Systems of Equations and Inequalities 77.1 Solve Linear Systems by Graphing

Transcription

1 Sstems of Equations and Inequalities 77. Solve Linear Sstems b Graphing 7.2 Solve Linear Sstems b Substitution 7.3 Solve Linear Sstems b Adding or Subtracting 7.4 Solve Linear Sstems b Multipling First 7.5 Solve Special Tpes of Linear Sstems 7.6 Solve Sstems of Linear Inequalities Before In previous chapters, ou learned the following skills, which ou ll use in Chapter 7: graphing linear equations, solving equations, determining whether lines are parallel, and graphing linear inequalities in two variables. Prerequisite Skills VOCABULARY CHECK Cop and complete the statement.. The least common multiple of 0 and 5 is?. 2. Two lines in the same plane are? if the do not intersect. SKILLS CHECK Graph the equation. (Review p. 225 for 7..) Solve the equation. (Review p. 48 for ) 7. 5m 4 2 m (z 5) z 5 6 Tell whether the graphs of the two equations are parallel lines. Eplain our reasoning. (Review p. 244 for 7.5.) , , , , Graph the inequalit. (Review p. 405 for 7.6.) < > 3 424

2 Now In Chapter 7, ou will appl the big ideas listed below and reviewed in the Chapter Summar on page 474. You will also use the ke vocabular listed below. Big Ideas Solving linear sstems b graphing 2 Solving linear sstems using algebra 3 Solving sstems of linear inequalities KEY VOCABULARY sstem of linear equations, p. 427 solution of a sstem of linear equations, p. 427 consistent independent sstem, p. 427 inconsistent sstem, p. 459 consistent dependent sstem, p. 459 sstem of linear inequalities, p. 466 solution of a sstem of linear inequalities, p. 466 graph of a sstem of linear inequalities, p. 466 Wh? You can use a sstem of linear equations to solve problems about traveling with and against a current. For eample, ou can write and solve a sstem of linear equations to find the average speed of a kaak in still water. Algebra The animation illustrated below for Eample 4 on page 446 helps ou answer this question: What is the average speed of the kaak in still water? You have to find the speed of the kaak in still water. Click the button that will produce an equation in one variable. Algebra at classzone.com Other animations for Chapter 7: pages 428, 435, 44, 446, 452, 459, and

3 Investigating Algebra ACTIVITY Use before Lesson Solving Linear Sstems Using Tables MATERIALS pencil and paper Q UESTION How can ou use a table to solve a linear sstem? A sstem of linear equations, or linear sstem, consists of two or more linear equations in the same variables. A solution of a linear sstem is an ordered pair that satisfies each equation in the sstem. You can use a table to find a solution to a linear sstem. EXPLORE Solve a linear sstem Bill and his brother collect comic books. Bill currentl has 5 books and adds 2 books to his collection ever month. His brother currentl has 7 books and adds 4 books to his collection ever month. Use the equations below to find the number of months after which Bill and his brother will have the same number of comic books in their collections Number of comic books in Bill s collection Number of comic books in his brother s collection STEP Make a table Cop and complete the table of values shown. STEP 2 Find a solution Find an -value that gives the same -value for both equations. STEP 3 Interpret the solution Use our answer to Step 2 to find the number of months after which Bill and his brother have the same number of comic books ?? 2?? 3?? 4?? 5?? DRAW CONCLUSIONS Use our observations to complete these eercises. When Bill and his brother have the same number of books in their collections, how man books will each of them have? 2. Graph the equations above on the same coordinate plane. What do ou notice about the graphs and the solution ou found above? Use a table to solve the linear sstem Chapter 7 Sstems of Equations and Inequalities

4 7. Solve Linear Sstems b Graphing Before You graphed linear equations. Now You will graph and solve sstems of linear equations. Wh? So ou can analze craft fair sales, as in E. 33. Ke Vocabular sstem of linear equations solution of a sstem of linear equations consistent independent sstem A sstem of linear equations, or simpl a linear sstem, consists of two or more linear equations in the same variables. An eample is shown below Equation Equation 2 A solution of a sstem of linear equations in two variables is an ordered pair that satisfies each equation in the sstem. One wa to find the solution of a linear sstem is b graphing. If the lines intersect in a single point, then the coordinates of the point are the solution of the linear sstem. A solution found using graphical methods should be checked algebraicall. E XAMPLE Check the intersection point Use the graph to solve the sstem. Then check our solution algebraicall Equation Equation 2 Solution The lines appear to intersect at the point (3, 2). CHECK Substitute 3 for and 2 for in each equation (2) 0 7 3(3) 2 2(2) c Because the ordered pair (3, 2) is a solution of each equation, it is a solution of the sstem. TYPES OF LINEAR SYSTEMS In Eample, the linear sstem has eactl one solution. A linear sstem that has eactl one solution is called a consistent independent sstem because the lines are distinct (are independent) and intersect (are consistent). You will solve consistent independent sstems in Lessons In Lesson 7.5 ou will consider other tpes of sstems. 7. Solve Linear Sstems b Graphing 427

5 KEY CONCEPT For Your Notebook Solving a Linear Sstem Using the Graph-and-Check Method STEP Graph both equations in the same coordinate plane. For ease of graphing, ou ma want to write each equation in slope-intercept form. STEP 2 Estimate the coordinates of the point of intersection. STEP 3 Check the coordinates algebraicall b substituting into each equation of the original linear sstem. E XAMPLE 2 Use the graph-and-check method Solve the linear sstem: Equation Equation 2 Solution STEP Graph both equations STEP 2 Estimate the point of intersection. The two lines appear to intersect at (4, 23). STEP 3 Check whether (4, 23) is a solution b substituting 4 for and 23 for in each of the original equations. Equation Equation (4) + (23) (23) c Because (4, 23) is a solution of each equation, it is a solution of the linear sstem. at classzone.com GUIDED PRACTICE for Eamples and 2 Solve the linear sstem b graphing. Check our solution Chapter 7 Sstems of Equations and Inequalities

6 E XAMPLE 3 Standardized Test Practice ELIMINATE CHOICES You can eliminate choice A because neither of the equations include the cost of a season pass. The parks and recreation department in our town offers a season pass for $90. As a season pass holder, ou pa $4 per session to use the town s tennis courts. Without the season pass, ou pa $3 per session to use the tennis courts. Which sstem of equations can be used to find the number of sessions of tennis after which the total cost with a season pass, including the cost of the pass, is the same as the total cost without a season pass? A 5 4 B C 5 3 D Solution Write a sstem of equations where is the total cost (in dollars) for sessions. EQUATION Total cost (dollars) 5 Cost per session (dollars/session) p Number of sessions (sessions) EQUATION p Total cost (dollars) 5 Cost for season pass (dollars) Cost per session (dollars/session) p Number of sessions (sessions) p c The correct answer is C. A B C D GUIDED PRACTICE for Eample 3 4. Solve the linear sstem in Eample 3 to find the number of sessions after which the total cost with a season pass, including the cost of the pass, is the same as the total cost without a season pass. 5. WHAT IF? In Eample 3, suppose a season pass costs $35. After how man sessions is the total cost with a season pass, including the cost of the pass, the same as the total cost without a season pass? 7. Solve Linear Sstems b Graphing 429

7 E XAMPLE 4 Solve a multi-step problem RENTAL BUSINESS A business rents in-line skates and biccles. During one da, the business has a total of 25 rentals and collects $450 for the rentals. Find the number of pairs of skates rented and the number of biccles rented. Solution STEP Write a linear sstem. Let be the number of pairs of skates rented, and let be the number of biccles rented Equation for number of rentals Equation for mone collected from rentals STEP 2 Graph both equations. STEP 3 Estimate the point of intersection. The two lines appear to intersect at (20, 5). STEP 4 Check whether (20, 5) is a solution (20) 30(5) c The business rented 20 pairs of skates and 5 biccles. GUIDED PRACTICE for Eample 4 6. WHAT IF? In Eample 4, suppose the business has a total of 20 rentals and collects $420. Find the number of biccles rented. 7. EXERCISES HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 5 and 3 5 STANDARDIZED TEST PRACTICE Es. 2, 6, 7, 27, 28, 29, and 32 5 MULTIPLE REPRESENTATIONS E. 35 SKILL PRACTICE. VOCABULARY Cop and complete: A(n)? of a sstem of linear equations in two variables is an ordered pair that satisfies each equation in the sstem. 2. WRITING Eplain how to use the graph-and-check method to solve a linear sstem of two equations in two variables. CHECKING SOLUTIONS Tell whether the ordered pair is a solution of the linear sstem. 3. (23, ); 4. (5, 2); 5. (22, ); Chapter 7 Sstems of Equations and Inequalities

8 EXAMPLE on p. 427 for Es MULTIPLE CHOICE Which ordered pair is a solution of the linear sstem 522 and ? A (22, 0) B (0, 22) C (2, 0) D (0, 2) 7. MULTIPLE CHOICE Which ordered pair is a solution of the linear sstem and ? A (23, 3) B (23, 6) C (3, 3) D (3, 6) SOLVING SYSTEMS GRAPHICALLY Use the graph to solve the linear sstem. Check our solution ERROR ANALYSIS Describe and correct the error in solving the linear sstem below Equation Equation = 3 3 = 6 The solution is (3, 2). EXAMPLE 2 on p. 428 for Es GRAPH-AND-CHECK METHOD Solve the linear sstem b graphing. Check our solution } } 5 3 } 25. } 2 } 2 52} } 3 5 } } OPEN ENDED Find values for m and b so that the sstem 5 3 } 5 2 and 5 m b has (5, 2) as a solution. 28. WRITING Solve the linear sstem shown b graphing. Eplain wh it is important to check our solution Equation Equation 2 7. Solve Linear Sstems b Graphing 43

9 29. EXTENDED RESPONSE Consider the equation 2} 6 5 } a. Solve the equation using algebra. b. Solve the linear sstem below using a graph. 52} 6 Equation 4 5 } 3 Equation 2 2 c. How is the linear sstem in part (b) related to the original equation? d. Eplain how to use a graph to solve the equation 2} } CHALLENGE The three lines given below form a triangle. Find the coordinates of the vertices of the triangle. Line : Line 2: 2 5 Line 3: PROBLEM SOLVING EXAMPLES 3 and 4 on pp for Es TELEVISION The graph shows a projection, from 990 on, of the percent of eighth graders who watch hour or less of television on a weekda and the percent of eighth graders who watch more than hour of television on a weekda. Use the graph to predict the ear when the percent of eighth graders who watch hour or less will equal the percent who watch more than hour. Percent of eighth graders p hour or less more than hour t Years since MULTIPLE CHOICE A car dealership is offering interest-free car loans for one da onl. During this da, a salesperson at the dealership sells two cars. One of his clients decides to pa off his $7,424 car in 36 monthl paments of $484. His other client decides to pa off his $5,840 car in 48 monthl paments of $330. Which sstem of equations can be used to determine the number of months after which both clients will have the same loan balance? A B , ,840 C ,840 D , , , CRAFTS Kirigami is the Japanese art of making paper designs b folding and cutting paper. A student sells small and large greeting cards decorated with kirigami at a craft fair. The small cards cost $3 per card, and the large cards cost $5 per card. The student collects $95 for selling a total of 25 cards. How man of each tpe of card did the student sell? WORKED-OUT SOLUTIONS on p. WS 5 STANDARDIZED TEST PRACTICE 5 MULTIPLE REPRESENTATIONS

10 34. FITNESS You want to burn 225 calories while eercising at a gm. The number of calories that ou burn per minute on different machines at the gm is shown below. Stair machine Elliptical trainer Stationar bike You burn 5 Cal/min. You burn 8 Cal/min. You burn 6 Cal/min. 35. a. Suppose ou have 40 minutes to eercise at the gm and ou want to use the stair machine and stationar bike. How man minutes should ou spend on each machine so that ou burn 225 calories? b. Suppose ou have 30 minutes to eercise at the gm and ou want to use the stair machine and the elliptical trainer. How man minutes should ou spend on each machine so that ou burn 225 calories? MULTIPLE REPRESENTATIONS It costs $5 for a earl membership to a movie club at a movie theater. A movie ticket costs $5 for club members and $8 for nonmembers. a. Writing a Sstem of Equations Write a sstem of equations that ou can use to find the number of movies viewed after which the total cost for a club member, including the membership fee, is the same as the cost for a nonmember. b. Making a Table Make a table of values that shows the total cost for a club member and a nonmember after paing to see, 2, 3, 4, 5, and 6 movies. c. Drawing a Graph Use the table to graph the sstem of equations. Under what circumstances does it make sense to become a movie club member? Eplain our answer b using the graph. 36. CHALLENGE With a minimum purchase of $25, ou can open a credit account with a clothing store. The store is offering either $25 or 20% off of our purchase if ou open a credit account. You decide to open a credit account. Should ou choose $25 or 20% off of our purchase? Eplain. MIXED REVIEW Solve the equation (p. 34) (p. 4) 39. 3( 2) 526 (p. 48) (p. 54) (p. 54) 42. 4( ) (p. 54) PREVIEW Prepare for Lesson 7.2 in Es Write the equation so that is a function of. (p. 84) EXTRA PRACTICE for Lesson 7., p. 944 ONLINE 7. QUIZ at classzone.com 433

11 Graphing Calculator ACTIVITY 7. Solving Linear Sstems b Graphing Use after Lesson 7. classzone.com Kestrokes Q UESTION How can ou use a graphing calculator to solve a linear sstem? E XAMPLE Solve a linear sstem Solve the linear sstem using a graphing calculator Equation Equation 2 STEP Rewrite equations Solve each equation for. STEP 2 Enter equations Press and enter the equations. Equation Equation } } } Y=-(5/2)X+3 Y2=(/3)X+(5/3) Y3= Y4= Y5= Y6= Y7= STEP 3 Displa graph Graph the equations using a standard viewing window. STEP 4 Find point of intersection Use the intersect feature to find the point where the graphs intersect. X= Y= The solution is about (0.47,.8). P RACTICE Solve the linear sstem using a graphing calculator Chapter 7 Sstems of Equations and Inequalities

12 7.2 Solve Linear Sstems b Substitution Before You solved sstems of linear equations b graphing. Now You will solve sstems of linear equations b substitution. Wh? So ou can find tubing costs, as in E. 32. Ke Vocabular sstem of linear equations, p. 427 KEY CONCEPT Solving a Linear Sstem Using the Substitution Method STEP For Your Notebook Solve one of the equations for one of its variables. When possible, solve for a variable that has a coefficient of or 2. STEP 2 Substitute the epression from Step into the other equation and solve for the other variable. STEP 3 Substitute the value from Step 2 into the revised equation from Step and solve. E XAMPLE Use the substitution method Solve the linear sstem: Equation 2 5 Equation 2 Solution STEP Solve for. Equation is alread solved for. STEP 2 Substitute 3 2 for in Equation 2 and solve for. 2 5 Write Equation 2. 2(3 2) 5 Substitute 3 2 for Simplif. Subtract 4 from each side. 5 Divide each side b 7. STEP 3 Substitute for in the original Equation to find the value of () c The solution is (, 5). CHECK Substitute for and 5 for in each of the original equations () 2 2(5) at classzone.com 7.2 Solve Linear Sstems b Substitution 435

13 E XAMPLE 2 Use the substitution method Solve the linear sstem: Equation Equation 2 CHOOSE AN EQUATION Equation was chosen in Step because has a coefficient of. So, onl one step is needed to solve Equation for. Solution STEP Solve Equation for Write original Equation Revised Equation STEP 2 Substitute for in Equation 2 and solve for Write Equation 2. 4(2 2 6) Substitute 2 6 for Distributive propert Simplif. Add 24 to each side. 5 2 Divide each side b 4. STEP 3 Substitute 2 for in the revised Equation to find the value of Revised Equation 5 2(2) 2 6 Substitute 2 for. 522 Simplif. c The solution is (22, 2). CHECK Substitute 22 for and 2 for in each of the original equations. Equation Equation (2) (22) 6(2) CHECK REASONABLENESS When solving a linear sstem using the substitution method, ou can use a graph to check the reasonableness of our solution. For eample, the graph at the right verifies that (22, 2) is a solution of the linear sstem in Eample GUIDED PRACTICE for Eamples and 2 Solve the linear sstem using the substitution method Chapter 7 Sstems of Equations and Inequalities

14 E XAMPLE 3 Solve a multi-step problem ANOTHER WAY For an alternative method for solving the problem in Eample 3, turn to page 442 for the Problem Solving Workshop. WEBSITES Man businesses pa website hosting companies to store and maintain the computer files that make up their websites. Internet service providers also offer website hosting. The costs for website hosting offered b a website hosting compan and an Internet service provider are shown in the table. Find the number of months after which the total cost for website hosting will be the same for both companies. Compan Set-up fee (dollars) Cost per month (dollars) Internet service provider Website hosting compan None Solution STEP Write a sstem of equations. Let be the total cost after months. Equation : Internet service provider Total cost 5 Set-up fee Cost per month p Number of months p Equation 2: Website hosting compan Total cost 5 Cost per month p Number of months p The sstem of equations is: Equation Equation 2 STEP 2 Substitute for in Equation and solve for Write Equation Substitute for Subtract 2.95 from each side Divide each side b 0.5. c The total cost will be the same for both companies after 20 months. GUIDED PRACTICE for Eample 3 4. In Eample 3, what is the total cost for website hosting for each compan after 20 months? 5. WHAT IF? In Eample 3, suppose the Internet service provider offers $5 off the set-up fee. After how man months will the total cost for website hosting be the same for both companies? 7.2 Solve Linear Sstems b Substitution 437

15 E XAMPLE 4 Solve a miture problem ANTIFREEZE For etremel cold temperatures, an automobile manufacturer recommends that a 70% antifreeze and 30% water mi be used in the cooling sstem of a car. How man quarts of pure (00%) antifreeze and a 50% antifreeze and 50% water mi should be combined to make quarts of a 70% antifreeze and 30% water mi? Solution STEP Write an equation for the total number of quarts and an equation for the number of quarts of antifreeze. Let be the number of quarts of 00% antifreeze, and let be the number of quarts a 50% antifreeze and 50% water mi. Equation : Total number of quarts 5 Equation 2: Number of quarts of antifreeze quarts of 00% antifreeze quarts of 50% 50% mi quarts of 70% 30% mi DRAW A DIAGRAM Each bar shows the liquid in each mi. The green portion shows the percent of the mi that is antifreeze. 00% 5 50% 70% p 0.5 p 5 0.7() The sstem of equations is: 5 Equation STEP 2 Solve Equation for. 5 Write Equation Equation Revised Equation STEP 3 Substitute 2 for in Equation 2 and solve for Write Equation 2. ( ) Substitute 2 for Solve for. STEP 4 Substitute 6.6 for in the revised Equation to find the value of c Mi 4.4 quarts of 00% antifreeze and 6.6 quarts of a 50% antifreeze and 50% water mi to get quarts of a 70% antifreeze and 30% water mi. GUIDED PRACTICE for Eample 4 6. WHAT IF? How man quarts of 00% antifreeze and a 50% antifreeze and 50% water mi should be combined to make 6 quarts of a 70% antifreeze and 30% water mi? 438 Chapter 7 Sstems of Equations and Inequalities

16 7.2 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 3 and 33 5 STANDARDIZED TEST PRACTICE Es. 2, 8, 29, 33, and 37. VOCABULARY Give an eample of a sstem of linear equations. 2. WRITING If ou are solving the linear sstem shown using the substitution method, which equation would ou solve for which variable? Eplain Equation Equation 2 EXAMPLE on p. 435 for Es. 3 8 EXAMPLE 2 on p. 436 for Es. 9 9 SOLVING LINEAR SYSTEMS Solve the linear sstem using substitution MULTIPLE CHOICE Which ordered pair is a solution of the linear sstem and ? A (6, 7) B (7, 6) C (7, ) D (, 7) 9. ERROR ANALYSIS Describe and correct the error in solving the linear sstem and Step Step 2 Step 3 The solution (9 2 3) is (6, ) SOLVING LINEAR SYSTEMS Solve the linear sstem using substitution } 2 } } } 8 3 } } } } 3 } Solve Linear Sstems b Substitution 439

17 29. WRITING Suppose ou solve a linear sstem using substitution. Eplain how ou can use a graph to check our solution. 30. CHALLENGE Find values of a and b so that the linear sstem shown has a solution of (29, 4). a b 526 Equation a 2 b 5256 Equation 2 PROBLEM SOLVING EXAMPLE 3 on p. 437 for Es FUNDRAISING During a football game, the parents of the football plaers sell pretzels and popcorn to raise mone for new uniforms. The charge $2.50 for a bag of popcorn and $2 for a pretzel. The parents collect $336 in sales during the game. The sell twice as man bags of popcorn as pretzels. How man bags of popcorn do the sell? How man pretzels do the sell? 32. TUBING COSTS The members of an outing club take a da-long tubing trip down a river. The compan that offers the tubing trip charges $5 to rent a tube for a person to use and $7.50 to rent a cooler tube, which is used to carr food and water in a cooler. The club members spend $360 to rent a total of 26 tubes. How man of each tpe of tube do the rent? 33. SHORT RESPONSE In the mobile shown, objects are attached to each end of a dowel. For the dowel to balance, the following must be true: p Weight hanging from point A 5 p Weight hanging from point B The weight of the objects hanging from point A is.5 pounds, and the weight of the objects hanging from point B is.2 pounds. The length of the dowel is 9 inches. How far from point A should the string be placed? Eplain. 34. MULTI-STEP PROBLEM Two swimming teams are competing in a 400 meter medle rela. During the last leg of the race, the swimmer in lane has a.2 second head start on the swimmer in lane 2, as shown. 2 Swimming at.8 m/sec with a.2 sec head start Swimming at.9 m/sec a. Let t be the time since the swimmer in lane 2 started the last leg. After how man seconds into the leg will the swimmer in lane 2 catch up to the swimmer in lane? b. Does the swimmer in lane 2 catch up to the swimmer in lane before the race ends? Eplain WORKED-OUT SOLUTIONS on p. WS 5 STANDARDIZED TEST PRACTICE

18 EXAMPLE 4 on p. 438 for E CHEMISTRY In our chemistr lab, ou have a bottle of % hdrochloric acid solution and a bottle of 5% hdrochloric acid solution. You need 00 milliliters of a 3% hdrochloric acid solution for an eperiment. How man milliliters of each solution do ou need to mi together? 36. MONEY Laura has $4.50 in dimes and quarters. She has 3 more dimes than quarters. How man quarters does she have? 37. SHORT RESPONSE A gazelle can run 73 feet per second for several minutes. A cheetah can run 88 feet per second, but it can sustain this speed for onl 20 seconds. A gazelle is 350 feet from a cheetah when both animals start running. Can the gazelle sta ahead of the cheetah? Eplain. at classzone.com 38. CHALLENGE A gardener needs 6 bushels of a potting medium of 40% peat moss and 60% vermiculite. He decides to add 00% vermiculite to his current potting medium that is 50% peat moss and 50% vermiculite. The gardener has 5 bushels of the 50% peat moss and 50% vermiculite mi. Does he have enough of the 50% peat moss and 50% vermiculite mi to make 6 bushels of the 40% peat moss and 60% vermiculite mi? Eplain. MIXED REVIEW Solve the proportion. Check our solution } 7 5 } 2 (p. 62) } q 5 5 } 3 (p. 68) 43. } } 0 (p. 62) 4. 3 } 5 } 26 (p. 68) 44. 2r 6 2 } 5 } 3z (p. 62) } 3 5 s } s 2 2 (p. 68) PREVIEW Prepare for Lesson 7.3 in Es Write two equations in standard form that are equivalent to the given equation. (p. 3) QUIZ for Lessons Solve the linear sstem b graphing. Check our solution. (p. 427) Solve the linear sstem using substitution. (p. 435) EXTRA PRACTICE for Lesson 7.2, p. 944 ONLINE QUIZ at classzone.com 44

19 LESSON 7.2 Using ALTERNATIVE METHODS Another Wa to Solve Eample 3, page 437 MULTIPLE REPRESENTATIONS In Eample 3 on page 437, ou saw how to solve the problem about website hosting b solving a linear sstem algebraicall. You can also solve the problem using a table. P ROBLEM WEBSITES Man businesses pa website hosting companies to store and maintain the computer files that make up their websites. Internet service providers also offer website hosting. The costs for website hosting offered b a website hosting compan and an Internet service provider are shown in the table. Find the number of months after which the total cost for website hosting will be the same for both companies. Compan Set-up fee Cost per month Internet service provider $0 $2.95 Website hosting compan None $22.45 M ETHOD Making a Table An alternative approach is to make a table. STEP Make a table for the total cost of website hosting for both companies. Include the set-up fee in the cost for the first month. STEP 2 Look for the month in which the total cost of the service from the Internet service provider and the website hosting compan is the same. This happens after 20 months. Months Internet service provider Website hosting compan $3.95 $ $53.90 $ $75.85 $67.35 A A A 9 $ $ $ $ $ $47.45 P RACTICE. TAXIS A tai compan charges $2.80 for the first mile and $.60 for each additional mile. Another tai compan charges $3.20 for the first mile and $.50 for each additional mile. After how man miles will each tai cost the same? Use a table to solve the problem. 2. SCHOOL PLAY An adult ticket to a school pla costs $5 and a student ticket costs $3. A total of $460 was collected from the sale of 20 tickets. How man student tickets were purchased? Solve the problem using algebra. Then use a table to check our answer. 442 Chapter 7 Sstems of Equations and Inequalities

20 Investigating g Algebra ACTIVITY Use before Lesson Linear Sstems and Elimination MATERIALS algebra tiles Q UESTION How can ou solve a linear sstem using algebra tiles? You can use the following algebra tiles to model equations. -tiles -tiles -tiles E XPLORE Solve a linear sstem using algebra tiles. Solve the linear sstem: Equation 5 3 Equation 2 STEP Model equations Model each equation using algebra tiles. Arrange the algebra tiles so that one equation is directl below the other equation. STEP 2 Add equations Combine the two equations to form one equation. Notice that the new equation has one positive -tile and one negative -tile. The -tiles can be removed because the pair of -tiles has a value of 0. STEP 3 Solve for Divide the remaining tiles into four equal groups. Each -tile is equal to two -tiles. So, STEP 4 Solve for To find the value of, use the model for Equation 2. Because 5 2, ou can replace the -tile with two -tiles. Solve the new equation for. So 5, and the solution of the sstem is (2, ). 5 DRAW CONCLUSIONS Use our observations to complete these eercises Use algebra tiles to model and solve the linear sstem REASONING Is it possible to solve the linear sstem and 2 5 using the steps shown above? Eplain our reasoning. 7.3 Solve Linear Sstems b Adding or Subtracting 443

21 7.3 Solve Linear Sstems b Adding or Subtracting Before You solved linear sstems b graphing and using substitution. Now You will solve linear sstems using elimination. Wh? So ou can solve a problem about arranging flowers, as in E. 42. Ke Vocabular sstem of linear equations, p. 427 When solving a linear sstem, ou can sometimes add or subtract the equations to obtain a new equation in one variable. This method is called elimination. KEY CONCEPT For Your Notebook Solving a Linear Sstem Using the Elimination Method STEP Add or subtract the equations to eliminate one variable. STEP 2 Solve the resulting equation for the other variable. STEP 3 Substitute in either original equation to find the value of the eliminated variable. E XAMPLE Use addition to eliminate a variable Solve the linear sstem: Equation Equation 2 ADD EQUATIONS When the coefficients of one variable are opposites, add the equations to eliminate the variable. Solution STEP Add the equations to eliminate one variable. STEP 2 Solve for STEP 3 Substitute 3 for in either equation and solve for Write Equation. 2 3(3) 5 Substitute 3 for. 5 Solve for. c The solution is (, 3). CHECK Substitute for and 3 for in each of the original equations () 3(3) 0 22() 5(3) Chapter 7 Sstems of Equations and Inequalities

22 E XAMPLE 2 Use subtraction to eliminate a variable Solve the linear sstem: Equation Equation 2 SUBTRACT EQUATIONS When the coefficients of one variable are the same, subtract the equations to eliminate the variable. Solution STEP Subtract the equations to eliminate one variable. STEP 2 Solve for STEP 3 Substitute 24 for in either equation and solve for Write Equation. 4(24) Substitute 24 for. 5 6 Solve for. c The solution is (24, 6). E XAMPLE 3 Arrange like terms Solve the linear sstem: Equation Equation 2 AVOID ERRORS Make sure that the equal signs are in the same column, just as the like terms are. Solution STEP Rewrite Equation 2 so that the like terms are arranged in columns STEP 2 Add the equations. STEP 3 Solve for STEP 4 Substitute 2 for in either equation and solve for Write Equation (2) 4 Substitute 2 for. 5 5 Solve for. c The solution is (2, 5). GUIDED PRACTICE for Eamples, 2, and 3 Solve the linear sstem Solve Linear Sstems b Adding or Subtracting 445

23 E XAMPLE 4 Write and solve a linear sstem KAYAKING During a kaaking trip, a kaaker travels 2 miles upstream (against the current) and 2 miles downstream (with the current), as shown. The speed of the current remained constant during the trip. Find the average speed of the kaak in still water and the speed of the current. Upstream: 3 hours DIRECTION OF CURRENT Downstream: 2 hours COMBINE SPEEDS When ou go upstream, the speed at which ou can travel in still water is decreased b the speed of the current. The opposite is true when ou go downstream. STEP Write a sstem of equations. First find the speed of the kaak going upstream and the speed of the kaak going downstream. Upstream: d 5 rt Downstream: d 5 rt 2 5 r p r p r 6 5 r Use the speeds to write a linear sstem. Let be the average speed of the kaak in still water, and let be the speed of the current. Equation : Going upstream Speed of kaak in still water 2 Speed of current 5 Speed of kaak going upstream Equation 2: Going downstream Speed of kaak in still water Speed of current 5 Speed of kaak going downstream 5 6 STEP 2 Solve the sstem of equations Write Equation. 5 6 Write Equation Add equations. 5 5 Solve for. Substitute5for in Equation 2 and solve for Substitute 5 for in Equation 2. 5 Subtract 5 from each side. c The average speed of the kaak in still water is 5 miles per hour, and the speed of the current is mile per hour. at classzone.com 446 Chapter 7 Sstems of Equations and Inequalities

24 GUIDED PRACTICE for Eample 4 7. WHAT IF? In Eample 4, suppose it takes the kaaker 5 hours to travel 0 miles upstream and 2 hours to travel 0 miles downstream. The speed of the current remains constant during the trip. Find the average speed of the kaak in still water and the speed of the current. 7.3 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 7 and 4 5 STANDARDIZED TEST PRACTICE Es. 2, 5, 22, 36, and 44 5 MULTIPLE REPRESENTATIONS E. 42. VOCABULARY Give an eample of a linear sstem in two variables that can be solved b first adding the equations to eliminate one variable. 2. WRITING Eplain how to solve the linear sstem shown using the elimination method Equation Equation 2 EXAMPLE on p. 444 for Es. 3 8 EXAMPLE 2 on p. 445 for Es. 9 5 USING ADDITION Solve the linear sstem using elimination USING SUBTRACTION Solve the linear sstem using elimination MULTIPLE CHOICE Which ordered pair is a solution of the linear sstem and ? A (22, 4) B (2, 24) C (4, 22) D (4, 2) EXAMPLE 3 on p. 445 for Es ARRANGING LIKE TERMS Solve the linear sstem using elimination MULTIPLE CHOICE Which ordered pair is a solution of the linear sstem and ? A (23, 24) B (3, 4) C (24, 3) D (4, 3) 7.3 Solve Linear Sstems b Adding or Subtracting 447

25 ERROR ANALYSIS Describe and correct the error in finding the value of one of the variables in the given linear sstem SOLVING LINEAR SYSTEMS Solve the linear sstem using elimination } }2 } } } } } 4 2 } } } } } WRITING AN EQUATION OF A LINE Use the following steps to write an equation of the line that passes through the points (, 2) and (24, 2). a. Write a sstem of linear equations b substituting for and 2 for in 5 m b and 24 for and 2 for in 5 m b. b. Solve the sstem of linear equations from part (a). What is the slope of the line? What is the -intercept? c. Write an equation of the line that passes through (, 2) and (24, 2). 35. GEOMETRY The rectangle has a perimeter P of 4 feet, and twice its length l is equal to less than 4 times its width w. Write and solve a sstem of linear equations to find the length and the width of the rectangle. P 5 4 ft l w 36. SHORT RESPONSE Find the solution of the sstem of linear equations below. Eplain our steps Equation Equation Equation CHALLENGE For a Þ 0, what is the solution of the sstem a and a ? 38. CHALLENGE Solve for,, and z in the sstem of equations below. Eplain our steps. 7 3z 5 29 Equation 3z Equation Equation WORKED-OUT SOLUTIONS on p. WS 5 STANDARDIZED TEST PRACTICE 5 MULTIPLE REPRESENTATIONS

26 PROBLEM SOLVING EXAMPLE 4 on p. 446 for Es ROWING During a practice, a 4 person crew team rows a rowing shell upstream (against the current) and then rows the same distance downstream (with the current). The shell moves upstream at a speed of 4.3 meters per second and downstream at a speed of 4.9 meters per second. The speed of the current remains constant. Use the models below to write and solve a sstem of equations to find the average speed of the shell in still water and the speed of the current. Upstream Speed of shell in still water 2 Speed of current 5 Speed of shell Downstream Speed of shell in still water Speed of current 5 Speed of shell 40. OIL CHANGE Two cars get an oil change at the same service center. Each customer is charged a fee (in dollars) for the oil change plus dollars per quart of oil used. The oil change for the car that requires 5 quarts of oil costs $ The oil change for the car that requires 7 quarts of oil costs $ Find the fee and the cost per quart of oil. 4. PHONES Cellular phone ring tones can be monophonic or polphonic. Monophonic ring tones pla one tone at a time, and polphonic ring tones pla multiple tones at a time. The table shows the ring tones downloaded from a website b two customers. Use the information to find the cost of a monophonic ring tone and a polphonic ring tone, assuming that all monophonic ring tones cost the same and all polphonic ring tones cost the same. Customer Monophonic ring tones Polphonic ring tones Total cost (dollars) 42. Julie Tate MULTIPLE REPRESENTATIONS For a floral arrangement class, Alicia has to create an arrangement of twigs and flowers that has a total of 9 objects. She has to pa for the twigs and flowers that she uses in her arrangement. Each twig costs $, and each flower costs $3. a. Writing a Sstem Alicia spends $5 on the twigs and flowers. Write and solve a linear sstem to find the number of twigs and the number of flowers she used. b. Making a Table Make a table showing the number of twigs in the arrangement and the total cost of the arrangement when the number of flowers purchased is 0,, 2, 3, 4, or 5. Use the table to check our answer to part (a). 7.3 Solve Linear Sstems b Adding or Subtracting 449

27 43. MULTI-STEP PROBLEM On a tpical da with light winds, the 800 mile flight from Charlotte, North Carolina, to Phoeni, Arizona, takes longer than the return trip because the plane has to fl into the wind. a. The flight from Charlotte to Phoeni is 4 hours 30 minutes long, and the flight from Phoeni to Charlotte is 4 hours long. Find the average speed (in miles per hour) of the airplane on the wa to Phoeni and on the return trip to Charlotte. b. Let s be the speed (in miles per hour) of the plane with no wind, and let w be the speed (in miles per hour) of the wind. Use our answer to part (a) to write and solve a sstem of equations to find the speed of the plane with no wind and the speed of the wind. 44. SHORT RESPONSE The students in the graduating classes at the three high schools in a school district have to pa for their caps and gowns. A cap-and-gown set costs dollars, and an etra tassel costs dollars. At one high school, students pa $3262 for 25 cap-and-gown sets and 72 etra tassels. At another high school, students pa $3346 for 22 capand-gown sets and 72 etra tassels. How much will students at the third high school pa for 28 cap-and-gown sets and 56 etra tassels? Eplain. 45. CHALLENGE A clothing manufacturer makes men s dress shirts. For the production process, an ideal sleeve length (in centimeters) for each shirt size and an allowable deviation (in centimeters) from the ideal length are established. The deviation is epressed as 6. For a specific shirt size, the minimum allowable sleeve length is 62.2 centimeters and the maimum allowable sleeve length is 64.8 centimeters. Find the ideal sleeve length and the allowable deviation. MIXED REVIEW Graph the equation (p. 225) (p. 225) ( 2) (p. 302) } 2 ( ) (p. 302) 3 Solve the linear sstem b graphing. Check our solution. (p. 427) PREVIEW Prepare for Lesson 7.4 in Es Find the least common multiple of the pair of numbers. (p. 90) 53. 9, , , Chapter 7 EXTRA Sstems PRACTICE of Equations for and Lesson Inequalities 7.3, p. 944 ONLINE QUIZ at classzone.com

28 7.4 Solve Linear Sstems b Multipling First Before You solved linear sstems b adding or subtracting. Now You will solve linear sstems b multipling first. Wh So ou can solve a problem about preparing food, as in E. 39. Ke Vocabular least common multiple, p. 90 In a linear sstem like the one below, neither variable can be eliminated b adding or subtracting the equations. For sstems like these, ou can multipl one or both of the equations b a constant so that adding or subtracting the equations will eliminate one variable The new sstem is equivalent to the original sstem. E XAMPLE Multipl one equation, then add Solve the linear sstem: Equation Equation 2 ANOTHER WAY You can also multipl Equation 2 b 3 and subtract the equations. Solution STEP Multipl Equation 2 b 23 so that the coefficients of are opposites (23) STEP 2 Add the equations STEP 3 Solve for. 52 STEP 4 Substitute 2 for in either of the original equations and solve for Write Equation (2) 5 5 Substitute 2 for. 2 (23) 5 5 Multipl Subtract 23 from each side. 5 4 Divide each side b 2. c The solution is (4, 2). CHECK Substitute 4 for and 2 for in each of the original equations. Equation Equation (4) 5(2) 0 9 2(4) 3(2) Solve Linear Sstems b Multipling First 45

29 MULTIPLYING BOTH EQUATIONS To eliminate one variable when adding or subtracting equations in a linear sstem, ou ma need to multipl both equations b constants. Use the least common multiple of the coefficients of one of the variables to determine the constants The least common multiple of 29 and 22 is 236. E XAMPLE 2 Multipl both equations, then subtract Solve the linear sstem: Equation Equation 2 Solution STEP Arrange the equations so that like terms are in columns Write Equation. ANOTHER WAY You can also multipl Equation b 3 and Equation 2 b 4. Then add the revised equations to eliminate Rewrite Equation 2. STEP 2 Multipl Equation b 2 and Equation 2 b 5 so that the coeffcient of in each equation is the least common multiple of 5 and 2, or STEP 3 Subtract the equations STEP 4 Solve for. 5 5 STEP 5 Substitute 5 for in either of the original equations and solve for Write Equation. 4(5) Substitute 5 for. 5 3 Solve for. c The solution is (5, 3). CHECK Substitute 5 for and 3 for in each of the original equations. Equation Equation (5) 5(3) (3) 0 3(5) at classzone.com GUIDED PRACTICE for Eamples and 2 Solve the linear sstem using elimination Chapter 7 Sstems of Equations and Inequalities

30 E XAMPLE 3 Standardized Test Practice Darlene is making a quilt that has alternating stripes of regular quilting fabric and sateen fabric. She spends $76 on a total of 6 ards of the two fabrics at a fabric store. Which sstem of equations can be used to find the amount (in ards) of regular quilting fabric and the amount (in ards) of sateen fabric she purchased? Sateen fabric costs $6 per ard. Quilting fabric costs $4 per ard. ELIMINATE CHOICES You can eliminate choice A because + cannot equal both 6 and 76. A 5 6 B C 5 76 D Solution Write a sstem of equations where is the number of ards of regular quilting fabric purchased and is the number of ards of sateen fabric purchased. Equation : Amount of fabric Amount of quilting fabric Amount of sateen fabric 5 Total ards of fabric 5 6 Equation 2: Cost of fabric Quilting fabric price (dollars/d) p Amount of quilting fabric (d) Sateen fabric price (dollars/d) p Amount of sateen fabric (d) 5 Total cost (dollars) 4 p 6 p 5 76 The sstem of equations is: 5 6 Equation Equation 2 c The correct answer is B. A B C D GUIDED PRACTICE for Eample 3 4. SOCCER A sports equipment store is having a sale on soccer balls. A soccer coach purchases 0 soccer balls and 2 soccer ball bags for $55. Another soccer coach purchases 2 soccer balls and 3 soccer ball bags for $89. Find the cost of a soccer ball and the cost of a soccer ball bag. 7.4 Solve Linear Sstems b Multipling First 453

31 CONCEPT SUMMARY For Your Notebook Methods for Solving Linear Sstems Method Table (p. 426) Eample When to Use When -values are integers, so that equal values can be seen in the table Graphing (p. 427) When ou want to see the lines that the equations represent 5 4 Substitution (p. 435) Addition (p. 444) Subtraction (p. 445) Multiplication (p. 45) = = = 23 3 = = = 7 When one equation is alread solved for or When the coefficients of one variable are opposites When the coefficients of one variable are the same When no corresponding coefficients are the same or opposites 7.4 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 5 and 39 5 STANDARDIZED TEST PRACTICE Es. 2, 8, 34, 4, and 42 5 MULTIPLE REPRESENTATIONS E. 40. VOCABULARY What is the least common multiple of 2 and 8? 2. WRITING Eplain how to solve the linear sstem using the elimination method Equation Equation 2 EXAMPLE on p. 45 for Es. 3 8 SOLVING LINEAR SYSTEMS Solve the linear sstem using elimination Chapter 7 Sstems of Equations and Inequalities

32 EXAMPLE 2 on p. 452 for Es SOLVING LINEAR SYSTEMS Solve the linear sstem using elimination MULTIPLE CHOICE Which ordered pair is a solution of the linear sstem and ? A (23, 22) B (23, 2) C (22, 23) D (2, 23) ERROR ANALYSIS Describe and correct the error when solving the linear sstem SOLVING LINEAR SYSTEMS Solve the linear sstem using an algebraic method } } } } 2 2 } } 3 2 }5 } GEOMETRY A rectangle has a perimeter of 8 inches. A new rectangle is formed b doubling the width w and tripling the length l, as shown. The new rectangle has a perimeter P of 46 inches. a. Write and solve a sstem of linear equations to find the length and width of the original rectangle. b. Find the length and width of the new rectangle. P 5 46 in. 3l 2w 34. WRITING For which values of a can ou solve the linear sstem a and without multipling first? Eplain. CHALLENGE Find the values of a and b so that the linear sstem has the given solution. 35. (4, 2) 36. (2, ) a 2 b 5 4 Equation b 2 a 5 0 Equation Solve Linear Sstems b Multipling First 455

33 PROBLEM SOLVING EXAMPLE 3 on p. 453 for Es BOOK SALE A librar is having a book sale to raise mone. Hardcover books cost $4 each and paperback books cost $2 each. A person spends $26 for 8 books. How man hardcover books did she purchase? 38. MUSIC A website allows users to download individual songs or an entire album. All individual songs cost the same to download, and all albums cost the same to download. Ran pas $4.94 to download 5 individual songs and album. Seth pas $22.95 to download 3 individual songs and 2 albums. How much does the website charge to download a song? an entire album? 39. FARM PRODUCTS The table shows the number of apples needed to make the apple pies and applesauce sold at a farm store. During a recent apple picking at the farm, 69 Grann Smith apples and 95 Golden Delicious apples were picked. How man apple pies and batches of applesauce can be made if ever apple is used? Tpe of apple Grann Smith Golden Delicious Needed for a pie 5 3 Needed for a batch of applesauce MULTIPLE REPRESENTATIONS Tickets for admission to a high school football game cost $3 for students and $5 for adults. During one game, $2995 was collected from the sale of 729 tickets. a. Writing a Sstem Write and solve a sstem of linear equations to find the number of tickets sold to students and the number of tickets sold to adults. b. Drawing a Graph Graph the sstem of linear equations. Use the graph to determine whether our answer to part (a) is reasonable. 4. SHORT RESPONSE A dim sum restaurant offers two sizes of dishes: small and large. All small dishes cost the same and all large dishes cost the same. The bills show the cost of the food before the tip is included. What will 3 small and 2 large dishes cost before the tip is included? Eplain. 42. OPEN ENDED Describe a real-world problem that can be solved using a sstem of linear equations. Then solve the problem and eplain what the solution means in this situation. 5 WORKED-OUT SOLUTIONS 456 Chapter 7 Sstems on p. WS of Equations and Inequalities 5 STANDARDIZED TEST PRACTICE 5 MULTIPLE REPRESENTATIONS

34 43. INVESTMENTS Matt invested $2000 in stocks and bonds. This ear the bonds paid 8% interest, and the stocks paid 6% in dividends. Matt received a total of $44 in interest and dividends. How much mone did he invest in stocks? in bonds? 44. CHALLENGE You drive a car 45 miles at an average speed r (in miles per hour) to reach our destination. Due to traffic, our average speed on the return trip is 3 } 4 r. The round trip took a total of hour 45 minutes. Find the average speed for each leg of our trip. MIXED REVIEW Graph the equation. (pp. 25, 225, 244) } 4 } } PREVIEW Prepare for Lesson 7.5 in Es Determine which lines are parallel. (p. 244) 5. a b c 52. (2, 5) (0, 3) (3, 3) (6, 3) 2 (0, ) 2 (3, ) c b (22, 2) 2 (0, ) a (2, 2) (0, 22) (6, 0) (5, 22.5) Solve the linear sstem using an method. (pp. 427, 435, 444, 45) QUIZ for Lessons Solve the linear sstem using elimination. (pp. 444, 45) } } EXTRA PRACTICE for Lesson 7.4, p. 944 ONLINE QUIZ at classzone.com 457

35 MIXED REVIEW of Problem Solving STATE TEST PRACTICE classzone.com Lessons MULTI-STEP PROBLEM Fling into the wind, a helicopter takes 5 minutes to travel 5 kilometers. The return flight takes 2 minutes. The wind speed remains constant during the trip. a. Find the helicopter s average speed (in kilometers per hour) for each leg of the trip. b. Write a sstem of linear equations that represents the situation. c. What is the helicopter s average speed in still air? What is the speed of the wind? 4. OPEN-ENDED Describe a real-world problem that can be modeled b a linear sstem. Then solve the sstem and interpret the solution in the contet of the problem. 5. SHORT RESPONSE A hot air balloon is launched at Kirb Park, and it ascends at a rate of 7200 feet per hour. At the same time, a second hot air balloon is launched at Newman Park, and it ascends at a rate of 4000 feet per hour. Both of the balloons stop ascending after 30 minutes. The diagram shows the altitude of each park. Are the hot air balloons ever at the same height at the same time? Eplain. 2. SHORT RESPONSE At a grocer store, a customer pas a total of $9.70 for.8 pounds of potato salad and.4 pounds of coleslaw. Another customer pas a total of $6.55 for pound of potato salad and.2 pounds of coleslaw. How much do 2 pounds of potato salad and 2 pounds of coleslaw cost? Eplain. 3. GRIDDED ANSWER During one da, two computers are sold at a computer store. The two customers each arrange pament plans with the salesperson. The graph shows the amount of mone (in dollars) paid for the computers after months. After how man months will each customer have paid the same amount? Amount paid (dollars) Months since purchase 6. EXTENDED RESPONSE A chemist needs 500 milliliters of a 20% acid and 80% water mi for a chemistr eperiment. The chemist combines milliliters of a 0% acid and 90% water mi and milliliters of a 30% acid and 70% water mi to make the 20% acid and 80% water mi. a. Write a linear sstem that represents the situation. b. How man milliliters of the 0% acid and 90% water mi and the 30% acid and 70% water mi are combined to make the 20% acid and 80% water mi? c. The chemist also needs 500 milliliters of a 5% acid and 85% water mi. Does the chemist need more of the 0% acid and 90% water mi than the 30% acid and 70% water mi to make this new mi? Eplain. 458 Chapter 7 Sstems of Equations and Inequalities

36 7.5 Solve Special Tpes of Linear Sstems Before You found the solution of a linear sstem. Now You will identif the number of solutions of a linear sstem. Wh? So ou can compare distances traveled, as in E. 39. Ke Vocabular inconsistent sstem consistent dependent sstem sstem of linear equations, p. 427 parallel, p. 244 A linear sstem can have no solution or infinitel man solutions. A linear sstem has no solution when the graphs of the equations are parallel. A linear sstem with no solution is called an inconsistent sstem. A linear sstem has infinitel man solutions when the graphs of the equations are the same line. A linear sstem with infinitel man solutions is called a consistent dependent sstem. E XAMPLE A linear sstem with no solution Show that the linear sstem has no solution Equation Equation 2 Solution REVIEW GRAPHING For help with graphing linear equations, see pp. 25, 225, and 244. METHOD Graphing Graph the linear sstem c The lines are parallel because the have the same slope but different -intercepts. Parallel lines do not intersect, so the sstem has no solution. METHOD 2 Elimination Subtract the equations. IDENTIFY TYPES OF SYSTEMS The linear sstem in Eample is called an inconsistent sstem because the lines do not intersect (are not consistent) This is a false statement. c The variables are eliminated and ou are left with a false statement regardless of the values of and. This tells ou that the sstem has no solution. at classzone.com 7.5 Solve Special Tpes of Linear Sstems 459

37 E XAMPLE 2 A linear sstem with infinitel man solutions Show that the linear sstem has infinitel man solutions Equation 5 } 2 Equation 2 2 Solution METHOD Graphing Graph the linear sstem c The equations represent the same line, so an point on the line is a solution. So, the linear sstem has infinitel man solutions. METHOD 2 Substitution IDENTIFY TYPES OF SYSTEMS The linear sstem in Eample 2 is called a consistent dependent sstem because the lines intersect (are consistent) and the equations are equivalent (are dependent). Substitute } 2 2 for in Equation and solve for Write Equation. 2 2 } Substitute } 2 2 for Simplif. c The variables are eliminated and ou are left with a statement that is true regardless of the values of and. This tells ou that the sstem has infinitel man solutions. GUIDED PRACTICE for Eamples and 2 Tell whether the linear sstem has no solution or infinitel man solutions. Eplain IDENTIFYING THE NUMBER OF SOLUTIONS When the equations of a linear sstem are written in slope-intercept form, ou can identif the number of solutions of the sstem b looking at the slopes and -intercepts of the lines. Number of solutions One solution No solution Infinitel man solutions Slopes and -intercepts Different slopes Same slope Different -intercepts Same slope Same -intercept 460 Chapter 7 Sstems of Equations and Inequalities

38 E XAMPLE 3 Identif the number of solutions Without solving the linear sstem, tell whether the linear sstem has one solution, no solution, or infinitel man solutions. a Equation b Equation Equation Equation 2 Solution a Write Equation in slope-intercept form Write Equation 2 in slope-intercept form. c Because the lines have the same slope and the same -intercept, the sstem has infinitel man solutions. b } 2 Write Equation in slope-intercept form } 5 Write Equation 2 in slope-intercept form. 2 c Because the lines have the same slope but different -intercepts, the sstem has no solution. E XAMPLE 4 Write and solve a sstem of linear equations ART An artist wants to sell prints of her paintings. She orders a set of prints for each of two of her paintings. Each set contains regular prints and gloss prints, as shown in the table. Find the cost of one gloss print. Regular Gloss Cost $ $55 Solution STEP Write a linear sstem. Let be the cost (in dollars) of a regular print, and let be the cost (in dollars) of a gloss print Cost of prints for one painting Cost of prints for other painting STEP 2 Solve the linear sstem using elimination (23) c There are infinitel man solutions, so ou cannot determine the cost of one gloss print. You need more information. GUIDED PRACTICE for Eamples 3 and 4 3. Without solving the linear sstem, tell whether it has one solution, no solution, or infinitel man solutions Equation Equation 2 4. WHAT IF? In Eample 4, suppose a gloss print costs $3 more than a regular print. Find the cost of a gloss print. 7.5 Solve Special Tpes of Linear Sstems 46

39 CONCEPT SUMMARY For Your Notebook Number of Solutions of a Linear Sstem One solution No solution Infinitel man solutions The lines intersect. The lines have different slopes. The lines are parallel. The lines have the same slope and different -intercepts. The lines coincide. The lines have the same slope and the same -intercept. 7.5 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. and 37 5 STANDARDIZED TEST PRACTICE Es. 3, 4, 24, 25, 32, 33, and 40. VOCABULARY Cop and complete: A linear sstem with no solution is called a(n)? sstem. 2. VOCABULARY Cop and complete: A linear sstem with infinitel man solutions is called a(n)? sstem. 3. WRITING Describe the graph of a linear sstem that has no solution. 4. WRITING Describe the graph of a linear sstem that has infinitel man solutions. INTERPRETING GRAPHS Match the linear sstem with its graph. Then use the graph to tell whether the linear sstem has one solution, no solution, or infinitel man solutions A. B. C Chapter 7 Sstems of Equations and Inequalities

40 EXAMPLES and 2 on pp for Es INTERPRETING GRAPHS Graph the linear sstem. Then use the graph to tell whether the linear sstem has one solution, no solution, or infinitel man solutions } ERROR ANALYSIS Describe and correct the error in solving the linear sstem below The lines do not intersect, so there is no solution. SOLVING LINEAR SYSTEMS Solve the linear sstem using substitution or elimination MULTIPLE CHOICE Which of the linear sstems has eactl one solution? A B C D MULTIPLE CHOICE Which of the linear sstems has infinitel man solutions? A B C D EXAMPLE 3 on p. 46 for Es IDENTIFYING THE NUMBER OF SOLUTIONS Without solving the linear sstem, tell whether the linear sstem has one solution, no solution, or infinitel man solutions Solve Special Tpes of Linear Sstems 463

41 32. OPEN ENDED Write a linear sstem so that it has infinitel man solutions, and one of the equations is OPEN ENDED Write a linear sstem so that it has no solution and one of the equations is REASONING Give a countereample for the following statement: If the graphs of the equations of a linear sstem have the same slope, then the linear sstem has no solution. 35. CHALLENGE Find values of p, q, and r that produce the solution(s). a. No solution b. Infinitel man solutions c. One solution of (4, ) p q 5 r Equation Equation 2 PROBLEM SOLVING EXAMPLE 4 on p. 46 for Es RECREATION One admission to a roller skating rink costs dollars and renting a pair of skates costs dollars. A group pas $243 for admission for 36 people and 2 skate rentals. Another group pas $8 for admission for 2 people and 7 skate rentals. Is there enough information to determine the cost of one admission to the roller skating rink? Eplain. 37. TRANSPORTATION A passenger train travels from New York Cit to Washington, D.C., then back to New York Cit. The table shows the number of coach tickets and business class tickets purchased for each leg of the trip. Is there enough information to determine the cost of one coach ticket? Eplain. Destination Coach tickets Business class tickets Mone collected (dollars) Washington, D.C ,860 New York Cit , PHOTOGRAPHY In addition to taking pictures on our digital camera, ou can record 30 second movies. All pictures use the same amount of memor, and all 30 second movies use the same amount of memor. The number of pictures and 30 second movies on 2 memor cards is shown. a. Is there enough information given to determine the amount of memor used b a 30 second movie? Eplain. b. Given that a 30 second movie uses 50 times the amount of memor that a digital picture uses, can ou determine the amount of memor used b a 30 second movie? Eplain. Size of card (megabtes) Pictures Movies WORKED-OUT SOLUTIONS on p. WS 5 STANDARDIZED TEST PRACTICE

42 39. MULTI-STEP PROBLEM Two people are training for a speed ice-climbing event. During a practice climb, one climber starts 5 seconds after the first climber. The rates that the climbers ascend are shown. a. Let d be the distance (in feet) traveled b a climber t seconds after the first person starts climbing. Write a linear sstem that models the situation. b. Graph the linear sstem from part (a). Does the second climber catch up to the first climber? Eplain. Climbs 0 feet ever 30 seconds Climbs 5 feet ever 5 seconds 40. EXTENDED RESPONSE Two emploees at a banquet facilit are given the task of folding napkins. One person starts folding napkins at a rate of 5 napkins per minute. The second person starts 0 minutes after the first person and folds napkins at a rate of 4 napkins per minute. a. Model Let be the number of napkins folded minutes after the first person starts folding. Write a linear sstem that models the situation. b. Solve Solve the linear sstem. c. Interpret Does the solution of the linear sstem make sense in the contet of the problem? Eplain. 4. CHALLENGE An airplane has an average air speed of 60 miles per hour. The airplane takes 3 hours to travel with the wind from Salem to Lancaster. The airplane has to travel against the wind on the return trip. After 3 hours into the return trip, the airplane is 20 miles from Salem. Find the distance from Salem to Lancaster. If the problem cannot be solved with the information given, eplain wh. MIXED REVIEW Solve the equation, if possible. (p. 54) c c 43. 3m m m 44. z 3 5 0(2z 3) ( 2 w) 5 4(w 2 5) PREVIEW Prepare for Lesson 7.6 in Es Solve the inequalit. Then graph our solution < 23 (p. 356) (p. 356) 48. } 4 > 22.5 (p. 363) (p. 363) (p. 369) > 3( 2 3) (p. 369) < 3 (p. 380) (p. 380) (p. 398) (p. 398) Graph the inequalit. (p. 405) 56. < < > 6.5 EXTRA PRACTICE for Lesson 7.5, p. 944 ONLINE QUIZ at classzone.com 465

43 7.6 Solve Sstems of Linear Inequalities Before You graphed linear inequalities in two variables. Now You will solve sstems of linear inequalities in two variables. Wh So ou can find a marching band s competition score, as in E. 36. Ke Vocabular sstem of linear inequalities solution of a sstem of linear inequalities graph of a sstem of linear inequalities A sstem of linear inequalities in two variables, or simpl a sstem of inequalities, consists of two or more linear inequalities in the same variables. An eample is shown. 2 > 7 Inequalit 2 < 8 Inequalit 2 A solution of a sstem of linear inequalities is an ordered pair that is a solution of each inequalit in the sstem. For eample, (6, 25) is a solution of the sstem above. The graph of a sstem of linear inequalities is the graph of all solutions of the sstem. KEY CONCEPT For Your Notebook Graphing a Sstem of Linear Inequalities STEP Graph each inequalit (as ou learned to do in Lesson 6.7). STEP 2 Find the intersection of the half-planes. The graph of the sstem is this intersection. E XAMPLE Graph a sstem of two linear inequalities Graph the sstem of inequalities. > Inequalit 3 6 Inequalit 2 REVIEW GRAPHING INEQUALITIES For help with graphing a linear inequalit in two variables, see p Solution Graph both inequalities in the same coordinate plane. The graph of the sstem is the intersection of the two half-planes, which is shown as the darker shade of blue. CHECK Choose a point in the dark blue region, such as (0, ). To check this solution, substitute 0 for and for into each inequalit. >? 0 2 2? 0 6 (0, ) >22 6 at classzone.com 466 Chapter 7 Sstems of Equations and Inequalities

44 THE SOLUTION REGION In Eample, the half-plane for each inequalit is shaded, and the solution region is the intersection of the half-planes. From this point on, onl the solution region will be shaded. E XAMPLE 2 Graph a sstem of three linear inequalities Graph the sstem of inequalities. 2 Inequalit > 22 Inequalit Inequalit 3 Solution Graph all three inequalities in the same coordinate plane. The graph of the sstem is the triangular region shown. The region is to the right of the line 522. The region is on and below the line The region is on and above the line 52. GUIDED PRACTICE for Eamples and 2 Graph the sstem of linear inequalities.. < > > 2 > 2 3 < 4 > 2 4 < 3 < 5 E XAMPLE 3 Write a sstem of linear inequalities Write a sstem of inequalities for the shaded region. REVIEW EQUATIONS OF LINES For help with writing an equation of a line, see pp. 283, 302, and 3. Solution INEQUALITY : One boundar line for the shaded region is 5 3. Because the shaded region is above the solid line, the inequalit is 3. INEQUALITY 2: Another boundar line for the shaded region has a slope of 2 and a -intercept of. So, its equation is 5 2. Because the shaded region is above the dashed line, the inequalit is > 2. 2 c The sstem of inequalities for the shaded region is: 3 Inequalit > 2 Inequalit Solve Sstems of Linear Inequalities 467

45 E XAMPLE 4 Write and solve a sstem of linear inequalities BASEBALL The National Collegiate Athletic Association (NCAA) regulates the lengths of aluminum baseball bats used b college baseball teams. The NCAA states that the length (in inches) of the bat minus the weight (in ounces) of the bat cannot eceed 3. Bats can be purchased at lengths from 26 to 34 inches. a. Write and graph a sstem of linear inequalities that describes the information given above. b. A sporting goods store sells an aluminum bat that is 3 inches long and weighs 25 ounces. Use the graph to determine if this bat can be used b a plaer on an NCAA team. Solution a. Let be the length (in inches) of the bat, and let be the weight (in ounces) of the bat. From the given information, ou can write the following inequalities: 2 3 The difference of the bat s length and weight can be at most The length of the bat must be at least 26 inches. WRITING SYSTEMS OF INEQUALITIES Consider the values of the variables when writing a sstem of inequalities. In man real-world problems, the values cannot be negative The length of the bat can be at most 34 inches. The weight of the bat cannot be a negative number. Graph each inequalit in the sstem. Then identif the region that is common to all of the graphs of the inequalities. This region is shaded in the graph shown. b. Graph the point that represents a bat that is 3 inches long and weighs 25 ounces. (3, 25) c Because the point falls outside the solution region, the bat cannot be used b a plaer on an NCAA team. 5 0 GUIDED PRACTICE for Eamples 3 and 4 Write a sstem of inequalities that defines the shaded region WHAT IF? In Eample 4, suppose a Senior League (ages 0 4) plaer wants to bu the bat described in part (b). In Senior League, the length (in inches) of the bat minus the weight (in ounces) of the bat cannot eceed 8. Write and graph a sstem of inequalities to determine whether the described bat can be used b the Senior League plaer. 468 Chapter 7 Sstems of Equations and Inequalities

46 7.6 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 3 and 39 5 STANDARDIZED TEST PRACTICE Es. 2, 2, 22, 33, and 40. VOCABULARY Cop and complete: A(n)? of a sstem of linear inequalities is an ordered pair that is a solution of each inequalit in the sstem. 2. WRITING Describe the steps ou would take to graph the sstem of inequalities shown. 2 < 7 Inequalit 3 Inequalit 2 CHECKING A SOLUTION Tell whether the ordered pair is a solution of the sstem of inequalities. 3. (, ) 4. (0, 6) 5. (3, 2) EXAMPLE on p. 466 for Es. 627 MATCHING SYSTEMS AND GRAPHS Match the sstem of inequalities with its graph > > 28 2 < 2 2 A. B. C. GRAPHING A SYSTEM Graph the sstem of inequalities. 9. > > 3 < 2 6 > 2. < < < < < > 26 2 > 2 EXAMPLE 2 on p. 467 for Es < > < MULTIPLE CHOICE Which ordered pair is a solution of the sstem and 2 > 2? A (, 2) B (4, ) C (2, 0) D (3, 2) 7.6 Solve Sstems of Linear Inequalities 469

47 EXAMPLE 2 on p. 467 for Es MULTIPLE CHOICE The graph of which sstem of inequalities is shown? A < 2 B < < > 6 C > 2 D > < > ERROR ANALYSIS Describe and correct the error in graphing this sstem of inequalities: < 3 Inequalit > 2 Inequalit 2 3 Inequalit 3 EXAMPLE 3 on p. 467 for Es WRITING A SYSTEM Write a sstem of inequalities for the shaded region GRAPHING A SYSTEM Graph the sstem of inequalities. 30. > 4 3. < < 9 > > SHORT RESPONSE Does the sstem of inequalities have an solutions? Eplain. 2 > 5 Inequalit 2 < Inequalit 2 CHALLENGE Write a sstem of inequalities for the shaded region described. 34. The shaded region is a rectangle with vertices at (2, ), (2, 4), (6, 4), and (6, ). 35. The shaded region is a triangle with vertices at (23, 0), (3, 2), and (0, 22) WORKED-OUT SOLUTIONS on p. WS 5 STANDARDIZED TEST PRACTICE

48 PROBLEM SOLVING EXAMPLE 4 on p. 468 for Es COMPETITION SCORES In a marching band competition, scoring is based on a musical evaluation and a visual evaluation. The musical evaluation score cannot eceed 60 points, the visual evaluation score cannot eceed 40 points. Write and graph a sstem of inequalities for the scores that a marching band can receive. 37. NUTRITION For a hiking trip, ou are making a mi of ounces of peanuts and ounces of chocolate pieces. You want the mi to have less than 70 grams of fat and weigh less than 8 ounces. An ounce of peanuts has 4 grams of fat, and an ounce of chocolate pieces has 7 grams of fat. Write and graph a sstem of inequalities that models the situation. 38. FISHING LIMITS You are fishing in a marina for surfperch and rockfish, which are two species of bottomfish. Gaming laws in the marina allow ou to catch no more than 5 surfperch per da, no more than 0 rockfish per da, and no more than 5 total bottomfish per da. a. Write and graph a sstem of inequalitites that models the situation. b. Use the graph to determine whether ou can catch surfperch and 9 rockfish in one da. 39. HEALTH A person s maimum heart rate (in beats per minute) is given b where is the person s age in ears (20 65). When eercising, a person should aim for a heart rate that is at least 70% of the maimum heart rate and at most 85% of the maimum heart rate. a. Write and graph a sstem of inequalitites that models the situation. b. A 40-ear-old person s heart rate varies from 04 to 20 beats per minute while eercising. Does his heart rate sta in the suggested target range for his age? Eplain. 40. SHORT RESPONSE A photograph shop has a self-service photo center that allows ou to make prints of pictures. Each sheet of printed pictures costs $8. The number of pictures that fit on each sheet is shown. a. You want at least 6 pictures of an size, and ou are willing to spend up to $48. Write and graph a sstem of inequalities that models the situation. b. Will ou be able to purchase 2 pictures that are 3 inches b 5 inches and 6 pictures that are 4 inches b 6 inches? Eplain. Surfperch Four 3 inch b 5 inch pictures fit on one sheet. Rockfish Two 4 inch b 6 inch pictures fit on one sheet. 7.6 Solve Sstems of Linear Inequalities 47

49 4. CHALLENGE You make necklaces and kechains to sell at a craft fair. The table shows the time that it takes to make each necklace and kechain, the cost of materials for each necklace and kechain, and the time and mone that ou can devote to making necklaces and kechains. Necklace Kechain Available Time to make (hours) Cost to make (dollars) a. Write and graph a sstem of inequalities for the number of necklaces and the number of kechains that ou can make under the given constraints. b. Find the vertices (corner points) of the graph. c. You sell each necklace for $0 and each kechain for $8. The revenue R is given b the equation R Find the revenue for each ordered pair in part (b). Which verte results in the maimum revenue? MIXED REVIEW PREVIEW Prepare for Lesson 8. in Es Evaluate the epression when 5 2 (p. 8) z 3 when z 5 2 (p. 8) c when c 5 8 (p. 64) when 526 (p. 88) w } w when w 5 5 (p. 03) Ï } when 5 44 (p. 0) Use an method to solve the linear sstem. (pp. 427, 435, 444, 45) QUIZ for Lessons Graph the linear sstem. Then use the graph to tell whether the linear sstem has one solution, no solution, or infinitel man solutions. (p. 459) Graph the sstem of linear inequalities. (p. 466) 4. > < 7 < < 4 7. < > 25 2 > 23 < EXTRA PRACTICE for Lesson 7.6, p. 944 ONLINE QUIZ at classzone.com

50 MIXED REVIEW of Problem Solving Lessons MULTI-STEP PROBLEM A minimum of 600 bricks and 2 bags of sand are needed for a construction job. Each brick weighs 2 pounds, and each bag of sand weighs 50 pounds. The maimum weight that a deliver truck can carr is 3000 pounds. a. Let be the number of bricks, and let be the number of bags of sand. Write a sstem of linear inequalities that models the situation. b. Graph the sstem of inequalities. c. Use the graph to determine whether 700 bricks and 20 bags of sand can be delivered in one trip. 2. MULTI-STEP PROBLEM Dana decides to paint the ceiling and the walls of a room. She spends $20 on 2 gallons of paint for the ceiling and 4 gallons of paint for the walls. Then she decides to paint the ceiling and the walls of another room using the same kinds of paint. She spends $60 for gallon of paint for the ceiling and 2 gallons of paint for the walls. a. Write a sstem of linear equations that models the situation. b. Is there enough information given to determine the cost of one gallon of each tpe of paint? Eplain. c. A gallon of ceiling paint costs $3 more than a gallon of wall paint. What is the cost of one gallon of each tpe of paint? 3. SHORT RESPONSE During a sale at a music and video store, all CDs are priced the same and all DVDs are priced the same. Karen bus 4 CDs and 2 DVDs for $78. The net da, while the sale is still in progress, Karen goes back and bus 2 CDs and DVD for $39. Is there enough information to determine the cost of CD? Eplain. STATE TEST PRACTICE classzone.com 4. SHORT RESPONSE Two airport shuttles, bus A and bus B, take passengers to the airport from the same bus stop. The graph shows the distance d (in miles) traveled b each bus t hours after bus A leaves the station. The distance from the bus stop to the airport is 25 miles. If bus A and bus B continue at the same rates, will bus B ever catch up to bus A? Eplain. Distance (miles) d EXTENDED RESPONSE During the summer, ou want to earn at least $200 per week. You earn $0 per hour working as a lifeguard, and ou earn $8 per hour working at a retail store. You can work at most 30 hours per week. a. Write and graph a sstem of linear inequalities that models the situation. b. If ou work 5 hours per week as a lifeguard and 5 hours per week at the retail store, will ou earn at least $200 per week? Eplain. c. You are scheduled to work 20 hours per week at the retail store. What is the range of hours ou can work as a lifeguard to earn at least $200 per week? 6. OPEN-ENDED Describe a real-world situation that can be modeled b a sstem of linear inequalities. Then write and graph the sstem of inequalities. 7. GRIDDED ANSWER What is the area (in square feet) of the triangular garden defined b the sstem of inequalities below? Bus A Bus B t Time (hours) Mied Review of Problem Solving 473

51 7 CHAPTER SUMMARY Big Idea BIG IDEAS Solving Linear Sstems b Graphing For Your Notebook The graph of a sstem of two linear equations tells ou how man solutions the sstem has. One solution No solution Infinitel man solutions The lines intersect. The lines are parallel. The lines coincide. Big Idea 2 Solving Linear Sstems Using Algebra You can use an of the following algebraic methods to solve a sstem of linear equations. Sometimes it is easier to use one method instead of another. Method Procedure When to use Substitution Solve one equation for or. Substitute the epression for or into the other equation. When one equation is alread solved for or Addition Subtraction Multiplication Add the equations to eliminate or. Subtract the equations to eliminate or. Multipl one or both equations b a constant so that adding or subtracting the equations will eliminate or. When the coefficients of one variable are opposites When the coefficients of one variable are the same When no corresponding coefficients are the same or opposites Big Idea 3 Solving Sstems of Linear Inequalities The graph of a sstem of linear inequalities is the intersection of the half-planes of each inequalit in the sstem. For eample, the graph of the sstem of inequalities below is the shaded region. 6 Inequalit < 2 Inequalit Inequalit Chapter 7 Sstems of Equations and Inequalities

52 7 CHAPTER REVIEW REVIEW KEY VOCABULARY classzone.com Multi-Language Glossar Vocabular practice sstem of linear equations, p. 427 solution of a sstem of linear equations, p. 427 consistent independent sstem, p. 427 inconsistent sstem, p. 459 consistent dependent sstem, p. 459 sstem of linear inequalities, p. 466 solution of a sstem of linear inequalities, p. 466 graph of a sstem of linear inequalities, p. 466 VOCABULARY EXERCISES. Cop and complete: A(n)? consists of two or more linear inequalities in the same variables. 2. Cop and complete: A(n)? consists of two or more linear equations in the same variables. 3. Describe how ou would graph a sstem of two linear inequalities. 4. Give an eample of a consistent dependent sstem. Eplain wh the sstem is a consistent dependent sstem. REVIEW EXAMPLES AND EXERCISES Use the review eamples and eercises below to check our understanding of the concepts ou have learned in each lesson of Chapter Solve Linear Sstems b Graphing pp E XAMPLE Solve the linear sstem b graphing. Check our solution Equation Equation 2 Graph both equations. The lines appear to intersect at (, 2). Check the solution b substituting for and 2 for in each equation () EXAMPLES and 2 on pp for Es. 5 7 EXERCISES Solve the linear sstem b graphing. Check our solution Chapter Review 475

53 7 CHAPTER REVIEW 7.2 Solve Linear Sstems b Substitution pp E XAMPLE Solve the linear sstem: Equation Equation 2 STEP Substitute 5 7 for in Equation and solve for Write Equation Substitute 5 7 for Solve for. STEP 2 Substitute 22 for in Equation 2 to find the value of (22) c The solution is (22, 23). Check the solution b substituting 22 for and 23 for in each of the original equations. EXAMPLES, 2, and 3 on pp for Es. 8 EXERCISES Solve the linear sstem using substitution ART Kara spends $6 on tubes of paint and disposable brushes for an art project. Each tube of paint costs $3, and each disposable brush costs $.50. Kara purchases twice as man brushes as tubes of paint. Find the number of brushes and the number of tubes of paint that she purchases. 7.3 Solve Linear Sstems b Adding or Subtracting pp E XAMPLE Solve the linear sstem: Equation Equation 2 STEP Add the equations to eliminate one variable. STEP 2 Solve for. STEP 3 Substitute 23 for in either equation and solve for Write Equation. 5 2 (23) 5 8 Substitute 23 for. 5 Solve for c The solution is (, 23). Check the solution b substituting for and 23 for in each of the original equations. 476 Chapter 7 Sstems of Equations and Inequalities

54 EXAMPLES, 2, and 3 on pp for Es. 2 7 EXERCISES Solve the linear sstem using elimination. classzone.com Chapter Review Practice Solve Linear Sstems b Multipling First pp E XAMPLE Solve the linear sstem: Equation Equation 2 STEP Multipl the first equation b (23) STEP 2 Add the equations STEP 3 Solve for. 5 5 STEP 4 Substitute 5 for in either of the original equations and solve for Write Equation. 2 2(5) 527 Substitute 5 for. c The solution is (3, 5). CHECK 5 3 Solve for. Substitute 3 for and 5 for in each of the original equations. Equation Equation (5) (3) EXAMPLES and 2 on pp for Es EXERCISES Solve the linear sstem using elimination CAR MAINTENANCE You pa $24.50 for 0 gallons of gasoline and quart of oil at a gas station. Your friend pas $22 for 8 gallons of the same gasoline and 2 quarts of the same oil. Find the cost of quart of oil. Chapter Review 477

55 7 CHAPTER REVIEW 7.5 Solve Special Tpes of Linear Sstems pp E XAMPLE Show that the linear sstem has no solution Equation 5 2 Equation 2 Graph the linear sstem. The lines are parallel because the have the same slope but different -intercepts. Parallel lines do not intersect, so the sstem has no solution EXAMPLES, 2, and 3 on pp for Es EXERCISES Tell whether the linear sstem has one solution, no solution, or infinitel man solutions. Eplain Solve Sstems of Linear Inequalities pp E XAMPLE Graph the sstem of linear inequalities. < 22 3 Inequalit 2 3 Inequalit 2 The graph of < 22 3 is the half-plane below the dashed line The graph of 2 3 is the half-plane on and above the solid line The graph of the sstem is the intersection of the two half-planes shown as the darker shade of blue. 2 3 < 22 3 EXAMPLES, 2, 3, and 4 on pp for Es EXERCISES Graph the sstem of linear inequalities. 28. < > > 4 2 < 4 3. MOVIE COSTS You receive a $40 gift card to a movie theater. A ticket to a matinee movie costs $5, and a ticket to an evening movie costs $8. Write and graph a sstem of inequalities for the number of tickets ou can purchase using the gift card. 478 Chapter 7 Sstems of Equations and Inequalities

56 7 Solve CHAPTER TEST the linear sstem b graphing. Check our solution Solve the linear sstem using substitution Solve the linear sstem using elimination Tell whether the linear sstem has one solution, no solution, or infinitel man solutions Graph the sstem of linear inequalities. 25. < > 4 > 2 > TRUCK RENTALS Carrie and Dave each rent the same size moving truck for one da. The pa a fee of dollars for the truck and dollars per mile the drive. Carrie drives 50 miles and pas $25. Dave drives 20 miles and pas $76. Find the amount of the fee and the cost per mile. 29. GEOMETRY The rectangle has a perimeter P of 58 inches. The length l is one more than 3 times the width w. Write and solve a sstem of linear equations to find the length and width of the rectangle. P 5 58 in. l w 30. COMMUNITY SERVICE A town committee has a budget of $75 to spend on snacks for the volunteers participating in a clean-up da. The committee chairperson decides to purchase granola bars and at least 50 bottles of water. Granola bars cost $.50 each, and bottles of water cost $.75 each. Write and graph a sstem of linear inequalities for the number of bottles of water and the number of granola bars that can be purchased. Chapter Test 479

57 7 Standardized TEST PREPARATION MULTIPLE CHOICE QUESTIONS If ou have difficult solving a multiple choice problem directl, ou ma be able to use another approach to eliminate incorrect answer choices and obtain the correct answer. P ROBLEM Which ordered pair is the solution of the linear sstem 5 } 2 and ? A (2, ) B (, 23) C (22, 2) D (4, 2) Method SOLVE DIRECTLY Use substitution to solve the linear sstem. STEP Substitute } for in the equation and solve for } } } STEP 2 Substitute 22 for in 5 } to find the 2 value of. 5 } 2 5 } 2 (22) 52 The solution of the sstem is (22, 2). The correct answer is C. ABCD Method 2 ELIMINATE CHOICES Substitute the values given in each answer choice for and in both equations. Choice A: (2, ) Substitute 2 for and for. 5 } } 2 (2) 2(2) 3() Choice B: (, 23) Substitute for and 23 for. 5 } } 2 () 23 5 } 2 Choice C: (22, 2) Substitute 22 for and 2 for. 5 } } 2 (22) 2(22) 3(2) The correct answer is C. A BCD 480 Chapter 7 Sstems of Equations and Inequalities

58 P ROBLEM 2 The sum of two numbers is 2, and the difference of the two numbers is 5. What are the numbers? A 25 and 4 B and 6 C 2 and 23 D 22 and 3 Method SOLVE DIRECTLY Write and solve a sstem of equations for the numbers. STEP Write a sstem of equations. Let and be the numbers. 52 Equation Equation 2 STEP 2 Add the equations to eliminate one variable. Then find the value of the other variable , so 5 2 STEP 3 Substitute 2 for in Equation and solve for. 2 52, so 523 The correct answer is C. A B C D Method 2 ELIMINATE CHOICES Find the sum and difference of each pair of numbers. Because the difference is positive, be sure to subtract the lesser number from the greater number. Choice A: 25 and 4 Sum: Difference: Choice B: and 6 Sum: Choice C: 2 and 23 Sum: 2 (23) 52 Difference: 2 2 (23) 5 5 The correct answer is C. A BCD PRACTICE Eplain wh ou can eliminate the highlighted answer choice.. The sum of two numbers is 27. One number is twice the other. What are the numbers? A 9 and 8 B 23 and 24 C 28 and 29 D 24 and Which ordered pair is a solution of the linear sstem and 52} 2 2 4? A 2 7 } 6, 2 7 } 3 2 B 2 23 } 3, } 3 2 C (23, 2) D (23, 23) 3. Long-sleeve and short-sleeve T-shirts can be purchased at a concert. A long-sleeve T-shirt costs $25 and a short-sleeve T-shirt costs $5. During a concert, the T-shirt vendor collects $845 from the sale of 44 T-shirts. How man short-sleeve T-shirts were sold? A 00 B 80 C 26 D 44 Standardized Test Preparation 48

59 7 Standardized TEST PRACTICE MULTIPLE CHOICE. Which ordered pair is the solution of the linear sstem 5 } and 5 } 3 4? 2 2 A 3, 5 } 2 2 B 23, 2 } How man solutions does the sstem of linear equations whose graph is shown have? C 3 } 2, 7 } 4 2 D (0, ) 2. How man solutions does the linear sstem and have? A 0 B C 2 D Infinitel man 3. The sum of two numbers is 23, and the difference of the two numbers is. What are the numbers? A 4 and 7 B 23 and 8 C 3 and 4 D 27 and 4 4. Which ordered pair is the solution of the sstem of linear equations whose graph is shown? A 0 B C 2 D Infinitel man 8. At a baker, one customer pas $5.67 for 3 bagels and 4 muffins. Another customer pas $6.70 for 5 bagels and 3 muffins. Which sstem of equations can be used to determine the cost (in dollars) of one bagel and the cost (in dollars) of one muffin at the baker? A 5 7 B C D The perimeter P (in feet) of each of the two rectangles below is given. What are the values of l and w? A (3, 2) B (0, 0) C (0, 2) D (23, ) 5. Which ordered pair is the solution of the linear sstem 3 52 and 52} 2 } 7? 2 2 P 5 24 ft l ft w ft P 5 42 ft (l 4) ft 2w ft A 5 }2, 2 7 } 2 2 B (, 24) C (2, 23) D (0, 2) 6. Which ordered pair is a solution of the sstem 2 22 and 23 4? A (0, 0) B (2, 22) A l 5 7 and w 5 5 B l 5 8 and w 5 4 C l 5 and w 5 0 D l 5 2 and w 5 9 C (22, 2) D (5, 24) 482 Chapter 7 Sstems of Equations and Inequalities

60 STATE TEST PRACTICE classzone.com GRIDDED ANSWER 0. What is the -coordinate of the solution of the sstem whose graph is shown? 22. What is the -coordinate of the solution of the sstem 5 } 3 8 and ? 2. A science museum charges one amount for admission for an adult and a lesser amount for admission for a student. Admission to the museum for 28 students and 5 adults costs $284. Admission for 40 students and 0 adults costs $440. What is the admission cost (in dollars) for one student? SHORT RESPONSE 3. Is it possible to find a value for c so that the linear sstem below has eactl one solution? Eplain Equation 52 5 } 3 c Equation 2 4. A rental car agenc charges dollars per da plus dollars per mile to rent an of the midsized cars at the agenc. The total costs for two customers are shown below. Customer Time (das) Distance (miles) Cost (dollars) Jackson Bree How much will it cost to rent a mid-sized car for 5 das and drive 250 miles? Eplain. EXTENDED RESPONSE 5. A baseball plaer s batting average is the number of hits the plaer has divided b the number of at-bats. At the beginning of a game, a plaer has a batting average of.360. During the game, the plaer gets 3 hits during 5 at-bats, and his batting average changes to.375. a. Write a sstem of linear equations that represents the situation. b. How man at-bats has the plaer had so far this season? c. Another plaer on the team has a batting average of.240 at the beginning of the same game. During the game, he gets 3 hits during 5 at-bats, and his batting average changes to.300. Has this plaer had more at-bats so far this season than the other plaer? Eplain. 6. A gardener combines fluid ounces of a 20% liquid fertilizer and 80% water mi with fluid ounces of a 5% liquid fertilizer and 95% water mi to make 30 fluid ounces of a 0% liquid fertilizer and 90% water mi. a. Write a sstem of linear equations that represents the situation. b. Solve the sstem from part (a). c. Suppose the gardener combines pure (00%) water and the 20% liquid fertilizer and 80% water mi to make the 30 fluid ounces of the 0% liquid fertilizer and 90% water mi. Is more of the 20% liquid fertilizer and 80% water mi used in this mi than in the original mi? Eplain. Standardized Test Practice 483

61 CUMULATIVE REVIEW Chapters 7 Evaluate the epression p (p. 8) (9 2 6) (p. 8) 3. 5[(6 2 2) 2 2 5] (p. 8) 4. Ï } 44 (p. 0) 5. 2Ï } 2500 (p. 0) 6. 6Ï } 400 (p. 0) Check whether the given number is a solution of the equation or inequalit. (p. 2) ; ; h 5 2; 0. g 2 3 > 2; ; p 5; 6 Simplif the epression. 3. 5( 2 ) 4 (p. 96) 4. 2w (w 2 2)3 (p. 96) 5. (g 2 )(24) 3g (p. 96) 6. 0h 2 25 } 5 (p. 03) } 27 (p. 03) m } 2 (p. 03) Solve the equation (p. 34) (p. 34) (p. 34) 22. } (p. 34) (p. 4) } (p. 4) 25. 3( 2 2) 525 (p. 48) 26. 3(5 2 7) (p. 54) (2 2 0) (p. 54) Graph the equation (p. 225) (p. 225) (p. 225) (p. 244) (p. 244) } 3 (p. 253) Write an equation of the line in slope-intercept form with the given slope and -intercept. (p. 283) 34. slope: slope: slope: 27 -intercept: 2 -intercept: 3 -intercept: 0 Write an equation in point-slope form of the line that passes through the given points. (p. 302) 37. (, 20), (25, 2) 38. (4, 7), (24, 3) 39. (29, 22), (26, 8) 40. (2, ), (, 23) 4. (2, 4), (8, 2) 42. (26, ), (3, 25) Solve the inequalit. Then graph our solution < 23 (p. 356) (p. 356) (p. 363) 46. } 24 > 7 (p. 363) < (p. 369) > 23 2 (p. 369) (p. 369) < (p. 369) < 3 2 < 5(p. 380) (p. 380) < 5 (p. 398) (p. 398) 484 Cumulative Review: Chapters 7

62 Solve the linear sstem using elimination. (p. 45) ART PROJECT You are making a tile mosaic on the rectangular tabletop shown. A bag of porcelain tiles costs $3.95 and covers 36 square inches. How much will it cost to bu enough tiles to cover the tabletop? (p. 28) 30 in. 24 in. 59. FOOD The table shows the changes in the price for a dozen grade A, large eggs over 4 ears. Find the average earl change to the nearest cent in the price for a dozen grade A, large eggs during the period (p. 03) Year Change in price for a dozen grade A, large eggs (dollars) HONEY PRODUCTION Honebees visit about 2,000,000 flowers to make 6 ounces of hone. About how man flowers do honebees visit to make 6 ounces of hone? (p. 68) 6. MUSIC The table shows the price p (in dollars) for various lengths of speaker cable. (p. 253) Length, l (feet) Price, p (dollars) a. Eplain wh p varies directl with l. b. Write a direct variation equation that relates l and p. 62. CURRENCY The table shows the echange rate between the currenc of Bolivia (bolivianos) and U.S. dollars from 998 to (p. 335) Year Bolivianos per U.S. dollar a. Find an equation that models the bolivianos per U.S. dollar as a function of the number of ears since 998. b. If the trend continues, predict the number of bolivianos per U.S. dollar in BATTERIES A manufacturer of nickel-cadmium batteries recommends storing the batteries at temperatures ranging from 220 C to 45 C. Use an inequalit to describe the temperatures (in degrees Fahrenheit) at which the batteries can be stored. (p. 380) Cumulative Review: Chapters 7 485