Systems of Linear Equations and Systems of Linear Inequalities Chapter 6

Systems of

Linear Equations

and Systems of

Linear Inequalities

Chapter 6

Perimeter, Value, Interest, and

Mixture Problems

Section 6.5

Lehmann, Elementary and Intermediate Algebra, 1edSection 6.5 Slide 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 MethodUsing a Five-Step Problem Solving Method

Process


Solving a System to Make a PredictionUsing 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.


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 FormulaValue Problems

Formula


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 ProblemValue Problems

Example

Solution


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)


Solution Continued


• Substitute 295 for T:

• Since the store sells 35 packs of string, the second equation is

• The system is


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:


Solution Continued


• Substitute 8 for y in equation (2) and solve for x


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 , which checks


Solution Continued

5 27 20 8 295


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


Example

Solution


• 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


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:


Solution Continued


Step 3: Solve the System.• Substitute m – 15 for b in equation (2)


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: dollars


Solution Continued

23 400 38 1600 70,000


• 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 SituationValue Problems

Solution


• 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


Summary


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.


Example


• 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:


Solution


• Solve for y

• Substitute 10,000 – x for y in T = 45x + 65y

• Equation of f is


Solution Continued


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


Example Continued

0 10,000.x

Solution


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


Example Continued

Solution

8500 20 8500 650,000 480,000f


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


Example Continued

Solution 11,000 20 11,000 650,000 430,000f


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


Example Continued

Solution


• 7500 $45 tickets and 10,000 – 7500 = 2500 $65 tickets would need to be sold for the profit to be $150,000


Solution Continued


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 RateInterest Problems

Definition


How much interest will a person earn by investing $3200 in an account at 4% simple interest for one year.

Interest from an InvestmentInterest Problems

Example


• 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)


Solution

Example


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%


Solution

Example Continued


x = 2y•Total interest is $218, so second equation is

•The system is


Solution Continued


Step 3: Solve the system.• Substitute 2y for x in equation (2)

• Substitute 2000 for y in equation (1), solve for x


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

• Substitute 2000 for y in equation (1), solve for x


Solution Continued


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 InterestInterest Problems

Example


• 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


Solution


• 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


Solution Continued


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


Example Continued

0 6000.x

Solution


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


Example Continued

Solution


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


Example Continued

Solution


• 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 ProblemsMixture Problems

Introduction

20.20 20%

10

80.80 80%

10


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 ProblemMixture Problems

Example

Solution


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


Solution Continued


•The system is

Step 3: Solve the system.•Solve equation (1) for y


Solution Continued


• Substitute 5 – x in the equation y = 5 – x, solve for x

• Substitute 4 for x in the equation y = 5 – x


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


Solution Continued


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


Example

Solution


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


Solution Continued


•The system is


Solution Continued


Step 3: Solve the system.•Solve equation (2) for x

•Substitute 6 for x in equation (1)


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


Solution Continued

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Systems of Linear Equations and Systems of Linear Inequalities Chapter 6