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Chapter 6 Systems of Linear Equations and Systems of Linear Inequalities Section 6.5 Perimeter, Value, Interest, and Mixture Problems Five-Step Problem-Solving Method Using a Five-Step Problem Solving Method Process 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… Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 3 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. Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 4 Total-Value Formula Va l u e P r o b l e m s Formula 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. Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 5 Solving a Value Problem Va l u e P r o b l e m s Example 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? Solution 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 Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 6 Solving a Value Problem Va l u e P r o b l e m s Solution Continued 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) Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 7 Solving a Value Problem Va l u e P r o b l e m s Solution Continued • Substitute 295 for T: • Since the store sells 35 packs of string, the second equation is • The system is Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 8 Solving a Value Problem Va l u e P r o b l e m s Solution Continued 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: Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 9 Solving a Value Problem Va l u e P r o b l e m s Solution Continued • Substitute 8 for y in equation (2) and solve for x Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 10 Solving a Value Problem Va l u e P r o b l e m s Solution Continued 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 5 27 20 8 295, which checks Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 11 Solving a Value Problem Va l u e P r o b l e m s Example 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 Solution Step 1: Define the variable. • Let b be the price of balcony seats Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 12 Solving a Value Problem Va l u e P r o b l e m s Solution Continued • 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 Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 13 Solving a Value Problem Va l u e P r o b l e m s Solution Continued • 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: Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 14 Solving a Value Problem Va l u e P r o b l e m s Solution Continued Step 3: Solve the System. • Substitute m – 15 for b in equation (2) Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 15 Solving a Value Problem Va l u e P r o b l e m s Solution Continued • 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: 23 400 38 1600 70,000 dollars Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 16 Using a Function to Model a Value Situation Va l u e P r o b l e m s Solution • 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: Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 17 Solving a Value Problem Va l u e P r o b l e m s Summary • 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 Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 18 Using a Function to Model a Value Situation Va l u e P r o b l e m s Example 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. Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 19 Using a Function to Model a Value Situation Va l u e P r o b l e m s Solution • 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: Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 20 Using a Function to Model a Value Situation Va l u e P r o b l e m s Solution Continued • Solve for y • Substitute 10,000 – x for y in T = 45x + 65y • Equation of f is Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 21 Using a Function to Model a Value Situation Va l u e P r o b l e m s Example Continued 2. Use a graphing calculator to sketch a graph of f for 0 x 10,000. What is the slope? What does it mean in this situation? Solution • Sketch f • Graph is decreasing-slope of –20 • If one more ticket is sold for $45, the revenue will decrease by $20 Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 22 Using a Function to Model a Value Situation Va l u e P r o b l e m s Example Continued 3. Find f(8500). What does it mean in this situation? Solution • f.; 8500 20 8500 650,000 480,000 • Means if 8500 tickets sell for $45 (and 1500 tickets sell for $65), total revenue is $480,000 Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 23 Using a Function to Model a Value Situation Va l u e P r o b l e m s Example Continued 4. Find f(11,000). What does it mean in this situation? Solution • f.; 11,000 20 11,000 650,000 430,000 • Means if 11,000 tickets sell for $45 total revenue is $430,000 • Since there are only 10,000 seats model breakdown has occurred Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 24 Using a Function to Model a Value Situation Va l u e P r o b l e m s Example Continued 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? Solution • • 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 Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 25 Using a Function to Model a Value Situation Va l u e P r o b l e m s Solution Continued • 7500 $45 tickets and 10,000 – 7500 = 2500 $65 tickets would need to be sold for the profit to be $150,000 Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 26 Principal, Interest, and Annual Interest Rate Interest Problems Definition 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. Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 27 Interest from an Investment Interest Problems Example How much interest will a person earn by investing $3200 in an account at 4% simple interest for one year. Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 28 Interest from an Investment Interest Problems Solution • Find 4% of 3200: 0.04(3200) = 128 • The person will earn $128 in interest Example 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) Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 29 Interest from an Investment Interest Problems Example Continued How much will the person have to invest in each account to earn a total of $218 in one year? Solution 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% Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 30 Interest from an Investment Interest Problems Solution Continued x = 2y • Total interest is $218, so second equation is • The system is Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 31 Interest from an Investment Interest Problems Solution Continued Step 3: Solve the system. • Substitute 2y for x in equation (2) • Substitute 2000 for y in equation (1), solve for x Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 32 Interest from an Investment Interest Problems Solution Continued 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 Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 33 Using a Function to Model a Situation Involving Interest Interest Problems Example 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. Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 34 Using a Function to Model a Situation Involving Interest Interest Problems Solution • 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 Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 35 Using a Function to Model a Situation Involving Interest Interest Problems Solution Continued • 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 Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 36 Using a Function to Model a Situation Involving Interest Interest Problems Example Continued 2. Use a graphing calculator to draw a graph of f for 0 x 6000. What is the slope of f? What does it mean in this situation? Solution • Graph increasing with slope 0.0375 • One more dollar invested at 6%, total interest increases by 3.75 cents Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 37 Using a Function to Model a Situation Involving Interest Interest Problems Example Continued 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. Solution • 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 Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 38 Using a Function to Model a Situation Involving Interest Interest Problems Example Continued 4. How much money should be invested in each account to earn $300 in one year? Solution • 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 Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 39 Introduction of Mixture Problems Mixture Problems Introduction • 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 2 • 0.20 20% of the limeade is lime juice 10 8 • The remaining 0.80 80% of limeade is water 10 Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 40 Solving a Mixture Problem Mixture Problems Example 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? 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 Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 41 Solving a Mixture Problem Mixture Problems Solution Continued 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 Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 42 Solving a Mixture Problem Mixture Problems Solution Continued • The system is Step 3: Solve the system. • Solve equation (1) for y Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 43 Solving a Mixture Problem Mixture Problems Solution Continued • Substitute 5 – x in the equation y = 5 – x, solve for x • Substitute 4 for x in the equation y = 5 – x Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 44 Solving a Mixture Problem Mixture Problems Solution Continued 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 Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 45 Solving a Mixture Problem Mixture Problems Example 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? 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 Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 46 Solving a Mixture Problem Mixture Problems Solution Continued 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 Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 47 Solving a Mixture Problem Mixture Problems Solution Continued • The system is Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 48 Solving a Mixture Problem Mixture Problems Solution Continued Step 3: Solve the system. • Solve equation (2) for x • Substitute 6 for x in equation (1) Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 49 Solving a Mixture Problem Mixture Problems Solution Continued 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 Section 6.5 Lehmann, Elementary and Intermediate Algebra, 1ed Slide 50