1. 7-4 Applications of Linear Systems

2. # of Coins: 12 coins total, adding quarters and dimes together Q + D = 12 Value of Coins: Have $1.95 total Quarters = $0.25 Dimes = $0.10.25Q +.10D = 1.95 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? System: Q + D = 12.25Q +.10D = 1.95 System: Q + D = 12.25Q +.10D = 1.95

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

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

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

6. # of Coins: 28 coins total, adding quarters and dimes together Q + D = 28 Value of Coins: Have $5.20 total Quarters = $0.25 Dimes = $0.10.25Q +.10D = 5.20 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? System: Q + D = 28.25Q +.10D = 5.20 System: Q + D = 28.25Q +.10D = 5.20

7. Example 4 Solution: Using substitution! System: Q + D = 28.25Q +.10D = 5.20 System: Q + D = 28.25Q +.10D = 5.20 “Easy” variable to solve for is in first equation. (D is “easy” too!) Pattern: 1, 2, 1 1. 1 Q + D = 28 -D Q = 28 - D 25(28 – D) + 10D = 520 2 Get rid of decimals 25Q + 10D = 520 *Multiply equation #2 by 100! 2. 700 – 25D + 10D = 520 700 – 15D = 520 -700 – 15D = -180 -15 D = 12

8. Example 4 Solution: D = 12 Pattern: 1, 2, 1 1 Q = 28 - D System: Q = 28 - D 25Q + 10D = 520 System: Q = 28 - D 25Q + 10D = 520 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 24kg 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. Fruit Drink A 4% Sugar Fruit Drink B 8% Sugar Mixed Fruit Drink 5% Sugar Fruit Drink (kg) Sugar (kg) A B24.04A.08B.05(24) Don’t forget to convert percents to decimals!

10. Example 5 Solution: System: A + B = 24.04A +.08B = 1.2 System: A + B = 24.04A +.08B = 1.2 18 kg of fruit drink A and 6 kg of fruit drink B.

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

12. Example 6 Solution: System: A + W = 480 A – W = 400 System: A + W = 480 A – W = 400 A + W = 480 Using elimination! A – W = 400 2A = 880 2 2 A = 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.