Applications of systems of linear equations with 2 variables

1. Applications of systems of linear equations with 2 variables

2. An interesting problem A 600-seat movie theater charges $5.50 admission for adults and $2.50 for children. If the theater is full, and $1911 is collected, how many adults and how many children are in the audience?

3. To solve a real-life problem like this, we need to • formulate it as a mathematical problem, • solve the mathematical problem, and • interpret the solution of that problem

4. There are quantities: the number of children and the number of adults two unknown two variables, Define them as say x and y, respectively. x= number of children in attendance y= number of adults in attendance

5. We obtain two equations: x+y=600 Since the theater is full, Since $1911 is collected, 2.50x+5.50y=1911 Observe that both equations are linear (in standard form) x+y = 600 2.50x+5.50y = 1911 We call such a set of equations a system of linear equations with two variables

6. Solving a system of linear equations with two variables: • by graphing (See page 170) • by substitution (See page 173) • by elimination (See page 174)

7. Example #1:Solve the system obtained for the movie theater problem by graphing. x+y = 600 2.50x+5.50y = 1911 x = 463y = 137 We conclude that there are 463 children and 137 adults in attendance.

8. Example #2:Your local grocery store does not mark prices on its goods. Your mate went to this store, bought three 1-lb packages of bacon and two cartons of eggs, and paid a total of $7.45. Not knowing that you mate went to the store, on your way from work, you went to the same store, purchased two 1-lb packages of bacon and three cartons of eggs, and paid a total of $6.45.Now you want to return two 1-lb packages of bacon and two cartons of eggs. Your friend, a former MA 110 student, says that if you first solve the system you can easily find how much will be refunded. 3x+2y =7.45 2x+3y =6.45 x= cost of 1-lb package of bacon y= cost 1 cartons of eggs Each carton of eggs costs $0.89 The refund is $5.56

9. Applications of linear equations We can use a 5-step procedure to solve real-life problem that involves a system of linear equations: Read the problem carefully to find what quantities are to be found Define the unknown quantities as variables Derive from the problem linear equations that relate the variables Solve the system of equations obtained in the previous step Conclude by interpreting the solution of the system

10. Example #3:An individual has a total of $1000 in two banks. The first bank pays 8% a year and the second bank pays 10% a year. If the person receives $86 of interest in the first year, how much must have been deposited at each bank? $700 was invested at the bank that gives 8% and $300 was invested at the other bank.

11. Example #4: A dietitian is planning a meal for a patient using oranges and strawberries. Each orange contains 1 g of fiber and 0.075 g vitamin C, while each cup of strawberries contains 2 g of fiber and 0.060 g of vitamin C. How much of each of these fruits should be included in the meal so that the patient receives a total of 8 g of fiber and 0.42g of vitamin C?

12. Summarize the information provided in a table. Look at the columns! Cups of strawberries Needed Oranges Fiber 1 2 8 0.075 0.060 0.42 Vitamin C 4 oranges and 2 cups of strawberries