7 4 Applications of Linear Systems Example 1

7 -4 Applications of Linear Systems

Example 1 Suppose you have just enough money, in coins, to pay for a loaf of bread priced at $1. 95. You have 12 coins, all quarters and dimes. Let Q equal the number of quarters and D equal the number of dimes. Write a system of equations to solve the problem. How many quarters do you have? Dimes? # of Coins: • 12 coins total, adding quarters and dimes together Value of Coins: • Have $1. 95 total • Quarters = $0. 25 • Dimes = $0. 10. 25 Q +. 10 D = 1. 95 Q + D = 12 System: Q + D = 12. 25 Q +. 10 D = 1. 95

Example 2 Several students decide to start a T-shirt company. After initial expenses of $280, they purchase each T-shirt wholesale for $3. 99. They sell each T-shirt for $10. 99. How many must they sell to break even? Let: T = T-shirts Expenses: • Spent $280 on miscellaneous supplies • Spent $3. 99 per shirt M = money Income: • Sell each shirt for $10. 99 (only income!) M = 3. 99 T + 280 hen w s an e me! m o n c e n k ev nses = i a e r B pe x e M r M= you M = 10. 99 T To System: Solve: M = 3. 99 T M = M+ 280 3. 99 TM+ =280 10. 99 T = 10. 99 T

Example 3 Suppose you are trying to decide whether to buy ski equipment. Typically, it costs you $60 a day to rent ski equipment and buy a lift ticket (the ticket is included in that rate). You can buy ski equipment for about $400. A lift ticket alone costs $35 for one day. How many days must you ski for it to be worth it to buy the equipment? (break-even point) Let: D = days Renting: • Spend $60 per day M = money Buying: • Spend $400 flat rate to buy equipment • Spend $35 per day M = 60 D M = 35 D + 400 en h w int is = the o p ven enting e k Brea st for r ! o g the c or buyin M f M= cost To System: Solve: M M == 60 D M ==35 D 35 D++400

l to at a u eq ns th uals em ea eq ! h g t er m rice n i tt oth ng p e S ch nti si ea e re rcha th e pu th Example 3 Solution: To Solve: M=M 60 D = 35 D + 400 Yo -35 D 25 D = 400 25 25 D = 16 uw ould for 16 d hav pric e e of ays for to ski the t ski e purcha he equ s al th quipm ing ren e ting e price nt to per o day f.

Example 4 You have 28 coins in your pocket, consisting of only quarters and dimes. If the total amount of money in your pocket is $5. 20, how many quarters and dimes do you have? # of Coins: • 28 coins total, adding quarters and dimes together Value of Coins: • Have $5. 20 total • Quarters = $0. 25 • Dimes = $0. 10. 25 Q +. 10 D = 5. 20 Q + D = 28 System: Q + D = 28. 25 Q +. 10 D = 5. 20

ing s U su ti bs ! it on “Easy” variable to solve for is in first equation. (D is “easy” too!) Example 4 Solution: tu System: Q + D = 28. 25 Q +. 10 D = 5. 20 2. 25 Q + 10 D = 520 1. Pattern: 1, 2, 1 1 Q + D = 28 -D -D Q = 28 - D *M u equ ltiply by 1 ation # 2 00! Get rid of decimals 2 25(28 – D) + 10 D = 520 700 – 25 D + 10 D = 520 700 – 15 D = 520 -700 – 15 D = -180 -15 D = 12

System: Pattern: 1, 2, 1 Example 4 Solution: Q = 28 - D 25 Q + 10 D = 520 D = 12 1 Q = 28 - D Q = 28 – 12 Q = 16 You and have 1 6 qu 1 2 poc d ket. imes i arters n yo ur

Example 5 Suppose you want to combine two types of fruit to drink to create 24 kg of a drink that will be 5% sugar by weight. Fruit drink A is 4% sugar by weight and fruit drink B is 8% sugar by weight. Don’t forget to convert percents to decimals! Fruit Drink (kg) Sugar (kg) Fruit Drink A Fruit Drink B Mixed Fruit 4% Sugar 8% Sugar Drink 5% Sugar A B 24 . 04 A . 08 B . 05(24)

Example 5 Solution: System: A + B = 24. 04 A +. 08 B = 1. 2 18 kg of fruit drink A and 6 kg of fruit drink B.

Example 6 A plane takes about 6 hours to fly you 2400 miles from NYC to Seattle. At the same time, your friend flies from Seattle to NYC. His plane travels with the same average airspeed, but his flight takes 5 hours. Find the average airspeed of the planes. Find the average wind speed. Let: A = airspeed W = wind speed So, Airspeedwhich is the speed ofplane an aircraft! is faster? Wind speed is the speed of the wind! Rate = airspeed + wind speed (faster!) faster! r=A+W d = (A + W)(t) W 2400 = (A + W)(5) 5 480 = A + W 5 Rate = airspeed – wind speed (slower!) slower! r=A–W d = (A – W)(t) W 2400 = (A – W)(6) 6 400 = A – W 6

! n tio na i m li ge in Us Example 6 Solution: System: A + W = 480 A – W = 400 2 A = 880 2 A 2 = 440 A + W = 480 440 + W = 480 -440 W = 40 The average airspeed of the planes is 440 mph and the average wind speed is 40 mph.