if it is incorrect. I can t eliminate. If I eliminate, I can t add the epressions that contain. The epressions 6 and 6 are additive inverses. 8. Circle the equation ou get after ou eliminate Solve the equation for. Use that value to solve for. Solve the equation for. Use the -value to solve for. 8? ? ? The solution of the sstem is (, 7 ). Problem Solving a Sstem b Subtracting Equations Got It? Washing cars and 3 trucks takes 130 min. Washing cars and 5 trucks takes 190 min. How long does it take to wash each tpe of vehicle? 11. Use the equations at the right for this sstem. c 1 3t c 1 5t Let c 5 time to wash a car. Let t 5 time to wash a truck. 1. Eliminate the variable c. Then solve for t. c 1 3t c 1 3t c 1 5t To subtract the second row from the first, change to opposite signs and add down. c 1 5? t ? t 560 t Lesson 6-3

11 13. Use the value of t to solve one of the original equations for c. Sample: c 1 3(30) c c 5 0 c It takes 30 min to wash a truck and 0 min to wash a car. Problem Solving a Sstem b Multipling Both Equations Got It? How can ou use the Multiplication Propert of Equalit to change the equations in this sstem in order to solve it using elimination? 15. How can ou eliminate the same variable from both equations? Eplanations ma var. Multipl the first equation b 3 and the second equation b. Then subtract the equations. 16. Write and solve a revised sstem Multipl b [ 3 ] Multipl b [ ] Subtract the two equations Problem 5 Solve for the variable. Substitute the variable and find the solution. The solution is { fg, f1g } 17. Show that the solution of the revised sstem is a solution of the original sstem. Substitute the solution into each equation () (1) Finding the Number of Solutions Got It? How man solutions does the sstem have? Solve the sstem () 1 3(1) Chapter 6 180

12 19. How man solutions does the sstem have? Eplain. The sstem has no solution. Eplanations ma var. Sample: When I solved the sstem, I got 0 5 5, which is a false statement. Lesson Check Do ou UNDERSTAND? Vocabular If ou add two equations in two variables and the sum is an equation in one variable, what method are ou using to solve the sstem? Eplain. 0. Add each sstem of two linear equations together. A B Which sstem of equations, A or B, has a sum that is an equation in one variable? Sstem A has a sum that is an equation in one variable.. How can ou use the equation in one variable to solve the sstem of equations? Answers ma var. Sample: I can find a value for that variable, and then substitute that value into one of the original equations to find the value of the other variable. 3. When ou use an equation in one variable to solve a sstem of equations, what method are ou using to solve the sstem? Eplain. Answers ma var. Sample: I am using elimination because one variable is eliminated when I add the two equations. Math Success Check off the vocabular words that ou understand. elimination method solution of a sstem of linear equations Rate how well ou can find the solution of a sstem of linear equations b elimination. Need to review Now I get it! Lesson 6-3

13 Applications of Linear Sstems 6- Vocabular Review 1. Multiple Choice Which equation shows what happens when ou use substitution to solve this sstem of equations? ( 1) 5 1 3( 1) ( 1) 5 1 Vocabular Builder intersection (noun) in The intersection of 1 = 3 and = 1 is (, 1). tur sek shun Definition: An intersection is where two or more lines or roads meet. HSM11_A1MC_060_T9109 Use Your Vocabular. The Ancient Spanish Monaster (shown b the star on map) in North Miami Beach, Florida, is thought to be the oldest building in the western hemisphere. Name two streets that intersect at the Ancient Spanish Monaster.. O O Blvd. ne Bisca St 5. 3 rd NE 167th St Hw e ii W.D Write the ordered pair for the point of intersection. If there is no point of intersection, write none. NE NE nd Ave Sample: NE 167th St. and W. Diie Hw. 3.. NE 171st St. HSM11_A1MC_060_T9110 O (0, 1) none (3, ) HSM11_A1MC_060_T9113 HSM11_A1MC_060_T9111 HSM11_A1MC_060_T911 Chapter 6 18 Math Usage: For two lines, the point of intersection is a point in common where the intersect, or meet.

14 Problem 1 Finding a Break-Even Point Got It? A puzzle epert wrote a new sudoku puzzle book. His initial costs are $86. Binding and packaging each book costs $.80. The price of the book is $. How man copies must be sold to break even? 6. Complete the model below. Relate number epenses = $86 1 $.80? of books income = $ sold? number of books sold Define Let = the number of books sold. Let = the number of dollars of epense or income. Write = = 7. Use substitution to solve the sstem of equations Start with the equation for epenses Substitute the income epression for Subtract 0.8 from each side Divide each side b The puzzle epert must sell 70 books to break even. Problem Identifing Constraints and Viable Solutions Got It? The zoo has two water tanks that are leaking. One tank contains 10 gal of water and is leaking at at a constant rate of gal/h. The second tank contains 6 gal of water and is leaking at a constant rate of gal/h. When will the tanks have the same amount of water? Eplain. 9. How much water would leak from Tank 1 in hours? gal in 3 hours? 6 gal How are ou getting our answer? Eplanations ma var. Sample: The rate of the leak is gal/hr, so multipl the rate b the number of hours. What algebraic epression could represent how much water is lost from Tank 1 in hours? What algebraic epression could represent how much water is lost from Tank in hours? 10. If there are 10 gallons in Tank 1, how much will be left in hours? 6 gal in 3 hours? gal How are ou getting our answer? Eplanations ma var. Sample: Subtract the amount leaked in or 3 hours from the total number of gallons. 183 Lesson 6-

15 11. If represents the number of gallons of water left in the tanks after hours, write two equations that represent how much water is left in Tank 1 and Tank after hours. Tank 1: 5 10 Tank : Solve the equations for and b substituting the epression for from the first equation for in the second equation; then find and substitute the value of to find Does the answer make sense? Eplain () 5 1 Eplanations ma var. Sample: No, because it is not possible to have hours. Problem 3 Solving a Wind or Current Problem Got It? You row upstream at a speed of mi/h. You travel the same distance downstream at a speed of 5 mi/h. What would be our rowing speed in still water? What is the speed of the current? 1. Complete the model below. Define Let a 5 speed of the boat in still water. Let c 5 the speed of the current. Relate Write speed of boat in still water 1 a 1 c speed of the current 15. Now solve the sstem of equations. Solutions ma var. Sample: a 1 c 5 5 a c 5 a 5 7 a downstream speed 16. Your rowing speed in still water is 3.5 mi/h. The speed of the current is 1.5 speed of boat in still water speed of the current 5 upstream speed 5 5 a c 5 mi/h. Substitute 3.5 for a in a 1 c c 5 5 c Chapter 6 18

16 Lesson Check Do ou UNDERSTAND? Reasoning Which method would ou use to solve the following sstem? Eplain Draw a line from each statement in Column A to the most appropriate method for solving a sstem of equations in Column B. Column A Column B One equation is alread solved for one of the variables. Substitution Elimination A visual displa of the equations is needed. Graphing The coefficients of one variable are able to be made the same or opposite. 18. Circle the method ou would use to solve the given sstem. Graphing Elimination Substitution 19. Eplain wh ou chose that method. Eplanations ma var. Sample: The equations have coefficients of that _ could be opposites. So, I would multipl the first equation b and the _ second equation b 3 to eliminate. _ Math Success Check off the vocabular words that ou understand. intersection sstem of linear equations substitution elimination Rate how well ou can appl sstems of equations. Need to review Now I get it! Lesson 6-

17 6-5 Linear Inequalities Vocabular Review A solution is a value or values which, when substituted for a variable, make an equation true. Determine the solution of each inequalit below. Circle our answer , 37. m , 0 m. 6 m, 6 3. Multiple Choice How man solutions does this sstem of equations have? infinitel man Vocabular Builder linear inequalit (noun) LIN ee ur in ee KWAL uh tee linear inequalit $ 9 Definition: Replacing the equals sign in a linear equation with an inequalit smbol makes a linear inequalit in two variables. Main Idea: A linear inequalit in two variables has an infinite number of solutions, each an ordered pair that makes the inequalit true. Eample: $ 9 is a linear inequalit in two variables. Use Your Vocabular For Eercises 7, write an inequalit using the smbol and the equation , m $ # Circle the linear inequalities in two variables. 6 S 18 m R 55 3 L 6 K # Chapter 6 186

18 Problem 1 Identifing Solutions of a Linear Inequalit Got It? Is (3, 6) a solution of K 3 1? 9. Complete the steps to determine whether (3, 6) is a solution. # 3? 1 Write the inequalit. 6 3? 3 1 Substitute. 6 1 Multipl. 6 6 Simplif. 10. Is (3, 6) a solution? Yes / No For each inequalit below, determine whether (3, 6) is a solution # 3 6 S S 10 False 6 K 3? 3 6 K 9 6 K 7 True Yes / No Yes / No Problem Graphing an Inequalit in Two Variables Got It? What is the graph of K 1 1 1? Underline the correct word to complete each sentence. 13. The smbol # means greater / less than or equal to. 1. The boundar line will be dashed / solid. 15. Graph the boundar line on the grid. 16. The point (0, 0) is not on the line. Check to see whether that point is a solution of the inequalit. 0 1? K Circle the true statement. Then shade the graph. The side of the line that contains The side of the line that does NOT (0, 0) should be shaded. contain (0, 0) should be shaded. O 187 Lesson 6-5

19 Problem 3 Graphing a Linear Inequalit in One Variable Got It? What is the graph of the inequalit R 5? Of K? 18. Label each graph. O O O O tpe of boundar line dashed solid dashed solid equation of boundar line inequalit S L 3 R K Describe the graph of each inequalit. Use the words boundar line, dashed or solid, and shade or shaded in each description.,5 The boundar line 55 is dashed; the shaded side does not contain (0, 0). # The boundar line 5 will be solid and the shaded side will contain (0, 0). Problem Rewriting to Graph an Inequalit Got It? For a part, ou can spend no more than $1 on nuts. Peanuts cost $/lb. Cashews cost $/lb. What are three possible combinations of peanuts and cashews ou can bu? 0. Let 5 the number of pounds of peanuts and let 5 the number of pounds of cashews. 1. Complete the steps to solve the inequalit for. 1 # 1 Write the inequalit. 1 # 1 1 Subtract the same amount from both sides. # 1 1 Divide each side b the same amount. # Simplif.. Graph the inequalit on the grid. 3. Write three possible combinations of peanuts and cashews that ou can bu. Answers ma var. Sample: lb of peanuts and 1 lb of cashews, 1 lb of peanuts and 1 lb of cashews, or 0 lb of peanuts and 3 lb of cashews Cashews (lb) O Peanuts (lb) Chapter 6 188

20 When a linear inequalit is solved for, the direction of the inequalit smbol determines which side of the boundar line to shade. If the smbol is, or #, shade below the boundar line. If the smbol is. or $, shade above it. Problem 5 Writing an Inequalit From a Graph Got It? You are writing an inequalit from a graph. The boundar line is dashed and has slope 1 3 and -intercept. The area above the line is shaded. What inequalit should ou write?. Write the equation of the boundar line in slope-intercept form. 5. Circle the smbol ou will use in the inequalit , #. $ 6. Now write the complete inequalit. S 1 3 Lesson Check Do ou UNDERSTAND? Writing To graph the inequalit R 3 1 3, do ou shade above or below the boundar line? Eplain. Answers ma var. Samples are given. 7. Describe how to determine which side of the boundar line should be shaded. Test a point that is not on the line, such as (0, 0). If (0, 0) is a solution, then so are all the points on the same side of the boundar line as (0, 0). 8. Do ou shade above or below the boundar line for, 3 1 3? Eplain. I would shade below because (0, 0) is a solution and (0, 0) is below the boundar line. Math Success Check off the vocabular words that ou understand. linear inequalit Need to review solution of an inequalit Rate how well ou can graph a linear inequalit in two variables Now I get it! 189 Lesson 6-5

21 6-6 Sstems of Linear Inequalities Vocabular Review 1. Circle the sstem of equations , 10. How can ou tell if an ordered pair is a solution to a sstem of equations? Answers ma var. Sample: An ordered pair is a solution to a sstem of equations if it makes all of the equations true. Vocabular Builder boundar line bown dree Related Words: bounds (noun or verb), border (noun or verb), bounded (adjective) O Definition: A boundar is something that divides one item from another. Math Usage: The graph of a linear inequalit in two variables is a region bounded b a line. All points on one side of the boundar line are solutions. HSM11_A1MC_0606_T91 Use Your Vocabular 3. Complete each sentence with bounds, boundar, or border. The landscaper planted a 9 of flowers along the sidewalk. border The ball bounced out of 9. bounds The 9 line on the map between countries is shown. boundar. Circle the ordered pair that is on the boundar of 1 # 5. (9, 0) Chapter 6 (0, 6) (6, 1) (10, 5) 190 (5, 3) boundar (noun)

22 Problem 1 Graphing a Sstem of Inequalities Got It? What is the graph of the sstem? L K 5. The boundar line for $ 1 5 will be solid / dashed. The boundar line for 3 1 # will be solid / dashed. 6. Write the inequalit 3 1 # in slope-intercept form. # 3? 7. Graph the inequalities on the coordinate grid at the right. 8 8 O 8 8 Problem Writing a Sstem of Inequalities From a Graph Got It? What sstem of inequalities is represented b the graph? O 8. Find the inequalit shown b the dashed line. 9. Find the inequalit shown b the solid line. Determine the equation of the dashed line The shaded region is above / below the Determine the equation of the solid line. The shaded region is above / below the line. The boundar line is solid / dashed. line. The boundar line is solid / dashed. Circle the inequalit smbol ou should use to write the inequalit for this graph. Circle the inequalit smbol ou should use to write the inequalit for this graph.., # $., # $ 10. Write the sstem of inequalities represented b the graph., # Lesson 6-6

23 Problem 3 Using a Sstem of Inequalities Got It? You want to build a fence for a rectangular dog run. You want the run to be at least 10 ft wide. The run can be at most 50 ft long. You have 16 ft of fencing. What is a graph showing the possible dimensions of the dog run? Remember, a dog run must have fencing on all four sides. 11. Follow the steps to write a sstem of inequalities and graph it. 1 Define what each variable represents. Let 5 the width of the dog run. Let 5 the length of the dog run. Write a sstem of inequalities In slope-intercept form, this is Graph the sstem of inequalities Wh is the graph onl in the first quadrant? Length (ft) O Width (ft) Answers ma var. Sample: The graph is onl in the first quadrant because the values are about length, and length cannot be negative. 13. Use the graph to write possible dimensions of the dog run. Answers ma var. Samples given. width (): 0 feet width (): 30 feet width (): 5 feet length (): 3 feet length (): 30 feet length (): 38 feet Chapter 6 19

24 Lesson Check Do ou UNDERSTAND? Reasoning Suppose ou are graphing a sstem of two linear inequalities, and the boundar lines for the inequalities are parallel. Does that mean that the sstem has no solution? Eplain. 1. Draw a line from each graph to the corresponding sstem of linear inequalities. # $3 1 6 $3 1 6 $ #3 1 3 $ Which of the sstems have solutions? The sstems shown b the second graph and the third graph have solutions. The sstem shown b the first graph does not. 16. Would the sstem of # and #3 1 3 have a solution? If so, describe the solution. Yes. Answers ma var. Sample: The solution would be K Is it possible for a sstem of two linear inequalities with parallel boundar lines to have a solution? 18. Is it possible for a sstem of two linear inequalities with parallel boundar lines to have no solution? Math Success Check off the vocabular words that ou understand. sstem of linear inequalities Rate how well ou can solve a sstem of linear inequalities. Need to review solution of a sstem of linear inequalities Now I get it! Yes / No Yes / No 193 Lesson 6-6