11.4 System of Linear Equations and Problem Solving

Transcription

1 11.4 System of Linear Equations and Problem Solving Learning Objectives: A. Use a system of equations to solve problems. Problem Solving Steps: 1. UNDERSTAND the problem by do the following: Read and reread the problem. Identify what is given and what is the question. Choose two variables to represent the two unknowns being asked. Construct a drawing if needed. 2. TRANSLATE the problem into two equations. 3. SOLVE the system of equations. 4. INTERPRET the results: Check the proposed solution in the stated problem and state your conclusion. Class notes: A. Finding Unknown Numbers Example 1. The difference of two numbers is 12. Twice the smaller decreased by the larger is 20. Find the two numbers. B. Coin Problems Total Value = numbers of coins value of each coin Example 2. Tom has $ 1.10 in quarters and nickels. How many quarters and nickels does he have if he has 14 coins in total? 1

2 C. Mixture Problems Formula: Amount of solution = number of liters percent of the solution Example 3. A pharmacist wants to make 50 liters of a 60% alcohol solution. She currently has a 20% alcohol solution and a 70% alcohol solution. How many liters of a 20% alcohol solution and a 70% alcohol solution she needs to make 50 liters of a 60% alcohol solution? D. Investment Problems I = Pr t where I = interest earn, P = principal, r = interest rate, t = time (in year) Example 4. Lit invested $6000, part at 6% and the rest at 4%. How much is invested at each rate if the annual income from the two investments is $290? P r t = I Account 1 = Account 2 = 2

3 11.4 Exercises 1. The sum of two numbers is 56. Their difference is 12. What are the numbers? 2. The sum of two numbers is 56. Their difference is 4. Find the two numbers. 3. The sum of two numbers is 44. The second number is 5 more than twice the first. Find the numbers. 4. The difference between two numbers is 16. Five times the smaller is the same as 8 less than twice the larger. Find the numbers. 5. Two shirts and three pairs of socks cost $31. Three shirts and two pairs of socks cost $29. Find the cost of each. 6. Admission prices at a local weekend fair were $ 5 for children and $7 for adults. The total money collected was $3379, and 587 people attended the fair. How many children and how many adults attended the fair? 7. The price of admission for a concert was $9 for adults and $4 for children. Altogether, 1770 tickets were sold, and the resulting revenue was $14,680. How many adults and how many children attended the concert? 8. At school, two photography packages are available. Package A contains 1 class picture and 10 wallet size pictures for $19. Package B contains 2 class pictures and 15 wallet size pictures for $31. Find the cost of a class picture and the cost of a walletsize picture. 9. Tim has $ 1.10 in quarters and nickels. How many quarters and nickels does she have if he has 14 coins in total? 10. Todd has 27 total coins in his bank, all dimes and quarters. The coins have a total value of $4.95. How many of each coin does he have? 11. Nary has $2.20 in dimes and nickels. She has 26 coins in total. How many dimes and nickels does she have? 12. Lit invested $6000, some part of it at 6% and the rest at 4%. How much is invested at each rate if the annual income from the two investments is $290? 13. A broker invested a total of $4500 in two different stocks. One stock earned 9% per year. The other earned 6% per year. If $360 was earned from the investment, how much money was invested in each? 14. A chemist has one solution that is 10% iodine and another solution that is 50% iodine. How much of each solution should the chemist use to get 100 ml of a mixture that is 20% iodine? 15. A pharmacist wants to make 50 liters of a 60% alcohol solution but she only has a 20% and a 70% alcohol solution. How many liters of a 20% and a 70% alcohol solution does she need to make 50 liters of a 60% alcohol solution? 3

4 16. A chemist has one solution that is 20% alcohol and another that is 60% alcohol. How much of each solution should the chemist use to get 100 ml of a solution that is 52% alcohol? 17. The perimeter of a rectangle is 58 inches. The base is 5 inches more than three times the height. Find the length of the height and the length of the base. 18. The perimeter of a rectangle is 54 cm. Two times the height is 3 cm more than the base. Find the length of the height and length of the base. 19. The sum of the legs of a right triangle is 17 inches. The longer leg is 2 more than twice the shorter. Find the length of each leg. 20. Two angles are complementary. The larger angle is 6 less than 5 times the smaller angle. Find the measure of each angle. 4

5 Some challenging questions: 1. The two longest rivers in the world are the Nile and the Amazon. Together, they are 8108 miles long. The Amazon is 156 miles shorter than the Nile. Find the length of each river. 2. A zoo has several flamingos and several giraffes. They have 30 eyes and 44 legs. How many flamingos and how many giraffes are in the zoo? 3. Solve the following puzzle. Same shapes must have the same values; different shapes must have different values. 4. How much is a helmet, a bell, and a lock? 5. Mr. Seibold has 6 daughters. Each daughter is 4 years older than her next younger sister. The oldest daughter is 3 times as old as her youngest sister. How old is each of the daughters? 6. As you know, one way to tell an insect from a spider is to count its legs. All insects have six legs, and all spiders have eight legs. So if some insects and spiders went to a dance, and there were 48 dancing legs, how many insects and how many spiders were at the dance if at least one insect and one spider went? 5

6 7. Which is worth more, a SMILE or a FROWN? The costs of combinations of frowns, smiles, and flowers are shown. How much is a smile worth? How much is a frown worth? And a flower? + + = $ = $ = $ = $37 = = = $52 $50 $42 6