1. Systems of Linear Equations and Systems of Linear Inequalities Chapter 6

2. Perimeter, Value, Interest, and Mixture Problems Section 6.5

3. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 3 To solve some problems in which we want to find two quantities, it is useful to perform the following five steps: Step 1: Define each variable. For each quantity that we are trying to find, we usually define a variable to be that unknown quantity. Step 2: Write a system of two equations. We find a system of two equations by using the variables from step 1. We can usually write both equations either… Five-Step Problem-Solving Method Using a Five-Step Problem Solving Method Process

4. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 4 Solving a System to Make a Prediction Using a Five-Step Problem Solving Method Process Continued...by translating into mathematics the information stated in the problem or by making a substitution into a formula. Step 3: Solve the system. We solve the system of equations from step 2. Step 4: Describe each result. We use a complete sentence to describe the found quantities. Step 5: Check. We reread the problem and check the quantities we found agree with the given info.

5. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 5 If n objects each have a value v, then their total value T is given by T = vn In words: The total value is equal to the value of one object times the number of objects. Total-Value Formula Value Problems Formula

6. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 6 A music store charges $5 for a six-string pack of electric-guitar strings and $20 for a four-string pack of electric-bass strings. If the store sells 35 packs of strings for a total revenue of $295, how many packs of each type of string were sold? Step 1: Define the variable. Let x be the number of packs of guitar strings sold Let y be the number of packs for bass string sold Solving a Value Problem Value Problems Example Solution

7. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 7 Step 2: Write a system of two equations. Revenue from guitar strings is the price per pack times the number of packs sold: 5x Revenue from the bass strings is the price per pack time the number of packs sold: 20y Add both revenues to find total revenue T (dollars) Solving a Value Problem Value Problems Solution Continued

8. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 8 Substitute 295 for T: Since the store sells 35 packs of string, the second equation is The system is Solving a Value Problem Value Problems Solution Continued

9. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 9 Step 3: Solve the System. We can use the elimination method Multiply both sides of equation (2) by –5 Add the left sides and add the right sides of the equations and solve for y: Solving a Value Problem Value Problems Solution Continued

10. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 10 Substitute 8 for y in equation (2) and solve for x Solving a Value Problem Value Problems Solution Continued

11. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 11 Step 4: Describe each result. 27 guitar strings and 8 bass strings sold Step 5: Check. Sum of 27 and 8 is 35, which is the total number of strings sold Revenue from 27 packs of guitar and 8 packs of bass strings, which checks Solving a Value Problem Value Problems Solution Continued

12. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 12 The American Analog Set will play at an auditorium that has 400 balcony seats and 1600 main-level seats. If tickets for balcony seats will cost $15 less than tickets for main-level seats, what should the price be for each type of ticket so that the total revenue from a sellout performance will be $70,000 Step 1: Define the variable. Let b be the price of balcony seats Solving a Value Problem Value Problems Example Solution

13. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 13 Let m be the price for main-level seats, both in dollars Step 2: Write a system of two equations. Tickets for balcony seats will cost $15 less than tickets for main-level seats Solving a Value Problem Value Problems Solution Continued

14. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 14 Total revenue is $70,000 Second equation is Units on both sides of the equation are in dollars This suggest that our work is corret The system is: Solving a Value Problem Value Problems Solution Continued

15. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 15 Step 3: Solve the System. Substitute m – 15 for b in equation (2) Solving a Value Problem Value Problems Solution Continued

16. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 16 Substitute 38 for m in equation (1) Solve for b: Step 4: Describe each result. Balcony seats priced at $23,Main-level at $38 Step 5: Check. Difference in the price is: 38 – 23 = 15 Total revenue is: dollars Solving a Value Problem Value Problems Solution Continued

17. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 17 Add the revenues from the general and reserve tickets to find the total revenue T We now have T in terms of x and y We want T in terms for just x Total number of tickets sold for a sell out is 10,000: Using a Function to Model a Value Situation Value Problems Solution

18. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 18 Last 2 example analyzed one aspect of a situation by working with linear equations We want to analyze many aspects of a certain situation It can help to use a system to find a linear function Use function to analyze the situation in various ways Solving a Value Problem Value Problems Summary

19. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 19 A 10,000 seat amphitheater will sell general-seat tickets at $45 and reserve-seat tickets for $65 for a Foo Fighters concert. Let x and y be the number of tickets that will sell for $45 and $65, respectively. Assume that the show will sell out. 1. Find T = f(x) be the total revenue (in dollars) from selling the $45 and $65 tickets. Find the equation of f. Using a Function to Model a Value Situation Value Problems Example

20. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 20 Add the revenues from the general and reserve tickets to find the total revenue T We now have T in terms of x and y We want T in terms for just x Total number of tickets sold for a sell out is 10,000: Using a Function to Model a Value Situation Value Problems Solution

21. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 21 Solve for y Substitute 10,000 – x for y in T = 45x + 65y Equation of f is Using a Function to Model a Value Situation Value Problems Solution Continued

22. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 22 2.Use a graphing calculator to sketch a graph of f for What is the slope? What does it mean in this situation? Sketch f Graph is decreasing-slope of –20 If one more ticket is sold for $45, the revenue will decrease by $20 Using a Function to Model a Value Situation Value Problems Example Continued Solution

23. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 23 3.Find f(8500). What does it mean in this situation?.; Means if 8500 tickets sell for $45 (and 1500 tickets sell for $65), total revenue is $480,000 Using a Function to Model a Value Situation Value Problems Example Continued Solution

24. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 24 4.Find f(11,000). What does it mean in this situation?.; Means if 11,000 tickets sell for $45 total revenue is $430,000 Since there are only 10,000 seats model breakdown has occurred Using a Function to Model a Value Situation Value Problems Example Continued Solution

25. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 25 5.The total cost of the production is $350,000. How many of each type of ticket must be sold to make a profit of $150,000? Profit of $150,000, revenue needs to be 350,000 + 150,000 = 500,000 dollars Substitute 500,000 for T in the equation T = – 20x + 650,000 and solve for x Using a Function to Model a Value Situation Value Problems Example Continued Solution

26. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 26 7500 $45 tickets and 10,000 – 7500 = 2500 $65 tickets would need to be sold for the profit to be $150,000 Using a Function to Model a Value Situation Value Problems Solution Continued

27. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 27 Money deposited in an account such as a savings account, CD, or mutual fund is called the principle. A person invest money in hopes of later getting back the principal plus additional money called the interest. The annual interest rate is the percentage of the principle that equals the interest earned per year. Principal, Interest, and Annual Interest Rate Interest Problems Definition

28. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 28 How much interest will a person earn by investing $3200 in an account at 4% simple interest for one year. Interest from an Investment Interest Problems Example

29. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 29 Find 4% of 3200: 0.04(3200) = 128 The person will earn $128 in interest A person plans to invest twice as much money in an Elfun Trust account at 2.7% annual interest and in a Vanguard Morgan account at 5.5% annual interest. Both interest rates are 5-year averages. (continue) Interest from an Investment Interest Problems Solution Example

30. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 30 How much will the person have to invest in each account to earn a total of $218 in one year? Step 1: Define each variable. Let x be money (in dollars) invested at 2.7% and y be invested at 5.5% annual interested Step 2: Write a system of two inequalities. Invests twice as much in 2.7% account than 5.5% Interest from an Investment Interest Problems Solution Example Continued

31. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 31 x = 2y Total interest is $218, so second equation is The system is Interest from an Investment Interest Problems Solution Continued

32. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 32 Step 3: Solve the system. Substitute 2y for x in equation (2) Substitute 2000 for y in equation (1), solve for x Interest from an Investment Interest Problems Solution Continued

33. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 33 Step 4: Describe each result. Person should invest $4000 at 2.7% and $2000 at 5.5% annual interest Step 5: Check. Note that 4000 is twice 2000, which checks Total interest is which also checks Substitute 2000 for y in equation (1), solve for x Interest from an Investment Interest Problems Solution Continued

34. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 34 A person plans to invest a total of $6000 in a Gabelli ABC mutual fund that has a 3-year average annual interest rate of 6% and in a Presidential Bank Internet CD account at 2.25% annual interest. Let x and y be the money (in dollars) invested in the mutual fund and CD, respectively. 1. Let I = f(x) be the total interest (in dollars) earned from investing the $6000 for one year. Find the equation of f. Using a Function to Model a Situation Involving Interest Interest Problems Example

35. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 35 Interest earned from investing x dollars in account at 6 annual interest is 0.06x Interest earned from investing y dollars in account at 2.25% annual interest is 0.0225y Add two interest earnings gives total interest earned Using a Function to Model a Situation Involving Interest Interest Problems Solution

36. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 36 We describe I in terms of just x Person plans to invest $6000 Isolating y Substitute 6000 – x for y in I = 0.06x + 0.0225y Using a Function to Model a Situation Involving Interest Interest Problems Solution Continued

37. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 37 2.Use a graphing calculator to draw a graph of f for What is the slope of f? What does it mean in this situation? Graph increasing with slope 0.0375 One more dollar invested at 6%, total interest increases by 3.75 cents Using a Function to Model a Situation Involving Interest Interest Problems Example Continued Solution

38. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 38 3.Use a graphing calculator to create a table of values of f. Explain how such a table could help the person decide how much money to invest in each account. May want to know how much risk to take This gives possible interest earnings so clearer idea of how much money to invest in each Using a Function to Model a Situation Involving Interest Interest Problems Example Continued Solution

39. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 39 4.How much money should be invested in each account to earn $300 in one year? Substitute 300 for I: I = 0.0375x + 135, solve for x Should invest $4400 in Gabelli mutual fund and 6000 – 4400 = 1600 dollars in Presidential CD Using a Function to Model a Situation Involving Interest Interest Problems Example Continued Solution

40. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 40 Chemist, cooks, pharmacist, mechanics all mix different substances (typically liquids) Suppose 2 ounces of lime juice is mixes with 8 ounces of water to make 10 ounces of unsweetened limeade of the limeade is lime juice The remaining of limeade is water Introduction of Mixture Problems Mixture Problems Introduction

41. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 41 A chemist needs 5 quarts of a 17% acid, but he has a 15% acid solution and a 25% acid solution. How many quarts of the 15% acid solution should he mix with the 25% acid solution to make 5 quarts of a 17% acid solution? Step 1: Define the variables. Let x be the number of quarts of 15% acid solution and y be the number of quarts of 25% acid solution Solving a Mixture Problem Mixture Problems Example Solution

42. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 42 Step 2: Write a system of two equations. Wants 5 quarts of the total mixture, first equation: x + y = 5 The amount of pure acid doesn’t change despite the distribution of the two variables Sum of the amounts of pure acid in both 15% acid solution and 25% acid solution is equal to the amount of pure acid in the desired mixture Solving a Mixture Problem Mixture Problems Solution Continued

43. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 43 The system is Step 3: Solve the system. Solve equation (1) for y Solving a Mixture Problem Mixture Problems Solution Continued

44. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 44 Substitute 5 – x in the equation y = 5 – x, solve for x Substitute 4 for x in the equation y = 5 – x Solving a Mixture Problem Mixture Problems Solution Continued

45. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 45 Step 4: Describe each result. 4 quarts of the 15% acid solution 1 quart of the 25% acid solution Step 5: Check Compute total amount of pure acid Compute amount of pure acid in the 5 quarts Solving a Mixture Problem Mixture Problems Solution Continued

46. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 46 A chemist needs 8 cups of a 15% alcohol solution but has only a 20% alcohol solution. How much 20% solution and water should she mix to form the desired 8 cups of 15% solution? Step 1: Define the variables. Let x be the number of cups of 20% alcohol solution and y be number of cups of water Solving a Mixture Problem Mixture Problems Example Solution

47. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 47 Step 2: Write a system of two equations. Wants 8 cups of the total mixture, first equation: x + y = 8 No alcohol in water Second equation: amount of pure alcohol in the 20% alcohol solution is equal to the amount of pure alcohol in the desired mixture Solving a Mixture Problem Mixture Problems Solution Continued

48. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 48 The system is Solving a Mixture Problem Mixture Problems Solution Continued

49. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 49 Step 3: Solve the system. Solve equation (2) for x Substitute 6 for x in equation (1) Solving a Mixture Problem Mixture Problems Solution Continued

50. Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5Slide 50 Step 4: Describe each result. Chemist needs to mix 6 cups of the 20% solution with 2 cups of water Step 5: Check. 6 +2 = 8, which checks with 8 cups of 15% solution 6 cups of 20% solution: 6(0.20) = 1.2 cups 8 cups of 15% solution: 8(0.15) = 1.2 cups Amounts of pure alcohol in 20% and 15% checks Solving a Mixture Problem Mixture Problems Solution Continued