Applications of Linear Systems (For help, go to Lesson 2-5.) 1.Two trains run on parallel tracks. The first train leaves a city hour before the second train. The first train travels at 55 mi/h. The second train travels at 65 mi/h. Find how long it takes for the second train to pass the first train. 2.Luis and Carl drive to the beach at an average speed of 50 mi/h. They return home on the same road at an average speed of 55 mi/h. The trip home takes 30 min. less. What is the distance from their home to the beach? 1212 ALGEBRA 1 LESSON

Applications of Linear Systems 1. 55t = 65(t – 0.5) 55t = 65t – = 10t 3.25 = t It takes 3.25 hours for the second train to pass the first train. 2.50t = 55(t – 0.5) 50t = 55t – = 5t 5.5 = t 50t = 50(5.5) = 275 It is 275 miles to the beach from their home. ALGEBRA 1 LESSON 9-3 Solutions 9-3

Define:Let a = volume of theLet b = volume of the 50% solution.25% solution. Relate:volume of solutionamount of acid Write: a + b = a b = 0.4(10) Applications of Linear Systems A chemist has one solution that is 50% acid. She has another solution that is 25% acid. How many liters of each type of acid solution should she combine to get 10 liters of a 40% acid solution? Step 1:Choose one of the equations and solve for a variable. a + b = 10Solve for a. a = 10 – bSubtract b from each side. ALGEBRA 1 LESSON

Applications of Linear Systems (continued) Step 2:Find b. 0.5a b = 0.4(10) 0.5(10 – b) b = 0.4(10)Substitute 10 – b for a. Use parentheses. 5 – 0.5b b = 0.4(10) Use the Distributive Property. 5 – 0.25b = 4 Simplify. –0.25b = –1 Subtract 5 from each side. b = 4 Divide each side by –0.25. Step 3:Find a. Substitute 4 for b in either equation. a + 4 = 10 a = 10 – 4 a = 6 To make 10 L of 40% acid solution, you need 6 L of 50% solution and 4 L of 25% solution. ALGEBRA 1 LESSON

Define:Let p = the number of pages. Let d = the amount of dollars of expenses or income. Relate:Expenses are per-pageIncome is price expenses plustimes pages typed. computer purchase. Write: d = 0.5 p d = 5.5 p Applications of Linear Systems Suppose you have a typing service. You buy a personal computer for $1750 on which to do your typing. You charge $5.50 per page for typing. Expenses are $.50 per page for ink, paper, electricity, and other expenses. How many pages must you type to break even? ALGEBRA 1 LESSON

Applications of Linear Systems (continued) Choose a method to solve this system. Use substitution since it is easy to substitute for d with these equations. d = 0.5p Start with one equation. 5.5p = 0.5p Substitute 5.5p for d. 5p = 1750Solve for p. p = 350 To break even, you must type 350 pages. ALGEBRA 1 LESSON

Define:Let A = the airspeed.Let W = the wind speed. Relate: with tail wind with head wind (rate)(time) = distance (A + W) (time) = distance (A + W) (time) = distance Write:(A + W)5.6 = 2800 (A + W)6.8 = 2800 Applications of Linear Systems Suppose it takes you 6.8 hours to fly about 2800 miles from Miami, Florida to Seattle, Washington. At the same time, your friend flies from Seattle to Miami. His plane travels with the same average airspeed, but this flight only takes 5.6 hours. Find the average airspeed of the planes. Find the average wind speed. ALGEBRA 1 LESSON

Applications of Linear Systems (continued) Solve by elimination. First divide to get the variables on the left side of each equation with coefficients of 1 or –1. (A + W)5.6 = 2800A + W = 500Divide each side by 5.6. (A – W)6.8 = 2800A – W 412Divide each side by 6.8. Step 1:Eliminate W. A + W = 500 A – W = 412Add the equations to eliminate W. 2A + 0 = 912 Step 2:Solve for A. A = 456Divide each side by 2. Step 3:Solve for W using either of the original equations. A + W = 500 Use the first equation W = 500 Substitute 456 for A. W = 44 Solve for W. The average airspeed of the planes is 456 mi/h. The average wind speed is 44 mi/h. ALGEBRA 1 LESSON

Applications of Linear Systems 1.One antifreeze solution is 10% alcohol. Another antifreeze solution is 18% alcohol. How many liters of each antifreeze solution should be combined to create 20 liters of antifreeze solution that is 15% alcohol? 2.A local band is planning to make a compact disk. It will cost $12,500 to record and produce a master copy, and an additional $2.50 to make each sale copy. If they plan to sell the final product for $7.50, how many disks must they sell to break even? 3.Suppose it takes you and a friend 3.2 hours to canoe 12 miles downstream (with the current). During the return trip, it takes you and your friend 4.8 hours to paddle upstream (against the current) to the original starting point. Find the average paddling speed in still water of you and your friend and the average speed of the current of the river. Round answers to the nearest tenth. 7.5 L of 10% solution; 12.5 L of 18% solution 2500 disks still water: 3.1 mi/h; current: 0.6 mi/h ALGEBRA 1 LESSON