III. Solving Applications: Systems of Two Equations

Transcription

1 III. Solving Applications: Systems of Two Equations Problem Solving Strategy Step 1: Familiarize. Step 2: Write system. Step 3: Solve the system. Read & reread problem. Organize info. Make a drawing. Label Unknowns - Find the question. Step 4: Check Did you answer the question completely (Reread question). Does answer make sense (Reread problem). Step 5: State answer using English. 1

2 Preview of Interest Problems Consider the following: Suppose you invest $1500 in a savings account that gives you 4% annual interest on your money. 1) How do you calculate interest? 2) How much total interest will you make at the end of the first year? 3) How much money is in this savings account after the first year? 2

3 Preview of Interest Problems Consider the following: Martha invest money into two different saving accounts. She deposits $1500 into an account paying at an interest rate of 4% per year and $2500 at an interest rate of 6% per year. 1) How much was Martha's total investment? 2) How much interest did Martha make on each account in the first year? 3) What was her total interest earned in the first year? 4) What will her total balance for the two accounts be at the end of the first year? 3

4 Preview of Interest Problems Consider the following: Martha invest money into two different saving accounts. She deposits x dollars into an account paying at an interest rate of 4% per year and y dollars at an interest rate of 6% per year. 1) How much was Martha's total investment? 2) How much interest did Martha make on each account in the first year? 3) What was her total interest earned in the first year? 4) What will her total balance for the two accounts be at the end of the first year? 4

5 Exercise: Use Systems to solve. Joe Huckle won $100,000 in a lottery. He invested part of the money at 10% annual interest and the other part at 12% annual interest. His total income from the two investments is $11,000. How much is invested at each rate? 1) How much was Joe's total investment? How much did he invest in each account? 2) How much interest did Joe make on each account in the first year? What is the total interest paid in the first year. 3) Write out the system of two equations and two unknowns and solve for the amount invested at each rate. 5

6 Exercise: Use Systems to solve. Steve plans to invest some of his California Lottery winnings. He invest part into a safe money-market fund that earns 2.5% annual interest and the rest in a more risky international fund that is expected to earn 9% annual interest. He intends to invest twice the amount of money in the money-market fund than the international fund. How much money is to be invested into each fund if he will earn a total of $560 at the end of the first year? 1) Define your unknown variables (what do they mean) 2) Create your system: Investments Interest 6

7 Exercise: Use Systems to solve. Tala took out two loans from the bank. She took out four times as much from the loan that had 5% interest than she did from the loan asking 6% interest. If the total interest she owed after 1 year was $520, how much did she borrow from each loan? 1) Define your unknown variables (what do they mean) 2) Create your system: Loan Interest 7

8 Exercise: Use Systems to solve. Claudia has $1.70 all in dimes and nickels. She has a total of 22 coins. How many of each kind does she have? The system used to solve this is: 1. What does d mean? What does n mean? 2. What does.10d mean? What does.05 mean? (fill in the table to find out) d 0.10d n 0.05n 8

9 Exercise: Use Systems to solve. Claudia has $1.70 all in dimes and nickels. She has a total of 22 coins. How many of each kind does she have? The system used to solve this is: 3. What does d + n = 22 mean? How many possible (d,n) pairs solve this? 4. What does 0.10d n = 1.70 mean? How many possible (d,n) pairs solve this? 5. How many solutions does the system have? Find the solution. 9

10 Chapter 3 Homework Part I: Introduction to Linear Systems 1) Determine if the point (3,-7) is a solution to the system 2) Solve the system by graphing: 3) Fill in the blank: When we are graphing to solve a system of two equations, if there is no solution, the lines will be. 10

11 Part II: Algebraic Methods to Solve Systems 4) Solve the system by substitution method: 7) Solve the system by elimination method: 5) Solve the system by substitution method: 8) Solve the system by elimination method: 6) Solve the system by substitution method: 9) Solve the system by elimination method: 11

12 10) Solve the system by elimination method: 13) Solve using the method of your choice: 11) Solve using the method of your choice: 14) Solve using the method of your choice: 12) Solve using the method of your choice: 15) Solve using the method of your choice: 12

13 16) Solve using the method of your choice: 18) Solve using the method of your choice: 17) Solve using the method of your choice: 13

14 Part III: Solving Applications 19) Mr. Simone deposits $8000 in one interest account and $2000 in a second interest account. The interest rate on the $8000 account is 2% more than the rate on the $2000 account. If the total yearly amount of interest on the two accounts is $578, find the interest rate on each account. 20) Jenny has $2000 to invest. He would like to earn $ per year in interest. How much should he invest at 6% if the rest is to be invested at 7%? 21) Maria has 19 coins that total $1.35. If the coins are all nickels and dimes, how many of each type does she have? 14

15 22) Suppose Bill has 21 coins totaling $3.45. If he has only dimes and quarters, how many of each kind does he have? Part IV: Systems of Linear Inequalities 23) Graph the system of inequalities and find the coordinates of any vertices formed: 24) Graph the system of inequalities and find the coordinates of any vertices formed: 15

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