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Solving Application Problems with Linear Systems

Learn how to use linear systems to solve real-world application problems involving two variables. Understand the steps to define variables, set up equations, solve the system, and find the answer.

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Solving Application Problems with Linear Systems

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  1. 4.4 – Applications of systems with two variables

  2. Using Linear Systems to Solve Application Problems: • 1. Define the variables. There will be two unknown values that you are trying to find. Give each one a variable to represent them in the system of equations. You can use x and y or any other variable as long as you explicitly state what each variable represents. • 2. Find two equations in the problem. This can be challenging. Somewhere in the problem you will be given two relationships among the variables. Use these relationships to set up two linear equations. • 3. Solve the system. Use substitution, linear combination, or Cramer’s rule to solve the system and find the value of both variables. • 4. Answer the question. Make sure to state the value of each variable with it’s appropriate units.

  3. 1. A coin bank contains 30 coins, all dimes and quarters, worth $5.70. How many dimes and how many quarters are in the bank • Define the variables. • Find two equations in the problem. • Solve the system. • Answer the question.

  4. 2. PHS collected $760 from 200 people that attended the school play. If each adult ticket cost $5 and if each student ticket cost $3, how many students and how many adults attended the play? 3. At a restaurant, four hamburgers and two orders of fries cost $27.10. Three hamburgers and four orders of fries cost $25.20. If all hamburgers cost the same price and all orders of fries cost the same price, find the cost of each.

  5. Wind Speed and water current • Tail Wind: a wind blowing in the same direction as the one in which the airplane is heading • Head Wind: a wind blowing in the direction opposite to the one in which the airplane is heading. • Wind Speed: the speed of the wind • Air Speed: the speed of the airplane in still air. • Ground Speed: the speed of the airplane relative to the ground. • With a tail wind, and airplane’s ground speed is the of its air speed and the wind speed. • With a head wind, the ground speed is the between the air speed and the wind speed. sum difference

  6. 4. With a given head wind, a certain airplane can travel 3600 km in 9 h. Flying in the opposite direction with the same wind blowing, the airplane can fly the same distance in 1 h less. Find the airplane’s speed and the wind speed.

  7. The same is true when working with a boat and water current. When a boat is traveling upstream, it is going against the water current. When it is going downstream, it is going with the water current. • Questions 5-10: Let b be the speed (in km/h) of a boat in still water and c be the speed (in km/h) of the current in a river. Translate each sentence into an equation involving b and c. • 5. The speed of the boat going upstream was 14 km/h. • 6. The speed of the boat going downstream was 16 km/h. • 7. The boat traveled 6 km upstream in 30 min. • 8. The boat traveled 6 km downstream in 20 min. • 9. With the current 2 km/h faster, the boat could travel 5 km downstream in 15 minutes. • 10. If the speed of the current were twice as great, a trip of 6 km upstream would take 40 minutes. • 11. Use the relationships in #7 and #8 to determine the speed of the boat in still water and the speed of the current.

  8. Solutions • 5. 6. • 7. 8. • 9. • 10. • 11. boat speed = 15 km/h, current speed= 3 km/h

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