12.1 Creating Systems of Linear Equations

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Name Class Date 12.1 Creating Sstems of Linear Equations Essential Question: How do ou use sstems of linear equations to model and solve real-world problems?

After leaving her hotel, a student is walking to a café to have breakfast and do some sightseeing. On the wa, she passes two stores that rent biccles. The first shop charges an initial fee of $7.

1 Name Class Date 12.1 Creating Sstems of Linear Equations Essential Question: How do ou use sstems of linear equations to model and solve real-world problems? Resource Locker Eplore Creating Linear Sstem Models b Changing Parameters Investigate how a sstem of equations can help ou compare and interpret situations where rates of change affect the outcome. After leaving her hotel, a student is walking to a café to have breakfast and do some sightseeing. On the wa, she passes two stores that rent biccles. The first shop charges an initial fee of $7. and $2.75 for each hour. The second shop charges a flat fee of $3. per hour. Over breakfast, the student needs to decide which rental agenc to use. How should she start? A Begin b finding functions that represent the cost of each rental. Let ƒ (t) represent the cost of renting a biccle for t hours from the first shop, and let g (t) represent the cost of renting a biccle from the second shop. Image Credits: James Marshall/Corbis B C D E F G H What is the initial cost of renting a biccle from the first shop? This represents the The slope of this linear model represents the The slope of the first model is. This makes ƒ (t) =. of the model for the first business. Similarl, the function modeling the cost for renting a biccle from the second shop is g (t) =. of the cost, as a function of time. Once the student decides the length of time she plans to spend on her bike ride, she can solve the linear sstem of the two functions, which compan she wants to rent a biccle., to determine from Module Lesson 1

2 Reflect 1. Discussion Under what conditions would this tpe of real-world situation have no solution? 2. Discussion Under what conditions would this tpe of real-world situation have infinitel man solutions? Eplain 1 Creating Linear Sstem Models from Verbal Descriptions Often, a compan will charge a start-up fee for its services, followed b a monthl or per unit cost. This can be written as a linear function in slope-intercept form. When the costs of the same services from two different companies must be compared, the variable amount for each must represent the same thing, and both models should produce values with the same tpe of unit. For eample, if one function models earl income in terms of thousands of dollars and the other function models monthl income in terms of hundreds of dollars, the comparison will not be accurate. Eample 1 Determine when the cost of the two services will be the same amount, and what the price will be. Video streaming service Atomic Stream charges $1 for membership and $1. for each movie download. Blitz Video charges $15 for a membership and $.5 per movie download. How man movies would ou need to download for the services to have identical costs? What is that cost? Analze Information Identif the important information. Atomic Stream has a membership fee. Atomic Stream has a per download fee. Blitz Video has a membership fee. Blitz Video has a per download fee. Formulate a Plan Create two functions to model the cost of each service, A () and B (), where represents the. The solution can be found b setting up an equation so that the function A () is to the function B () and then solving for. Module Lesson 1

3 Solve The model for Atomic Stream is A () =. The model for Blitz Video is B () =. The two functions are. Solve using substitution. You can use substitution because ou are solving for the value where A () B (). = A ( ) =.5 = 5 = = A () = The cost of each service is when movies are screened. Justif and Evaluate It is reasonable to epect the cost of the services to be the same after a number of uses. The businesses are in the same market but can appeal to different customers. Atomic Stream is more affordable for customers who stream movies a month, while Blitz Video is a better deal for people who stream movies a month. Your Turn 3. One cable television provider has a $6 setup fee and charges $8 per month, and another cable provider has a $16 equipment fee and charges $7 per month. 4. The Strauss famil is deciding between two lawn-care services. Green Lawn charges a $49 startup fee plus $29 per month. Yard Guard charges a $25 startup fee plus $37 per month. Module Lesson 1

4 Eplain 2 Creating Linear Sstem Models from Tables Sometimes, there is not enough information to model an equation. Businesses ma have a table of rates posted to eplain their pricing. To compare the cost of items or services from two or more businesses, table entries can be used to create and solve a linear sstem. Eample 2 Use the cost tables for two services to create a linear sstem of equations. Then solve the sstem to determine when the cost of the two services will be equal. Two garden suppl companies deliver mulch according to the following table. Mulch (Cubic Yards) Yard Depot Lawn & Garden 1 $6 $8 2 $9 $15 3 $12 $13 4 $15 $155 Yard Depot Use points (1, 6) and (2, 9). 9-6 m = _ 2-1 = 3 Write the equation. - 6 = 3 ( - 1) = Lawn & Garden Use points (1, 8) and (2, 15) m = _ 2-1 = 25 Write the equation. - 8 = 25 ( - 1) = The sstem of equations is: ƒ () = 3s + 3 g () = Solve for when ƒ () = g () to find the amount of cubic ards,, for which both companies charge the same amount. 3s + 3 = Both companies charge $18 for 5 cubic ards of mulch. = 5 ƒ (5) = 3 (5) + 3 = 18 Module Lesson 1

5 B The table shows canoe rental prices for two companies. Time t (in hours) Canoe Depot Paddle and Oar 1 $14 $2 2 $19 $23 3 $24 $26 Canoe Depot Use points (1, ) and (2, ). - m = = 2-1 Write the equation. - = ( - 1) = + Paddle & Oar Use points (1, ) and (2, ). - m = = 2-1 Write the equation. - = ( - 1) = + C () = The sstem of equations is + P () = + Solve for when C () = P () to find the number of hours,, for which both canoe rental places charge the same amount. + = + = C ( ) = Both companies charge $ for hours of canoe rental. Your Turn 5. Two garden suppl companies deliver pea stone according to the following table. Pea Stone (in cubic ards) Yard Depot Lawn & Garden 1 $75 $45 2 $11 $85 3 $145 $125 Yard Depot Lawn & Garden Module Lesson 1

6 6. Two beachfront stores rent surfboards according to the following table. Time t (in hours) Hang Ten Waverider 1 $28 $46 2 $48 $63 3 $68 $8 4 $88 $97 Hang Ten Waverider Eplain 3 Creating Linear Sstem Models from Graphs In newspapers and magazines, information is often displaed in the form of a graph. You can use the graph of a linear sstem to write the function models that are represented. Eample 3 Use the graph to make a linear model of each function. Describe the meaning of the terms in the models. Then create the linear sstem, and state what the solution represents. Cost ($) (, 4) h(t) (4, 12) g(t) (, 2) Time Bowled (hours) The sstem of equations is: g () = h () = g () The -intercept b is 4. Initial cost is $4. Use (, 4) and (4, 12) : m = _ = 2 Charge per hour: $2./h = The solution (4, 12) represents the same charge of $12 for 4 hours that both bowling alles charge. h () The -intercept b is 2. Initial cost is $2. Use (, 2) and (4, 12) : m = _ = 2.5 Charge per hour: $2.5/h = Module Lesson 1

7 B Cost ($) g() (7, 15) f() (, 8) (, 1) Trail Mi (pounds) f () The -intercept b is. Initial cost is $. Use (, ) and (7, 15) : 15 - m = _ = - Rate of change: $ /lb g () The -intercept b is. Initial cost is $. Use (, ) and (7, 15) : 15 - m = _ = - Rate of change: $ /lb = + = + ƒ () = The sstem of equations is + g () = + The solution (7, 15) represents the same charge of $ for pounds of trail mi. Your Turn 7. Use the graph to make a linear model of each function. Describe the meaning of terms in the models. Then create thelinear sstem and state what the solution represents. f () g () Cabin rental cost ($) g(t) f(t) (, 57) (7, 64) (, 15) Number of das Module Lesson 1

8 Elaborate 8. When writing a linear model of a situation, what does the slope represent? 9. Discussion Compare and contrast the sstem of equations that can be determined from a verbal description of a relationship, a table of values, and a graph. 1. Essential Question Check-In How do ou use sstems of linear equations to model and solve realworld problems? Evaluate: Homework and Practice 1. In the graph shown, what do the parameters of each line represent? Distance Traveled in Yards f(t) g(t) Time (min) Set up and solve a sstem of equations to solve the problem. Online Homework Hints and Help Etra Practice 2. Case wants to bu a gm membership. One gm has a $15 joining fee and costs $35 per month. Another gm has no joining fee and costs $6 per month. When would Case pa the same amount to be a member of either gm? How much would he pa? Module Lesson 1

3. A jar contains n nickels and d dimes. There are 2 coins in the jar, and the total value of the coins is $1.4. How man nickels and how man dimes are in the jar? 4.

9 3. A jar contains n nickels and d dimes. There are 2 coins in the jar, and the total value of the coins is $1.4. How man nickels and how man dimes are in the jar? 4. Helene invested a total of $1 in two simple-interest bank accounts. One account paid 5% annual interest; the other paid 6% annual interest. The total amount of interest she earned after 1 ear was $58. Find the amount invested in each account. 5. A local bos club sold 176 bags of mulch and made a total of $52. It sold two tpes of mulch: hardwood for $3.5 a bag and pine bark for $2.75 a bag. How man bags of each kind of mulch did it sell? Image Credits: bonchan/ Shutterstock 6. The school band sells carnations on Valentine s Da for $2 each. It bus the carnations from a florist for $.5 each, plus a $16 deliver charge. When will the cost of the carnations be equal to the revenue from selling them? How man carnations does it need to sell to reach this point? Use the given cost tables for the same product from two different companies to create a linear sstem. Then solve the sstem to determine when the cost of the product will be the same and what the price will be. 7. Two online spice retailers sell paprika b the pound using the following pricing chart. Paprika(lb) ispice Spice Magic 1 $15.75 $ $27.5 $ $39.25 $ $51 $57 Module Lesson 1

8. Two online retailers sell organic vanilla etract b the ounce using the following pricing chart. Vanilla Etract (oz) Chef Mate Grocer Gourmet 2 $12.5 $17 3 $17.25 $21 4 $22 $25 5 $26.75 $29 9.

10 8. Two online retailers sell organic vanilla etract b the ounce using the following pricing chart. Vanilla Etract (oz) Chef Mate Grocer Gourmet 2 $12.5 $17 3 $17.25 $21 4 $22 $25 5 $26.75 $29 9. Two dr cleaning companies offer a home pick-up and deliver service. The monthl cost depends on the number of garments laundered, and is shown in the following table. Number of Garments Compan 1 Compan 2 5 $55.25 $ $75.5 $ $95.75 $ A small town in the mountains needs to bu road salt for the coming winter. It has found two companies that use the following pricing table. Road Salt (tons) Compan 1 Compan 2 5 $1775 $275 1 $335 $4 15 $4925 $ A restaurant needs to stock paper towels in its kitchen and bathrooms. It has found two vendors using the following case price chart. Paper Towels (cases) Restaurant Warehouse Suppl Side 5 $3.2 $22.2 Image Credits: J/J Images - J Morrill Photo/Photolibrar/Gett Images 1 $48.15 $ $66.1 $62.1 Module Lesson 1

11 Use the graph to make a linear model of each function. Describe the meaning of the terms in the models. Then create the linear sstem, and state what the solution represents g(t) 1 g(t) Apples picked 5 (, 36) f(t) (5, 61) water (gal) 5 (, 22) f(t) (6, 64) (, 21) 5 1 Time (h) (, 4) 5 1 Time (min) g(t) (8, 74) f(t) Billing charges ($) 5 (, 26) Cost ($) 4 2 (, 125) g(t) f(t) (6, 23) (, 2) 5 1 Call time (min) (, 2) 5 1 Film length (min) Module Lesson 1

12 16. Two office suppl stores sell their brand of cop paper b the pound. One compan offers a flat rate shipping charge and the other offers free shipping. Use the graph provided to construct a linear sstem to model this situation. Solve the sstem to determine the amount of cop paper for which the cost is the same at both stores. Use the graph to verif that our answer is reasonable. 2 g() (45, 161.5) f() Cost ($) 1 (, 6) (4, 12) (1, 39) 25 5 Amount of cop paper (lb) 17. Find the Error A student is given the following problem: A painter can bu 5-gallon containers of paint from two different stores based on the following pricing table. Containers of Paint Compan A Compan B 2 $512. $ $94. $ $1296. $ $1688. $1658. The student s work is shown Compan A: m = = _ = 196 Compan B: m = _ = _ = 13 A () = b B () = 13 + b 512 = 196 (2) + b 834 = 13 (2) + b 12 = b 628 = b A () = = B () = = 58 = 5.5 He concludes that the answer is unreasonable because both of the prices for Compan A at = 4 and = 6 are greater than those of Compan B. After the two functions intersect, the prices for Compan B should be lower than the prices for Compan A. Is the student correct? If not, eplain his error and find the correct solution. Module Lesson 1

13 H.O.T. Focus on Higher Order Thinking 18. Two friends, Jorge and Mark, are taking a trip to the mountains for a camping trip, but the are not leaving together. Both friends separatel record the distance the each have traveled from home ever hour on the second da. The distance each friend is from his home can be modeled b a linear function of the hours spent traveling. Determine if the linear sstem created b the models for each situation has a unique solution with a positive value for time. a. Jorge and Mark each leave at the same time. Yes No b. Jorge travels 25 miles on the first da and drives 65 miles per hour on the second da. Mark travels 1 miles on the first da and also drives 65 miles per hour on the second da. c. Jorge travels 25 miles on the first da and drives 65 miles per hour on the second da. Mark travels 1 miles on the first da and drives 45 miles per hour on the second da. d. Jorge traveled 45 miles on the first da and drives 55 miles per hour on the second da. Mark arrived at the campsite on the first da after traveling 3 miles. e. Jorge gets sick before the trip and doesn t get in touch with Mark. Mark travels 4 miles on the first da and drives 67 miles per hour on the second da. Yes Yes Yes Yes No No No No 19. Communicate Mathematical Ideas Given a set of data measuring the distance two planes have traveled after takeoff as a function of when the both passed over the same point, how would ou find when the have both traveled the same distance since takeoff? 2. Analze Relationships How can ou use the slope and the -intercept of each model in a linear sstem to determine whether or not there will be a solution? Module Lesson 1

Lesson Performance Task A famil is going on vacation and the need to bring their dog to a kennel. Alpha Kennel charges an initial fee of $75 and a dail rate of $3.

14 Lesson Performance Task A famil is going on vacation and the need to bring their dog to a kennel. Alpha Kennel charges an initial fee of $75 and a dail rate of $3. Beta Kennel charges a flat fee of $34.95 a da. Find linear functions modeling the cost of boarding a dog for n das in each kennel. Set up and solve a sstem of linear equations. Then interpret the solution. What if the famil has two dogs? Alpha Kennel runs a special where ou receive a 1% discount if ou board more than one pet. Modif the linear models to give the price of boarding two dogs. Set up, solve, and interpret the linear sstem covering this case. Image Credits: taviphoto/shutterstock Module Lesson 1

Math Chapters 1 and 2 review

Math Chapters 1 and 2 review Math 100 - Chapters 1 and 2 review Name Solve the problem. 1) A person puts some warm turke leftovers in the refrigerator. Let T represent the temperature of the turke at M minutes after putting the turke