12.3 Modeling with Linear Systems

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1 Name Class Date 1.3 Modeling with Linear Sstems Essential Question: How can ou use sstems of linear equations or inequalities to model and solve contetual problems? Resource Locker Eplore Modeling Real-World Constraints with Sstems Real-world situations can often be modeled b sstems of equations. Usuall, information about prices and the total number of items purchased is given, and the sstem is solved to find the number of each item purchased. Houghton Mifflin Harcourt Publishing Compan Image Credits: Dann Clifford/Alam Joe goes to the store to bu jeans and some T-shirts. The jeans cost $ each and the T-shirts cost $ each. If Joe spends $1 on 5 items, how man pairs of jeans and how man T-shirts did he bu? A B C D E F G Write an epression to represent the amount that Joe spent on pairs of jeans. Write an epression to represent the amount that Joe spent on T-shirts. Now write an equation that represents the total amount spent on jeans and T-shirts. Amount spent on jeans + Amount spent on T-shirts = + = What variable represents the number of jeans purchased? What variable represents the number of T-shirts purchased? Total amount spent Write an equation to represent the total number of items purchased. Write the sstem that represents the situation. Reflect = 1 1. What units are associated with the two epressions that ou wrote in steps A and B? = 5. When ou add the units for the epressions representing the amounts spent on jeans and T-shirts, what units do ou get for the total amount spent? Module Lesson 3

2 Eplain 1 Modeling Real-World Constraints with Sstems of Linear Equations You can model real-world constraints, such as the number of items needed and the amount of mone one has to spend, with sstems of linear equations. Eample 1 Write a sstem of equations to represent the situation, and then solve the sstem. A Bobb will bu coffee and hot chocolate for his co-workers. Each cup of coffee costs $.5 and each cup of hot chocolate costs $1.5. If he pas a total of $15.75 for cups, how man of each did he bu? Create a table to organize the information. Coffee Hot Chocolate Total Number of Cups c h Cost $.5c $1.5h $15.75 Use the information to write a sstem of equations..5c + 1.5h = Total amount spent on c cups of coffee and h cups of hot chocolate c + h = Total number of cups bought Multipl the second equation b -.5 to get opposite coefficients for c. -.5(c + h = ) -.5c -.5h = 1 Add the new equation to the first equation..5c + 1.5h = (-.5c -.5h = -1) -.75h = -.5 Solve for h. -.75h = -.5 h = 3 Substitute the value found for h back into one of the original equations and solve for c. c + h = c + 3 = Houghton Mifflin Harcourt Publishing Compan Image Credits: Fotofermer/Shutterstock c = 5 So Bobb bought 5 cups of coffee and 3 cups of hot chocolate. Module 1 55 Lesson 3

3 B A student is buing pens and markers for school. Packs of pens cost $.75 each and packs of markers cost $3.5 each. If she bought a total of packs and spent $17.5, how man of each did she bu? Create a table to organize the information. Pens Markers Total Number of packs p m Cost $17.5 Use the information to write a sstem of equations. + = 17.5 Total amount spent on p packs of pens and m packs of markers p + m = Total number of packs bought Multipl the second equation b to get opposite coefficients for p. (p + m = ) p + m = Add the new equation to the first equation. p + m = p + m = m = Solve for m, the number of markers. m = Substitute the value found for m back into one of the original equations and solve for p. p + m = Houghton Mifflin Harcourt Publishing Compan p + = p = So the student bought packs of pens and packs of markers. Reflect 3. What s another possible wa to solve the problem? Module Lesson 3

4 Your Turn Write a sstem of equations to represent the situation, and then solve the sstem.. A compan has to bu computers and printers. Each computer costs $55 and each printer costs $39. If the compan spends $1 and bus a total of 1 machines, how man of each did it bu? Eplain Modeling Real-World Constraints with Sstems of Linear Inequalities You can use a sstem of linear inequalities and its graph to model man real-world situations. Eample Set up and solve the sstem of linear equalities. A Number of shorts Sue is buing T-shirts and shorts. T-shirts cost $1 and shorts cost $1. She plans on spending no more than $17 and bu at least 5 items. Show and describe all combinations of the number of T-shirts and shorts she could bu. First write the sstem. Let represent the number of T-shirts, and let represent the number of shorts. + 5 She wants to bu at least 5 items She wants to spend no more than $17. Graph the sstem of inequalities: T-shirts and Shorts 5 7 Number of T-shirts The possible solutions are where the shaded regions overlap. So, a possible solution is 5 T-shirts and shorts. Substitute this value into the inequalities to make sure it is a reasonable solution.? ? 1 (5) + 1 () The result is two inequalities that are true, so this is a reasonable answer. Houghton Mifflin Harcourt Publishing Compan Module 1 5 Lesson 3

5 B John has to bu two different kinds of rope. Rope A costs $. per foot and Rope B costs $.9 per foot. John needs to bu at least 15 feet of rope, but he wants to spend no more than $1. Show and describe all combinations of the number of feet of each tpe of rope John can bu. First write the sstem. Let represent the amount of Rope A, and let represent the amount of Rope B Graph the sstem. Rope B (ft) Buing Rope Rope A (ft) Describe the solutions to the sstem. The possible solutions are Your Turn Write a sstem of inequalities for the given situation and graph the sstem. Then determine if the point (, ) is a solution to the sstem. 5. A student has to bu graph paper and printer paper. The printer paper costs $ a pack, while the graphing paper costs $3 a pack. She wants to bu at least packs of paper but wants to spend at most $7. Buing Paper Houghton Mifflin Harcourt Publishing Compan Graphing paper (packs) Printer paper (packs) Module 1 51 Lesson 3

6 . Now assume that she wants to bu at least 7 packs and will spend at most $3. Graphing paper (packs) Buing Paper Printer paper (packs) Elaborate 7. Is it possible for a sstem of two linear inequalities to onl have one solution?. Wh can t a sstem of inequalities be solved using the same methods as solving sstems of equations? 9. Essential Question Check-In When writing a sstem of equations or inequalities from a situation, how do ou know that ou have possibl written the sstem correctl? Evaluate: Homework and Practice Write a sstem of equations that corresponds to the situation. Do not solve. Online Homework Hints and Help Etra Practice 1. Lisa spends part of her ear as a member of a gm. She then finds a better deal at another gm, so she cancels her membership with the first gm and spends the rest of the ear with the second gm. The membership to the first gm costs $75 per month, while the membership for the second gm costs $5 per month. She ends up spending a total of $775 over the course of the ear. Number of Months Cost First Gm Second Gm Total Houghton Mifflin Harcourt Publishing Compan Module 1 5 Lesson 3

7 . Jack is selling tickets to an event. Attendees can either bu a general admission ticket or a VIP ticket. The general adimission tickets are $ and the VIP tickets are $9. He doesn t know how man of each tpe he has sold, but he knows he sold a total of 9 tickets and made $1. Number of Tickets Cost General Admission VIP Total 3. There are adults and 3 children at a zoo. The zoo makes a total of $7 from the entrance fees, and the cost for an adult and a child to attend is $3. Price Revenue Adult Child Total. A local fish market is selling fish and lobsters b the pound. The fish costs $.5 a pound, while the lobster costs $9.5 a pound. The fish market sells 5.5 pounds and makes $ Houghton Mifflin Harcourt Publishing Compan Image Credits: Rand Duchaine/Alam Fish Lobster Total Weight Revenue 5. Jennifer has 1 nickels and dimes. The value of her coins is $1. Nickels Dimes Total Number of Coins Amount. The sum of 5 times one number and times a second number is 57. The sum of the two numbers is 1. First Number Second Number Total Value Value Module 1 53 Lesson 3

8 7. Gar goes to the grocer store to bu hot dogs and hamburgers for a cookout. He bus a total of packages for a total of $.5. A package of hot dogs costs $.9 and a package of hamburgers costs $5.9. Hot Dogs Hamburgers Total Value Amount. The sum of two numbers is, and the sum of times the first number and 3 times the second number is 15. First Number Second Number Total Value Value Find a sstem of equations that corresponds to the situation and then solve the resulting sstem. 9. Jan spends part of her ear as a member of a gm. She then finds a better deal at another gm, so she cancels her membership with the first gm and spends the rest of the ear with the second gm. The membership to the first gm costs $ per month, while the membership for the second gm costs $5 per month. If she ends up spending a total of $5 over the course of the ear, how much time did she spend at each gm? 1. John is selling tickets to an event. Attendees can either bu a general admission ticket or a VIP ticket. The general adimission tickets are $7 and the VIP tickets are $15. If he knows he sold a total of 33 tickets and made $73, how man of each tpe did he sell? Houghton Mifflin Harcourt Publishing Compan Module 1 5 Lesson 3

9 11. There are 15 adults and 5 children at a zoo. If the zoo makes a total of $51 from the entrance fees, and the cost of an adult and a child to attend is $31, how much does it cost each for a parent and a child? 1. A local fish market is selling fish and lobsters b the pound. The fish costs $5.5 a pound, while the lobster costs $1.5 a pound. The fish market sells.5 pounds and makes $15.5. Houghton Mifflin Harcourt Publishing Compan Image Credits: Melvn Longhurst/Alam 13. Nicole has 15 nickels and dimes. If the value of her coins is $1., how man of each coin does she have? Module 1 55 Lesson 3

10 1. The sum of times one number and 3 times a second number is. If the sum of the two numbers is 19, find the two numbers. 15. Meaghan goes to the grocer store to bu hot dogs and hamburgers for a cookout. She bus a total of packages for a total of $3.. If a package of hot dogs costs $.5 and a package of hamburgers costs $.9, determine how man packs of each she bought. 1. The sum of two numbers is 33, and the sum of 7 times the first number and 5 times the second number is 197. Houghton Mifflin Harcourt Publishing Compan Module 1 5 Lesson 3

11 Write the sstem of inequalities that represents the situation. Then graph the sstem and describe the solutions. Give one possible solution. 17. Angelique is buing towels for her apartment. She finds some green towels that cost $ each and blue towels that cost $1 each. She wants to bu at least towels but doesn t want to spend more than $7. How man of each towel can she purchase? Number of blue towels 1 Buing Towels 1 Number of green towels 1. The sum of two numbers is at least, and the sum of one of the numbers and 3 times the second number is no more than 15. Houghton Mifflin Harcourt Publishing Compan Values of second number Values of first number 19. The sum of two numbers is at most 1, and the sum of 3 times the first number and times the second number is at least. Values of second number Values of first number Module 1 57 Lesson 3

12 . Katie is purchasing plates and mugs for her house. She would like to bu at least items. Determine the possibilities if the plates cost $ each and the mugs cost $7 each, and she plans to spend no more than $11. Mugs Buing Plates and Mugs Plates 1. Christine is selling tickets at a museum. She knows that she has sold at least tickets. The adult tickets cost 1 dollars and the children s tickets cost 1 dollars. If she knows she has sold no more than $7 worth of tickets, what are the possible combinations? Children Selling Tickets 3 Adults. Mike is bringing cans and bottles to a reccling center. For a tpe A can or bottle he gets 5 cents, and for a tpe B can or bottle he gets 1 cents. He knows that he has redeemed at least 11 cans but has no more than 95 cents. What are the possible combinations? Tpe B Reccling Tpe A Houghton Mifflin Harcourt Publishing Compan Module 1 5 Lesson 3

13 H.O.T. Focus on Higher Order Thinking 3. Eplain the Error A student is given the following sstem. He graphs the sstem as shown and determines that a solution is (7, ). Where did the student go wrong? What should the correct answer be? + = + = Justif Reasoning Moll went shopping to bu jewelr. All of the earrings cost $15.5 and the necklaces cost $.75. If she spends $17.5 and bus 5 items, how man necklaces and pairs of earrings did she bu? Justif our answer. Houghton Mifflin Harcourt Publishing Compan 5. Check for Reasonableness Chris is at the florist and has to bu flowers. Arrangements of daisies are $.5 and arrangements of roses are $.5. He wants to spend less than $39 and wants to bu more than arrangements. What are possible combinations that Chris can bu? Check to make sure our answer is reasonable. Roses Flower Arrangements Daisies Module 1 59 Lesson 3

14 Lesson Performance Task Am is at the store to bu shirts and pants. The shirts cost $ each and the pants cost $5 each. She plans to spend no more than $ and bu at least 5 items. Find a possible combination of shirts and pants she can bu. How do ou know this is a solution? What are two possible was to show that this is a solution? Number of Pants Possible Options p s Number of Shirts Houghton Mifflin Harcourt Publishing Compan Module 1 57 Lesson 3