Applications of Linear Systems (For help, go to Lesson 2-5.) 1.Two trains run on parallel tracks....

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?

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ALGEBRA 1 LESSON 9-3ALGEBRA 1 LESSON 9-3

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Applications of Linear Systems

1. 55t = 65(t – 0.5)

55t = 65t – 32.5

32.5 = 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 – 27.5

27.5 = 5t

5.5 = t

50t = 50(5.5) = 275

It is 275 miles to the beach from their home.


Solutions

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Define: Let a = volume of the Let b = volume of the

50% solution. 25% solution.

Relate: volume of solution amount of acid

Write: a + b = 10 0.5 a + 0.25 b =

0.4(10)

Applications of Linear SystemsA 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 = 10 Solve for a. a = 10 – b Subtract b from each side.


9-3

Applications of Linear Systems(continued)

Step 2: Find b.

0.5a + 0.25b = 0.4(10) 0.5(10 – b) + 0.25b = 0.4(10) Substitute 10 – b for a.

Use parentheses. 5 – 0.5b + 0.25b = 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.


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Define: Let p = the number of pages.

Let d = the amount of dollars of expenses or income.

Relate: Expenses are per-page Income is priceexpenses plus times pages typed.computer purchase.

Write: d = 0.5 p + 1750 d = 5.5 p

Applications of Linear SystemsSuppose 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?


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Choose a method to solve this system. Use substitution since it is easy to substitute for d with these equations.

d = 0.5p + 1750 Start with one equation. 5.5p = 0.5p + 1750 Substitute 5.5p for d. 5p = 1750 Solve for p.

p = 350

To break even, you must type 350 pages.


9-3

Define: Let A = the airspeed. Let W = the wind speed.

Relate: with tail wind with head wind

(rate)(time) = distance (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 SystemsSuppose 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.


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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 = 2800 A + W = 500 Divide each side by 5.6.(A – W)6.8 = 2800 A – W 412 Divide each side by 6.8.

Step 1: Eliminate W. A + W = 500

A – W = 412 Add the equations to eliminate W. 2A + 0 = 912

Step 2: Solve for A. A = 456 Divide each side by 2.

Step 3: Solve for W using either of the original equations. A + W = 500 Use the first equation. 456 + 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.


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Applications of Linear Systems1. 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


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Applications of Linear Systems (For help, go to Lesson 2-5.) 1.Two trains run on parallel tracks....

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