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F.4 Present Value of Annuities and mortization
Example 1. Suppose you win a lottery worth $4,250,000 which is paid out with $170,000 payments
for the next 25 years. In order to make these payments to you, how much money must the lottery
commission have in an account now if the account earns interest at a rate of 3.25%/year compounded
annually?
Amortization
Example 2. What monthly payment is required to amortize a loan of $75,000 over 10 years if interest
at the rate of 5.25% per year, compounded monthly, is charged on the unpaid balance at the end of each
month?
Thaddeus made a down payment of $5000 towards the purchase of a new car. To pay the balance,
he secured a loan at the rate of 1.9% per year compounded monthly. Under the terms of his finance
agreement, he is required to make payments of $144/month for 36 months.
a) What is the cash price of the car?
b) How much total interest did Thaddeus pay on the loan?
Example 3. A family secured a 15-year bank loan of $162,600 to purchase a house. The bank charges
interest at a rate of 2.6% per year, compounded monthly.
a) What is their monthly payment?
b) How much total interest will they end up paying?
Outstanding principal is how much you still owe at a give point. To find the outstanding principal,
find the equity.
c) What is the outstanding principal after 10 years? In other words, how much do they still owe after
10 years.
Equity in a loan scenario is how much of the item you actually OWN. It is how much principal you
have paid on the original loan plus any down payment (what belongs to you). The interest you pay
does NOT count towards your equity.
d) What is their equity after 10 years?
Example 4. Find the amortization table for a $18000 loan amortized in five annual payments if the
interest rate is 4.9% per year compounded annually.
1
End of
Period
Payment
Payment
Toward
Interest
Payment
Toward
Principal
0
Outstanding
Principal
18000
1
2
3
4
5
Example 5. You have a $3000 credit card bill on a card that charges interest at a rate of 12% per year,
compounded monthly, on the unpaid balance.
a) If you do not make any additional purchases on the card and make a $71 payment each month, how
long will it take you to pay off your bill? How much total interest do you end up paying?
b) If you instead plan to pay off this credit card at the end of 3 years, how much will you have to pay
each month? How much of your first payment goes toward interest? How much of your first payment
goes toward principal (paying off your debt)?
4.3 Gauss Elimination for Systems of Linear Equations
Example 6. You have a total of $4000 on deposit with two savings institutions. Institution A pays
simple interest at the rate of 3% per year, whereas Institution B pays simple interest at the rate of 4%
per year. If you earn a total of $125 in interest during a single year, how much do you have on deposit
in each institution?
A linear equation in 2 variables is an equation of the form ax + by = c.
A linear equation in 3 variables is an equation of the form ax + by + cz = d.
System of linear equations:
a1,1 x + a1,2 y = b1
a2,1 x + a2,2 y = b2
To solve the system of equations means to find x, y that satisfy EVERY equation in the system.
Example 7. Determine the solutions to the following system of linear equations.
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a)
x + 2y = 5
2x + y = 4
b)
3x + 4y = 8
12x + 16y = 16
c)
2x + y = 2
6x + 3y = 6
Example 8. Determine the solutions to the following system of linear equations.
2x + 2y + 2z = 200
3x + 5y + 7z = 480
2x + y + 3z = 170
Example 9. Write the following system as an augmented matrix and determine the solutions to the
following system of linear equations.
x + 2y − 2z = 1
2x + 7y + 2z = −1
x + 6y + 7z = −3
Example 10. Write the following augmented matrix as a system of equations.


 1 0 −3 −4 




0 1 5
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Reducing Matrices using Your Calculator:
Step 1 Enter the matrix: Press 2nd x−1 . Scroll to the right to EDIT.Scroll down to your desired
matrix (say: 1: [A]). Press ENTER . Enter the dimensions of your matix, pressing ENTER
after each dimension. Enter each entry reading left to right and top to bottom, pressing ENTER
after each entry. Press 2nd MODE to return to the home screen.
Step 2 Press 2nd x−1 . Scroll to MATH.
Step 3 Select rref(. Press ENTER .
Step 4 Press 2nd x−1 . Select your desired matrix ([A]).
Step 5 Press )
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Step 6 Press ENTER .
Example 11. a) Solve the following systems of equations using your calculator.
2x − y − z = 0
3x + 2y + z = 7
x + 2y + 2z = 5
b) Solve the following systems of equations using your calculator.
3x − 9y + 6z = −12
x − 3y + 2z = −4
2x − 6y + 4z = 8
c) Solve the following systems of equations using your calculator.
3x − 4y = 10
−5x + 8y = −17
−3x + 12y = −12
Example 12. Write the following system as an augmented matrix and determine the solutions to the
following system of linear equations.
3x + 6y + 3z + 3u = 450
2x + 6y + 4z + 4u = 500
3y + 5z + 7u = 480
2y + 1z + 3u = 480
We now consider system of linear equations that may have no solution or have infinitely many solutions.
Nonetheless, we will see that Gauss elimination with backward substitution still provides the best and
most efficient way to solve these systems.
Example 13. Solve the following systems of equations.
a)
3x + 4y = 8
12x + 16y = 16
b)
2x + y = 2
6x + 3y = 6
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Example 14. A sporting goods stores sells footballs, basketballs, and volleyballs. A football costs $35, a
basketball costs $25, and a volleyball costs $15. On a given day, the store sold 5 times as many footballs
as volleyballs. They brought in a total of $3750 that day, and the money made from basketballs alone
was 4 times the money made from volleyballs alone. How many footballs, basketballs, and volleyballs
were sold?
Example 15. A furniture company makes loungers, chairs, and footstools out of wood, fabric, and
stuffing. The number of units of each of these materials needed for each of the products is given in the
table below. How many of each product can be made if there are 1110 units of wood, 880 units of fabric,
and 660 units of stuffing available? Just set up the problem. Let x, y, and z represent the number of
loungers, chairs, and footstools made respectively.
Wood Fabric
Stuffing
Lounger
40
40
20
Chair
30
20
20
Footstool
20
10
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4.4 Systems of Linear Equations with Non-Unique Solutions
Example 16. A person has 36 coins made up of nickels, dimes, and quarters. If the total value of the
coins is $4. How many of each type of coin does this person have? Let x, y, and z be the number of
nickels, dimes, and quarters the person has respectively
Example 17. An instructor wants to write a quiz with 9 questions where each question is worth 3, 4,
or 5 points based on difficulty. He wants the number of 3-point questions to be 1 more than the number
of 5-point questions, and he wants the quiz to be worth a total of 35 points. How many 3, 4, and 5
point questions could there be? Let x, y, and z be the number of 3-point, 4-point, and 5-point questions
respectively.
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