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7-4

7-4. Applications of Linear Systems.

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  1. 7-4 Applications of Linear Systems

  2. Example 1Suppose 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: Value of Coins: Have $1.95 total Quarters = $0.25 Dimes = $0.10 .25Q + .10D = 1.95 • 12 coins total, adding quarters and dimes together Q + D = 12 System: Q + D = 12 .25Q + .10D = 1.95

  3. 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 M = money Expenses: Income: Sell each shirt for $10.99 (only income!) M = 10.99T • Spent $280 on miscellaneous supplies • Spent $3.99 per shirt M = 3.99T + 280 System: M = 3.99T + 280 M = 10.99T To Solve: M = M 3.99T + 280 = 10.99T Break even means when your expenses = income! M = M

  4. Example 3Suppose 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) Renting: Buying: Spend $400 flat rate to buy equipment Spend $35 per day M = 35D + 400 Let: D = days M = money • Spend $60 per day M = 60D System: M = 60D M = 35D + 400 To Solve: M = M 60D = 35D + 400 Break-even point is when the cost for renting = the cost for buying! M = M

  5. Example 3 Solution: Setting them equal to each other means that the renting price equals the purchasing price! To Solve: M = M 60D = 35D + 400 60D = 35D + 400 -35D -35D You would have to ski for 16 days for the price of purchasing the ski equipment to equal the price of renting per day. 25D = 400 25 25 D = 16

  6. 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: Value of Coins: Have $5.20 total Quarters = $0.25 Dimes = $0.10 .25Q + .10D = 5.20 • 28 coins total, adding quarters and dimes together Q + D = 28 System: Q + D = 28 .25Q + .10D = 5.20

  7. “Easy” variable to solve for is in first equation. (D is “easy” too!) Example 4 Solution: Using substitution! *Multiply equation #2 by 100! System: Q + D = 28 .25Q + .10D = 5.20 Get rid of decimals 1. Pattern: 1, 2, 1 2. 25Q + 10D = 520 1 2 25(28 – D) + 10D = 520 700 – 25D + 10D = 520 700 – 15D = 520 Q + D = 28 -700 -700 -D -D – 15D = -180 Q = 28 - D -15 -15 D = 12

  8. System: Q = 28 - D 25Q + 10D = 520 Example 4 Solution: Pattern: 1, 2, 1 D = 12 1 Q = 28 - D Q = 28 – 12 Q = 16 You have 16 quarters and 12 dimes in your pocket.

  9. Example 5 Suppose you want to combine two types of fruit to drink to create 24kgof 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! B 24 A .04A .05(24) .08B

  10. Example 5 Solution: System: A + B = 24 .04A + .08B = 1.2 18 kg of fruit drink A and 6 kg of fruit drink B.

  11. Example 6 Aplane 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, which plane is faster? Airspeed is the speed of an aircraft! Wind speed is the speed of the wind! Rate = airspeed + wind speed (faster!) Rate = airspeed – wind speed (slower!) r = A + W r = A – W d = (A + W)(t) d = (A – W)(t) 2400 = (A + W)(5) 2400 = (A – W)(6) 5 5 6 6 480 = A + W 400 = A – W

  12. Example 6 Solution: Using elimination! System: A + W = 480 A – W = 400 A + W = 480 A + W = 480 440 + W = 480 A – W = 400 -440 -440 2A = 880 W = 40 2 2 A = 440 The average airspeed of the planes is 440 mph and the average wind speed is 40 mph.

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