The document outlines various mathematical problems involving systems of equations, including real-world applications in budgeting, ticket sales, travel, and geometry. It highlights the need for organized problem-solving skills and the significance of education in developing these skills. Multiple examples illustrate how algebra can be applied to real-life scenarios, emphasizing its importance in everyday decision-making.

1. Systems of Equations
1) A vendor sells hot dogs and bags of potato chips. A customer
buys 4 hot
dogs and 5 bags of potato chips for $12.00. Another customer
buys 3 hot
dogs and 4 bags of potato chips for $9.25. Find the cost of each
item.
1)
2) University Theater sold 556 tickets for a play. Tickets cost
$22 per adult
and $12 per senior citizen. If total receipts were $8492, how
many senior
citizen tickets were sold?
2)
3) A tour group split into two groups when waiting in line for
food at a fast
food counter. The first group bought 8 slices of pizza and 4 soft
drinks
for $36.12. The second group bought 6 slices of pizza and 6 soft
drinks
for $31.74. How much does one slice of pizza cost?
3)
4) Tina Thompson scored 34 points in a recent basketball game
without
making any 3-point shots. She scored 23 times, making several

2. free
throws worth 1 point each and several field goals worth two
points each.
How many free throws did she make? How many 2-point field
goals did
she make?
4)
5) Julio has found that his new car gets 36 miles per gallon on
the highway
and 31 miles per gallon in the city. He recently drove 397 miles
on 12
gallons of gasoline. How many miles did he drive on the
highway? How
many miles did he drive in the city?
5)
6) A textile company has specific dyeing and drying times for
its different
cloths. A roll of Cloth A requires 65 minutes of dyeing time and
50
minutes of drying time. A roll of Cloth B requires 55 minutes of
dyeing
time and 30 minutes of drying time. The production division
allocates
2440 minutes of dyeing time and 1680 minutes of drying time
for the
week. How many rolls of each cloth can be dyed and dried?
6)
7) A bank teller has 54 $5 and $20 bills in her cash drawer. The

3. value of the
bills is $780. How many $5 bills are there?
7)
8) Jamil always throws loose change into a pencil holder on his
desk and
takes it out every two weeks. This time it is all nickels and
dimes. There
are 2 times as many dimes as nickels, and the value of the dimes
is $1.65
more than the value of the nickels. How many nickels and dimes
does
Jamil have?
8)
9) A flat rectangular piece of aluminum has a perimeter of 60
inches. The
length is 14 inches longer than the width. Find the width.
9)
1
10) Jarod is having a problem with rabbits getting into his
vegetable garden,
so he decides to fence it in. The length of the garden is 8 feet
more than 3
times the width. He needs 64 feet of fencing to do the job. Find
the
length and width of the garden.
10)

4. 11) Two angles are supplementary if the sum of their measures
is 180°. The
measure of the first angle is 18° less than two times the second
angle.
Find the measure of each angle.
11)
12) The three angles in a triangle always add up to 180°. If one
angle in a
triangle is 72° and the second is 2 times the third, what are the
three
angles?
12)
13) An isosceles triangle is one in which two of the sides are
congruent. The
perimeter of an isosceles triangle is 21 mm. If the length of the
congruent sides is 3 times the length of the third side, find the
dimensions of the triangle.
13)
14) A chemist needs 130 milliliters of a 57% solution but has
only 33% and
85% solutions available. Find how many milliliters of each that
should be
mixed to get the desired solution.
14)
15) Two lines that are not parallel are shown. Suppose that the
measure of
angle 1 is (3x + 2y)°, the measure of angle 2 is 9y°, and the

5. measure of
angle 3 is (x + y)°. Find x and y.
15)
16) The manager of a bulk foods establishment sells a trail mix
for $8 per
pound and premium cashews for $15 per pound. The manager
wishes to
make a 35-pound trail mix-cashew mixture that will sell for $14
per
pound. How many pounds of each should be used?
16)
17) A college student earned $7300 during summer vacation
working as a
waiter in a popular restaurant. The student invested part of the
money at
7% and the rest at 6%. If the student received a total of $458 in
interest at
the end of the year, how much was invested at 7%?
17)
2
18) A retired couple has $160,000 to invest to obtain annual
income. They
want some of it invested in safe Certificates of Deposit yielding
6%. The
rest they want to invest in AA bonds yielding 11% per year.

6. How much
should they invest in each to realize exactly $15,600 per year?
18)
19) A certain aircraft can fly 1330 miles with the wind in 5
hours and travel
the same distance against the wind in 7 hours. What is the speed
of the
wind?
19)
20) Julie and Eric row their boat (at a constant speed) 40 miles
downstream
for 4 hours, helped by the current. Rowing at the same rate, the
trip back
against the current takes 10 hours. Find the rate of the current.
20)
21) Khang and Hector live 88 miles apart in southeastern
Missouri. They
decide to bicycle towards each other and meet somewhere in
between.
Hector's rate of speed is 60% of Khang's. They start out at the
same time
and meet 5 hours later. Find Hector's rate of speed.
21)
22) Devon purchased tickets to an air show for 9 adults and 2
children. The
total cost was $252. The cost of a child's ticket was $6 less than
the cost of
an adult's ticket. Find the price of an adult's ticket and a child's

7. ticket.
22)
23) On a buying trip in Los Angeles, Rosaria Perez ordered 120
pieces of
jewelry: a number of bracelets at $8 each and a number of
necklaces at
$11 each. She wrote a check for $1140 to pay for the order.
How many
bracelets and how many necklaces did Rosaria purchase?
23)
24) Natasha rides her bike (at a constant speed) for 4 hours,
helped by a
wind of 3 miles per hour. Pedaling at the same rate, the trip
back against
the wind takes 10 hours. Find find the total round trip distance
she
traveled.
24)
25) A barge takes 4 hours to move (at a constant rate)
downstream for 40
miles, helped by a current of 3 miles per hour. If the barge's
engines are
set at the same pace, find the time of its return trip against the
current.
25)
26) Doreen and Irena plan to leave their houses at the same
time, roller
blade towards each other, and meet for lunch after 2 hours on

8. the road.
Doreen can maintain a speed of 2 miles per hour, which is 40%
of Irena's
speed. If they meet exactly as planned, what is the distance
between
their houses?
26)
3
27) Dmitri needs 7 liters of a 36% solution of sulfuric acid for a
research
project in molecular biology. He has two supplies of sulfuric
acid
solution: one is an unlimited supply of the 56% solution and
the other
an unlimited supply of the 21% solution. How many liters of
each
solution should Dmitri use?
27)
28) Chandra has 2 liters of a 30% solution of sodium hydroxide
in a
container. What is the amount and concentration of sodium
hydroxide
solution she must add to this in order to end up with 6 liters of
46%
solution?
28)
29) Jimmy is a partner in an Internet-based coffee supplier. The

9. company
offers gourmet coffee beans for $12 per pound and regular
coffee beans
for $6 per pound. Jimmy is creating a medium-price product
that will
sell for $8 per pound. The first thing to go into the mixing bin
was 10
pounds of the gourmet beans. How many pounds of the less
expensive
regular beans should be added?
29)
30) During the 1998-1999 Little League season, the Tigers
played 57 games.
They lost 21 more games than they won. How many games did
they win
that season?
30)
31) The perimeter of a rectangle is 48 m. If the width were
doubled and the
length were increased by 24 m, the perimeter would be 112 m.
What are
the length and width of the rectangle?
31)
32) The perimeter of a triangle is 46 cm. The triangle is
isosceles now, but if
its base were lengthened by 4 cm and each leg were shortened
by 7 cm, it

10. would be equilateral. Find the length of the base of the original
triangle.
32)
33) The side of an equilateral triangle is 8 inches shorter than
the side of a
square. The perimeter of the square is 46 inches more than the
perimeter
of the triangle. Find the length of a side of the square.
33)
34) The side of an equilateral triangle is 2 inches shorter than
the side of a
square. The perimeter of the square is 30 inches more than the
perimeter
of the triangle. Find the length of a side of the triangle.
34)
4
Answer Key
Testname: SYSTEMS_OF_EQUATIONS
1) $1.75 for a hot dog; $1.00 for a bag of potato chips
2) 374 senior citizen tickets
3) $3.74 per slice of pizza
4) 12 free throws, 11 field goals
5) 180 miles on the highway, 217 miles in the city
6) 24 rolls of Cloth A, 16 rolls of Cloth B
7) 20 $5 bills
8) 11 nickels and 22 dimes

11. 9) 8 inches
10) length: 26 feet; width: 6 feet
11) first angle = 114°
second angle = 66°
12) 72°, 72°, 36°
13) 3 mm, 9 mm, 9 mm
14) 70 mL of 33%; 60 mL of 85%
15) x = 288
7
, y = 36
7
16) 5 pounds of trail mix
30 pounds of cashews
17) $2000
18) $120,000 at 11% and $40,000 at 6%
19) 38 mph
20) 3 mph
21) 6.6 mph
22) adult's ticket: $24; child's ticket: $18
23) 60 bracelets and 60 necklaces
24) 80 mi
25) 10 hr
26) 14 mi
27) 56% solution: 3 L; 21% solution: 4 L
28) 4 L of 54% solution
29) 20 lb
30) 18 games
31) Length: 16 m; width: 8 m
32) 8 cm

12. 33) 22 inches
34) 22 inches
5
Doe 1
Jane Doe
Professor Crawford
English 101
08 December 2017
Eisenhower: An American Hero
One of the most well-known American heroes of World War II
was Dwight D.
Eisenhower. He was a brilliant general and diplomat, possessing
enough tact to keep the Allies
united and ultimately win World War II. He also served as
President of the United States,
bringing peace to the nation. His autobiography was a best
seller, partly because of his good
writing skills. No one is perfect, and he was no exception, but
Eisenhower served his country

13. with everything he had, and America would not be the same
without him. Eisenhower is
remembered as an important figure in U.S. history because of
his military career, presidency, and
writing.
First, Eisenhower had a very long and successful military
career. According to historian
Stephen Ambrose, "He was the most successful general of the
greatest war ever fought" (p ar 45).
Despite having never experienced combat, he was named
Supreme Allied Commander in 1942.
He was known for being a great diplomat, a trait that served him
well as he had to negotiate a
successful working relationship with generals from many
different countries with many different
personalities. Ultimately, every campaign he was in charge of
was a success. The war was won,
and a part of that is due to Eisenhower's military career and
leadership.
Secondly, Dwight D. Eisenhower was a great president. In
1953, he was inaugurated as
the 34th President of the United States. He was good at nearly
everything he undertook, and
being president was no different. For example, "Eisenhower was

14. able to inspire considerable
Doe 2
trust and confidence, and to help stabilize and calm the country"
(Gilbert 1). He accomplished
many important things during his term, such as balancing the
budget, keeping the country out of
war, and enacting legislation for interstate highways. His
presidency went so well he was elected
to serve a second term despite worsening health. Eisenhower is
well remembered as a president,
and justly so.
Lastly, Eisenhower was a good writer. He was known as being
very good at writing
reports during his Army days, and he graduated first in his class
from the prestigious Army War
College. He ghostwrote a booklet on battle monuments that had
General Pershing’s name as the
author. Eisenhower retired from the U.S. Army in 1948 to focus
on writing. Soon the memoir he
had written, Crusade in Europe, about his time as Supreme
Allied Commander, became a

15. bestseller (Boyle 12). The book is a fascinating look into the
mind of the man who made some
very important decisions during World War II.
For all of these reasons, Dwight D. Eisenhower deserves to be
remembered as an
American hero and an important historical figure. He was a
general, president, and writer.
Granted, the United States has had a plethora of writers, many
generals, and a handful of
presidents, but Eisenhower is one of the rare ones who
encompassed all three roles. It would be
hard to find a person with even a passing knowledge of U.S.
history who had not at heard of his
name and the impact he had on this country. Eisenhower was a
humble hero, and he is an
inspiration to patriotic Americans everywhere.
Doe 3

16. Works Cited
Ambrose, Stephen E. "Eisenhower's Generalship." Parameters,
vol. 40, no. 4, 2011, pp. 90-98,
ProQuest Central; SciTech Premium Collection, https://search-
proquest-
com.ezproxy2.apus.edu/docview/867412831?accountid=8289.
Accessed 06DEC2017.
Boyle, P G. Eisenhower, Taylor and Francis, 2004. ProQuest
Ebook Central,
https://ebookcentral-proquest-
com.ezproxy1.apus.edu/lib/apus/detail.action?docID=4185965.
Accessed 08DEC2017.
Gilbert, Robert E. "Eisenhower's 1955 Heart Attack." Politics &
the Life Sciences, vol. 27, no. 1,
Mar. 2008, pp. 2-21. EBSCOhost,
search.ebscohost.com/login.aspx?direct=true&db=aph&AN=349
06464&site=ehost-
live&scope=site. Accessed 06DEC2017.
https://search-proquest-/
https://search-proquest-/

17. Forum: If Only I Had a System…
Applications of Systems of Linear Equalities
The Problem:
When students are surveyed about what makes a good math
Forum, at least half of the
responses involve
-life situations"
This Forum on applications of systems of equations addresses
both of these concerns.
Unfortunately, the typical postings are far from ideal.
This is an attempt to rectify the situation. Please read this in its
entirety before you
post your answer!

18. Pick-up games in the park vs. the NBA:
Shooting hoops in the park may be lots of fun, but it scarcely
qualifies as the precision
play of a well-coached team. On the one hand, you have
individuals with different
approaches and different skill levels, "doing their own thing"
within the general rules of
the game. On the other hand you have trained individuals, using
proven strategies and
basing their moves on fundamentals that have been practiced
until they are second
nature.
The purpose of learning algebra is to change a natural,
undisciplined approach to
individual problem solving into an organized, well-rehearsed
system that will work on
many different problems. Just like early morning practice, this
might not always be
pleasant; just like Michael Jordan, if you put in the time
learning how to do it correctly,
you will score big-time in the end.

19. But my brain just doesn't work that way. . .
Nonsense! This has nothing to do with how your brain works.
This is a matter of
learning to read carefully, to extract data from the given
situation and to apply a
mathematical system to the data in order to obtain a desired
answer. Anyone can learn
to do this. It is just a matter of following the system; much like
making cookies is a
matter of following a recipe.
"Pick-up Game" Math
It is appalling how many responses involve plugging in numbers
until it works.
work from there."
lved both cats and dogs so I took one of the
numbers, divided by
2 and then I experimented."
-life that hot dogs cost more than Coke, so I
crossed my fingers

20. and started with $0.50 for the Coke..."
The reason these "problem-solving" boards are moderated is so
that these creative
souls don’t get everyone else confused!
NBA Math
In more involved problems, where the answer might come out to
be something
irrational, like the square root of three, you are not likely to just
randomly guess the
correct answer to plug it in. To find that kind of answer by an
iterative process (plugging
and adjusting; plugging and adjusting; ...) would take lots of
tedious work or a computer.
Algebra gives you a relative painless way of achieving your
objective without wearing
your pencil to the nub.
The reason that all of the homework has involved x's and y's
and two equations, is that
we are going to solve these problems that way. Each of these
problems is a story about
two things, so every one of these is going to have an x and a y.
In some problems, it’s helpful to use different letters, to help
keep straight what the

21. variables stand for. For example, let L = the length of the
rectangle and W = the width.
The biggest advantage to this method is that when you have
found that w = 3 you are
more likely to notice that you still haven’t answered the
question, “What is the length of
the rectangle?”
Here are the steps to the solution process:
o one of these will be x
o the other will be y
L = ” )
o Unless it is your express purpose to drive your instructor right
over the
edge, make sure that your very first word is "Let"
“Let W = ” )

22. things.
o Use one of those relationships to write your first equation.
o Use the second relationship to write the second equation.
You will be using
either
o substitution
o or elimination - just like in the homework.

23. More examples…
For this problem, I'd use substitution to solve the system of
equations:
The length of a rectangle blah, blah, blah...
Let L = the length of the rectangle
... blah, blah, blah twice the width
Let W = the width of the rectangle
The length is 6 inches less than twice the width
L = 2W - 6
The perimeter of the rectangle is 56
2L + 2W =56

24. For this one, I'd use elimination to solve the system of
equations:
Blah, blah, blah bought 2 cokes...
Let x = the price of a coke
.. blah, blah, blah 4 hot dogs
Let y = the price of a hot dog
2 cokes plus 4 hot dogs cost 8.00
2x + 4y = 8.00
3 cokes plus 2 hot dogs cost 8.00
3x + 2y = 8.00

25. For this one, I'd use substitution to solve the system of
equations:
One number is blah, blah, blah...
Let x = the first number
...blah, blah, blah triple the second number
Let y = the second number
The first number is triple the second
x = 3y
The sum of the numbers is 24
x + y = 24

26. Checking your answers vs. Solving the problem
The problem: Two numbers add to give 4 and subtract to give 2.
Find the numbers.
Solving the problem:
Let x = the first number
Let y = the second number
Two numbers add to give 4: x + y = 4
Two numbers subtract to give 2: x - y = 2
Our two equations are: x + y = 4
x - y = 2 Adding the
equations we get
2x = 6
x = 3 The first number is
3.

27. x + y = 4 Substituting that
answer into equation 1
3 + y = 4
y = 1 The second
number is 1.
Checking the answers:
Two numbers add to give 4: 3 + 1 = 4
The two numbers subtract to give 2: 3 - 1 = 2
Do NOT demonstrate how to check the answers that are
provided and call that
demonstrating how to solve the problem!
Formulas vs. Solving equations
Formulas express standard relationships between measurements

28. of things in the real
world and are probably the mathematical tools that are used
most frequently in real-life
situations.
Solving equations involves getting an answer to a specific
problem, sometimes based
on real-world data, and sometimes not. In the process of solving
a problem, you may
need to apply a formula. As a member of modern society, it is
assumed that you know
certain common formulas such as the area of a square or the
perimeter of a rectangle. If
you are unsure about a formula, just Google it. Chances are
excellent it will be in one of
the first few hits.
If you are still baffled:
rked out in the PowerPoints in
the Other
Resources section of the Handy Helpers for Section 4.3.