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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