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Math Problems
Algebra 1
Solve a system of equations using substitution
Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Tom wants to make scrapbooks with old family photos. An online scrapbooking company charges
$
39
\$39
$39
for a basic book and
$
4
\$4
$4
per page. Meanwhile, a family friend is willing to make a scrapbook for
$
48
\$48
$48
plus
$
1
\$1
$1
per page. For a certain number of pages, the price would be equal. How much would that cost? How many pages would that be?
\newline
The scrapbook would cost
$
\$
$
_____ if it had _____ pages.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Mrs. Kerr is researching what it would cost to order flower arrangements for a fancy party. She wants one large centerpiece for the head table, and smaller arrangements for the smaller tables. Lakeside Florist charges
$
13
\$13
$13
for each smaller arrangement, plus
$
44
\$44
$44
for the large centerpiece. Emily's Flowers, in contrast, charges
$
40
\$40
$40
for the large centerpiece and
$
17
\$17
$17
per arrangement for the rest. If Mrs. Kerr orders a certain number of small arrangements, the cost will be the same at either flower shop. What would the total cost be? How many small arrangements would that be?
\newline
The cost will be
$
_
_
_
_
\$\_\_\_\_
$____
if Mrs. Kerr orders
_
_
_
\_\_\_
___
of the small arrangements.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Zack and Pete started out at their houses and are biking towards each other. Zack started out first, and has already gone
4
4
4
miles. He bikes at a constant speed of
3
3
3
miles per hour. Pete just left, and rides at
4
4
4
miles per hour. When the boys meet halfway between their houses, they will continue to the park together. How long will that take? How far will each boy have ridden?
\newline
In
_
\_
_
hours, both boys will have ridden
_
\_
_
miles.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
The freshman and sophomore classes at Arcadia High School are decorating floats for homecoming. The freshmen have already spent
$
117
\$117
$117
on their float, plus they need to buy floral sheeting that costs
$
87
\$87
$87
per roll. The sophomores, who have spent
$
180
\$180
$180
so far on theirs, still need to purchase vinyl grass at
$
78
\$78
$78
per roll. Both classes plan to buy the same number of rolls, since they have the same area to cover. By coincidence, the two floats will have the same total cost in the end. How much will each class spend in total? How many rolls will each class be buying?
\newline
The two classes will have each spend
$
\$
$
_____ in total by buying _____ rolls.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Two groups of volunteers are cleaning up the football stadium after the Homecoming game. Volunteers from the Band Booster Club have already cleaned
2
2
2
rows of bleachers and will continue to clean at a rate of
7
7
7
rows per minute. The leadership class has completed
9
9
9
rows and will continue working at
6
6
6
rows per minute. Once the two groups get to the point where they have cleaned the same number of rows, they will take a break and decide how to split up the remaining work. How long will that take? How many minutes will each group have cleaned by then?
\newline
In
_
\_
_
minutes, the groups will each cleaned
_
\_
_
rows each.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Wyatt and his little brother made up a game using coins. They flip the coins towards a cup and receive points for every one that makes it in. Wyatt starts with
18
18
18
points, and his little brother starts with
27
27
27
points. Wyatt gets
2
2
2
points for every successful shot, and his brother, since he is younger, gets
1
1
1
point for each successful shot. Eventually, the brothers will have a tied score in the game. How many additional shots will each brother have made? How many points will they both have?
\newline
Wyatt and his brother will have each made ___ shots, for a tied score of ___.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Kurt, an artist, plans to paint and sell some miniature paintings. He just bought some brushes for
$
18
\$18
$18
, and paint and canvas for each painting costs
$
79
\$79
$79
; he will sell each painting for
$
85
\$85
$85
. Once Kurt sells a certain number of his paintings, he will be breaking even. How many paintings will that be? How much will Kurt have earned?
\newline
Once Kurt has sold _____ paintings, his expenses and receipts will both equal
$
\$
$
_____.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
A fashion photographer needs to hire a stylist to prepare her models. Ted charges
$
163
\$163
$163
for showing up, plus
$
73
\$73
$73
per hour. Levi charges
$
169
\$169
$169
plus
$
67
\$67
$67
per hour. The photographer realizes that, given the expected duration of her photo shoot, either stylist would cost her the same amount. What would the duration be? What would the cost be?
\newline
If the shoot lasted for _____ hours, either stylist would cost
$
\$
$
_____.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Kiara has a home-based business making and selling scented soaps. She intially spent
$
64
\$64
$64
to purchase soap-making equipment, and the materials for each pound of soap cost
$
7
\$7
$7
. Kiara sells the soap for
$
9
\$9
$9
per pound. Eventually, she will sell enough soap to cover the cost of the equipment. How much soap will that be? What will be Kiara's total sales and costs be?
\newline
Once Kiara sells _____ pounds of soap, her sales and costs will both be
$
\$
$
_____.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
A fashion photographer needs to hire a stylist to prepare his models. Michelle charges
$
189
\$189
$189
for showing up, plus
$
73
\$73
$73
per hour. Vince charges
$
186
\$186
$186
plus
$
74
\$74
$74
per hour. The photographer realizes that, given the expected duration of his photo shoot, either stylist would cost him the same amount. What would the duration be? What would the cost be?
\newline
If the shoot lasted for _____ hours, either stylist would cost
$
\$
$
_____.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Isabella wants to buy a charm bracelet. Springfield Fine Jewelry charges
$
13
\$13
$13
per charm, plus
$
64
\$64
$64
for the bracelet. Patton Jewelers, in contrast, charges
$
12
\$12
$12
per charm and
$
67
\$67
$67
for the bracelet. If Isabella wants to add a certain number of charms to her bracelet, the cost will be the same at either jewelry shop. How many charms would that be? What would the total cost of the bracelet be?
\newline
If Isabella adds _____ charms to her bracelet, it will cost
$
\$
$
_____ at either shop.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Denise has a home-based business making and selling scented soaps. She intially spent
$
98
\$98
$98
to purchase soap-making equipment, and the materials for each pound of soap cost
$
8
\$8
$8
. Denise sells the soap for
$
15
\$15
$15
per pound. Eventually, she will sell enough soap to cover the cost of the equipment. How much soap will that be? What will be Denise's total sales and costs be?
\newline
Once Denise sells _____ pounds of soap, her sales and costs will both be
$
\$
$
_____.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
The Oakland Tigers are having team shirts made. One option is to pay Joey's Tees a
$
36
\$36
$36
setup fee and then buy the shirts for
$
7
\$7
$7
each. Another option is to go to City Printing, paying
$
46
\$46
$46
for a setup fee and an additional
$
6
\$6
$6
per shirt. The team parent in charge of the project notices that, with a certain number of shirts, the two options have the same cost. How many shirts is that? What is the cost?
\newline
If the Tigers have _____ shirts made, the cost is
$
\$
$
_____.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Mr. Russo is contemplating which chauffeured car service to take to the airport. The first costs
$
3
\$3
$3
up front and
$
4
\$4
$4
per mile. The second costs
$
13
\$13
$13
plus
$
2
\$2
$2
per mile. For a certain driving distance, the two companies charge the same total fare. What is the distance? What is the total fare?
\newline
For a driving distance of _____ miles, the total fare is
$
\$
$
_____.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
A fashion photographer needs to hire a stylist to prepare his models. Zach charges
$
168
\$168
$168
for showing up, plus
$
70
\$70
$70
per hour. Troy charges
$
178
\$178
$178
plus
$
68
\$68
$68
per hour. The photographer realizes that, given the expected duration of his photo shoot, either stylist would cost him the same amount. What would the duration be? What would the cost be?
\newline
If the shoot lasted for _____ hours, either stylist would cost
$
\$
$
_____.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Farid is going to hire a makeup artist for a fashion show and is comparing prices. Audrey charges
$
25
\$25
$25
as a booking fee and an additional
$
33
\$33
$33
per hour. Stanley charges
$
34
\$34
$34
per hour, plus a booking fee of
$
24
\$24
$24
. Depending on the length of the show, the cost could end up being the same for either artist. How long would the show be? What would the cost be?
\newline
If the show lasted for _____ hours, the cost would always be
$
\$
$
_____.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Nicole has a home-based business making and selling scented soaps. She intially spent
$
55
\$55
$55
to purchase soap-making equipment, and the materials for each pound of soap cost
$
4
\$4
$4
. Nicole sells the soap for
$
15
\$15
$15
per pound. Eventually, she will sell enough soap to cover the cost of the equipment. How much soap will that be? What will be Nicole's total sales and costs be?
\newline
Once Nicole sells _____ pounds of soap, her sales and costs will both be
$
\$
$
_____.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Steven and his cousin Carly are picking apples in their grand parents' orchard. Steven has filled
15
15
15
baskets with apples and is filling them at a rate of
4
4
4
baskets per hour. Carly has
11
11
11
full baskets and will continue picking at
8
8
8
baskets per hour. Once the cousins get to the point where they have filled the same number of baskets, they will carry them to the barn and then eat lunch. How long will that take? How much fruit will they have picked by then?
\newline
In hours, the cousins will each have filled baskets with apples.
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Isabel tried to evaluate an expression step by step.
\newline
(
2
+
8
)
+
8
=
2
+
(
8
+
8
)
Step
1
=
2
+
0
Step 2
=
2
Step
3
\begin{array}{l} (2+8)+8 \\ =2+(8+8) \quad \text { Step } 1 \\ =2+0 \quad \text { Step 2 } \\ =2 \quad \text { Step } 3 \\ \end{array}
(
2
+
8
)
+
8
=
2
+
(
8
+
8
)
Step
1
=
2
+
0
Step 2
=
2
Step
3
\newline
Find Isabel's first mistake.
\newline
Choose
1
1
1
answer:
\newline
(A) Changing the groups changes the sum's value.
\newline
(B)
8
8
8
plus
8
8
8
equals
16
16
16
, not
0
0
0
.
\newline
(C)
2
2
2
plus
0
0
0
equals
0
0
0
, not
2
2
2
.
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Amar tried to evaluate an expression step by step.
\newline
−
2
+
5
−
(
−
3
)
=
−
2
+
5
−
3
Step
1
=
−
2
+
2
Step 2
=
0
Step
3
\begin{array}{l} -2+5-(-3) \\ =-2+5-3 \quad \text { Step } 1 \\ =-2+2 \quad \text { Step 2 } \\ =0 \quad \text { Step } 3 \\ \end{array}
−
2
+
5
−
(
−
3
)
=
−
2
+
5
−
3
Step
1
=
−
2
+
2
Step 2
=
0
Step
3
\newline
Find Amar's first mistake.
\newline
Choose
1
1
1
answer:
\newline
(A) Amar was supposed to add
3
3
3
, not subtract
3
3
3
.
\newline
(B)
5
5
5
minus
3
3
3
equals
−
2
-2
−
2
, not
2
2
2
.
\newline
(C)
−
2
-2
−
2
plus
2
2
2
equals
−
4
-4
−
4
, not
0
0
0
.
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Because of a problem in the program, the timer in a video player did not begin counting until the video had been playing for several seconds. The player began counting at
0
0
0
seconds, even though the video had already played
190
190
190
frames. The video plays
25
25
25
frames per second.
\newline
How many frames had the video already played when the time was equal to
−
3
2
5
-3 \frac{2}{5}
−
3
5
2
seconds?
\newline
frames
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Ashley recently opened a store that uses only natural ingredients. She wants to advertise her products by distributing bags of samples in her neighborhood. It takes one person
2
3
\frac{2}{3}
3
2
of a minute to prepare one bag.
\newline
How many hours will it take Ashley and
4
4
4
of her friends to prepare
1575
1575
1575
bags of samples?
\newline
hours
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Sophia has a bag with
3
3
3
purple marbles and
4
4
4
blue marbles in it. She is going to randomly draw a marble from the bag
280
280
280
times, putting the marble back in the bag after each draw.
\newline
Complete the following statement with the best prediction.
\newline
Sophia will draw a purple marble...
\newline
Choose
1
1
1
answer:
\newline
(A) Exactly
120
120
120
times
\newline
(B) Close to
120
120
120
times but probably not exactly
120
120
120
times
\newline
(C) Exactly
160
160
160
times
\newline
(D) Close to
160
160
160
times but probably not exactly
160
160
160
times
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Scarlett is playing a video game. She spends
900
900
900
minerals to create
18
18
18
workers. Each worker costs the same number of minerals.
\newline
Write an equation to describe the relationship between
m
m
m
, the amount of minerals, and
w
w
w
, the number of workers.
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A unicorn daycare center requires there to be
2
2
2
supervisors for every
18
18
18
baby unicorns.
\newline
Write an equation that shows the relationship between
n
n
n
, the number of supervisors, and
u
u
u
, the number of baby unicorns.
\newline
Please note that this is a magical daycare center, so fractional supervisors are allowed.
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Esther the Clown does face painting at the city carnival. She paints
7
7
7
faces every
21
21
21
minutes and spends the same amount of time painting each face.
\newline
Write an equation that relates
f
f
f
, the number of faces she paints, and
m
m
m
, the time she spends painting in minutes.
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Tara is taller than Jeff. Jeff is
120
120
120
centimeters tall.
\newline
Write an inequality that compares Tara's height in centimeters,
h
h
h
, to Jeff's height.
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Jake is younger than Sophie. Sophie is
14
14
14
years old.
\newline
Write an inequality that compares Jake's age in years,
j
j
j
, to Sophie's age.
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Dominic sent a chain letter to his friends, asking them to forward the letter to more friends.
\newline
The relationship between the elapsed time
t
t
t
, in hours, since Dominic sent the letter, and the number of people,
P
hour
(
t
)
P_{\text {hour }}(t)
P
hour
(
t
)
, who receive the email is modeled by the following function:
\newline
P
hour
(
t
)
=
18
⋅
(
1.05
)
t
P_{\text {hour }}(t)=18 \cdot(1.05)^{t}
P
hour
(
t
)
=
18
⋅
(
1.05
)
t
\newline
Complete the following sentence about the daily rate of change in the number of people who receive the email.
\newline
Round your answer to two decimal places.
\newline
Every day, the number of people who receive the email grows by a factor of
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Luke and Owen have
$
100
\$ 100
$100
each. Their friend offered to invest their money, promising to return a sum
r
r
r
times as great as what they invested. Luke was suspicious, so he invested
$
10
\$ 10
$10
only, but Owen invested his entire
$
100
\$ 100
$100
. Fortunately, the friend did indeed return a sum
r
r
r
times as great to each.
\newline
They decided to make another investment. This time, Owen invested all of the money returned to him, and Luke invested the money returned to him and the remaining
$
90
\$ 90
$90
. Again, they got a sum
r
r
r
times as great as what they invested. In the end, Owen had
$
337.50
\$ 337.50
$337.50
more than Luke.
\newline
Write an equation in terms of
r
r
r
that models the situation.
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n
=
(
1.96
σ
E
)
2
n=\left(\frac{1.96 \sigma}{E}\right)^{2}
n
=
(
E
1.96
σ
)
2
\newline
Elon is planning a study of hours of internet use per month per household. He uses the given equation to determine the minimum number of randomly selected households,
n
n
n
, that he needs to include in his sample given a standard deviation of
σ
\sigma
σ
hours and a maximum allowable error of
E
E
E
hours. Which of the following equations correctly gives the standard deviation in terms of the maximum allowable error and the minimum number of randomly selected households?
\newline
Choose
1
1
1
answer:
\newline
(A)
σ
=
E
n
1.96
\sigma=\sqrt{\frac{E n}{1.96}}
σ
=
1.96
E
n
\newline
(B)
σ
=
E
n
1.96
\sigma=\frac{E \sqrt{n}}{1.96}
σ
=
1.96
E
n
\newline
(C)
σ
=
1.96
n
π
\sigma=\sqrt{\frac{1.96 n}{\pi}}
σ
=
π
1.96
n
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Kudzu is a vine that was introduced to the United States from Japan in
1876
1876
1876
as an ornamental plant. Starting in the year
1935
1935
1935
, kudzu was planted throughout the Southeast to combat soil erosion. By
1946
1946
1946
, kudzu covered approximately
3
3
3
million acres of land in the Southeast. Sixty years later, about
7
7
7
million acres were covered by kudzu. Which of the following functions best models the amount of kudzu, in millions of acres,
t
t
t
years after
1946
?
1946 ?
1946
?
\newline
Choose
1
1
1
answer:
\newline
(A)
K
(
t
)
=
3
⋅
(
1.014
)
t
K(t)=3 \cdot(1.014)^{t}
K
(
t
)
=
3
⋅
(
1.014
)
t
\newline
(B)
K
(
t
)
=
3
⋅
(
0.014
)
t
K(t)=3 \cdot(0.014)^{t}
K
(
t
)
=
3
⋅
(
0.014
)
t
\newline
(C)
K
(
t
)
=
3
+
(
1.014
)
t
K(t)=3+(1.014) t
K
(
t
)
=
3
+
(
1.014
)
t
\newline
(D)
K
(
t
)
=
3
+
(
0.014
)
t
K(t)=3+(0.014) t
K
(
t
)
=
3
+
(
0.014
)
t
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During last year's basketball season, Jackson attempted
150
150
150
free throws of which he made
120
120
120
. So far this season, he's attempted
24
24
24
and made
16
16
16
. A player's free-throw percentage is defined to be the percent of free-throws made out of the number of free-throws attempted. Assuming calculations began with the start of last year's season, which of the following best approximates Jackson's overall freethrow percentage to date?
\newline
Choose
1
1
1
answer:
\newline
(A)
67
%
67 \%
67%
\newline
(B)
73
%
73 \%
73%
\newline
(C)
78
%
78 \%
78%
\newline
(D)
80
%
80 \%
80%
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One cookbook recommends that a person can substitute
1
1
1
tablespoon (
T
b
s
p
\mathrm{Tbsp}
Tbsp
) of dried mint leaves for
1
4
\frac{1}{4}
4
1
cup (c) of fresh mint leaves. The salad recipe calls for
2
2
2
T
b
s
p
\mathrm{Tbsp}
Tbsp
of fresh mint leaves. How many
T
b
s
p
\mathrm{Tbsp}
Tbsp
of dried mint leaves could a person substitute into the recipe?
\newline
(There are
16
16
16
T
b
s
p
\mathrm{Tbsp}
Tbsp
in
1
c
1 \mathrm{c}
1
c
.)
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A train traveling at a speed of
s
s
s
miles per hour applies its brakes before a buffer stop. Assuming
d
≥
0
d \geq 0
d
≥
0
and
s
≥
0
s \geq 0
s
≥
0
, the distance,
d
d
d
, in yards from the train to the buffer stop once the train comes to rest is:
\newline
d
=
0.5
(
−
s
2
−
1.2
s
+
184
d=0.5\left(-s^{2}-1.2 s+184\right.
d
=
0.5
(
−
s
2
−
1.2
s
+
184
\newline
Which of the following equivalent expressions for
d
d
d
contains the traveling speed of the train, as a constant or coefficient, for which the train rests right at the buffer stop after applying its brakes?
\newline
Choose
1
1
1
answer:
\newline
(A)
−
0.5
s
2
−
0.6
s
+
92.3
-0.5 s^{2}-0.6 s+92.3
−
0.5
s
2
−
0.6
s
+
92.3
\newline
(B)
−
0.5
(
s
−
13
)
(
s
+
14.2
)
-0.5(s-13)(s+14.2)
−
0.5
(
s
−
13
)
(
s
+
14.2
)
\newline
(C)
−
0.5
(
s
+
0.6
)
2
+
92.48
-0.5(s+0.6)^{2}+92.48
−
0.5
(
s
+
0.6
)
2
+
92.48
\newline
(D)
(
6.5
−
0.5
s
)
(
s
+
14.2
)
(6.5-0.5 s)(s+14.2)
(
6.5
−
0.5
s
)
(
s
+
14.2
)
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For a particular delivery, a courier travels by bicycle at a speed of
v
v
v
miles per hour for a distance of
1
1
1
.
3
3
3
miles. After making the delivery, the courier travels the same distance back, but travels
2
2
2
miles per hour faster than on the way to the delivery. If the courier spent
0
0
0
.
2
2
2
hours travelling to and from the delivery, which of the following equations could be used to determine the speed of the bicycle on the way to the delivery?
\newline
Choose
1
1
1
answer:
\newline
(A)
0.2
v
2
−
2.2
v
−
2.6
=
0
0.2 v^{2}-2.2 v-2.6=0
0.2
v
2
−
2.2
v
−
2.6
=
0
\newline
(B)
1.3
v
2
−
2
v
−
0.2
=
0
1.3 v^{2}-2 v-0.2=0
1.3
v
2
−
2
v
−
0.2
=
0
\newline
(C)
0.2
(
v
2
−
3
)
+
2.6
=
0
0.2\left(v^{2}-3\right)+2.6=0
0.2
(
v
2
−
3
)
+
2.6
=
0
\newline
(D)
3
(
0.2
−
v
)
(
2.6
−
v
)
=
0
3(0.2-v)(2.6-v)=0
3
(
0.2
−
v
)
(
2.6
−
v
)
=
0
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Chang is measuring the speeds of stars as they travel around a black hole. She notices that the speed,
s
s
s
, in kilometers per second
t
t
t
days after March
1
st
1^{\text {st }}
1
st
is given by:
\newline
s
(
t
)
=
100
+
30
t
−
(
t
−
2
)
2
s(t)=100+30 t-(t-2)^{2}
s
(
t
)
=
100
+
30
t
−
(
t
−
2
)
2
\newline
Which of the following expressions for the star's speed is equivalent to the given expression and contains the maximum speed of the star as a constant or coefficient?
\newline
Choose
1
1
1
answer:
\newline
(A)
96
+
34
t
−
t
2
96+34 t-t^{2}
96
+
34
t
−
t
2
\newline
(B)
−
(
t
−
17
)
2
+
385
-(t-17)^{2}+385
−
(
t
−
17
)
2
+
385
\newline
(C)
−
(
t
−
34
)
2
+
385
-(t-34)^{2}+385
−
(
t
−
34
)
2
+
385
\newline
(D)
104
+
30
t
−
t
2
104+30 t-t^{2}
104
+
30
t
−
t
2
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The width of a rectangular box is
2
c
m
2 \mathrm{~cm}
2
cm
less than its length. The height of the box is
8
c
m
8 \mathrm{~cm}
8
cm
. If the box has a length of
x
x
x
centimeters, which of the following functions represents the surface area,
S
S
S
, in square centimeters?
\newline
Choose
1
1
1
answer:
\newline
(A)
S
=
x
2
+
14
x
−
16
S=x^{2}+14 x-16
S
=
x
2
+
14
x
−
16
\newline
(B)
S
=
2
x
2
+
28
x
−
32
S=2 x^{2}+28 x-32
S
=
2
x
2
+
28
x
−
32
\newline
(C)
S
=
16
+
2
x
−
x
2
S=16+2 x-x^{2}
S
=
16
+
2
x
−
x
2
\newline
(D)
S
=
32
+
4
x
−
2
x
2
S=32+4 x-2 x^{2}
S
=
32
+
4
x
−
2
x
2
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The denarius was a unit of currency in ancient Rome. Suppose it costs the Roman government
10
10
10
denarii per day to support
4
4
4
legionaries and
4
4
4
archers. It only costs
5
5
5
denarii per day to support
2
2
2
legionaries and
2
2
2
archers. Use a system of linear equations in two variables.
\newline
Can we solve for a unique cost for each soldier?
\newline
Choose
1
1
1
answer:
\newline
(A) Yes; a legionary costs
1
1
1
denarius per day to support, and an archer costs
2
2
2
denarii per day to support.
\newline
(B) Yes; a legionary costs
2
2
2
denarii per day to support, and an archer costs
4
3
\frac{4}{3}
3
4
denarii per day to support.
\newline
(C) No; the system has many solutions.
\newline
(D) No; the system has no solution.
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The denarius was a unit of currency in ancient Rome. Suppose it costs the Roman government
10
10
10
denarius per day to support
3
3
3
legionaries and
3
3
3
archers. It only costs
3
3
3
denarius per day to support one legionary and one archer. Use a system of linear equations in two variables.
\newline
Can we solve for a unique cost for each soldier?
\newline
Choose
1
1
1
answer:
\newline
(A) Yes; a legionary costs
1
1
1
denarius per day to support, and an archer costs
2
2
2
denarius per day to support.
\newline
(B) Yes; a legionary costs
2
2
2
denarius per day to support, and an archer costs
4
3
\frac{4}{3}
3
4
denarius per day to support.
\newline
(C) No; the system has many solutions.
\newline
(D) No; the system has no solution.
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Quinn has a large family. She has
4
4
4
cousins who live in Texas,
3
3
3
cousins who live in Nebraska, and the other
9
9
9
cousins who live in Michigan.
\newline
What is the ratio of Quinn's cousins who live in Texas to her total number of cousins?
\newline
Choose
1
1
1
answer:
\newline
(A)
3
3
3
to
16
16
16
\newline
(B)
4
4
4
to
16
16
16
\newline
(C)
7
7
7
to
16
16
16
\newline
(D)
9
9
9
to
16
16
16
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A washing machine is being redesigned to handle a greater volume of water. One part is a pipe with a radius of
3
3
3
centimeters
(
c
m
)
(\mathrm{cm})
(
cm
)
and a length of
11
c
m
11 \mathrm{~cm}
11
cm
. It gets replaced with a pipe of radius
4
c
m
4 \mathrm{~cm}
4
cm
, and the same length. The new pipe can hold
w
π
w \pi
w
π
more cubic centimeters
(
c
m
3
)
\left(\mathrm{cm}^{3}\right)
(
cm
3
)
of water than the old pipe, where
w
w
w
is a constant. What is the value of
w
w
w
?
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Solve the system of equations.
\newline
2
x
−
9
y
=
14
x
=
−
6
y
+
7
x
=
□
y
=
□
\begin{array}{l} 2 x-9 y=14 \\ x=-6 y+7 \\ x=\square \\ y=\square \end{array}
2
x
−
9
y
=
14
x
=
−
6
y
+
7
x
=
□
y
=
□
Get tutor help
Solve the system of equations.
\newline
−
4
x
+
7
y
=
20
y
=
3
x
+
15
x
=
□
y
=
□
\begin{array}{l} -4 x+7 y=20 \\ y=3 x+15 \\ x=\square \\ y=\square \end{array}
−
4
x
+
7
y
=
20
y
=
3
x
+
15
x
=
□
y
=
□
Get tutor help
Solve the system of equations.
\newline
−
4
x
+
3
y
=
−
2
y
=
x
−
1
x
=
□
y
=
□
\begin{array}{l} -4 x+3 y=-2 \\ y=x-1 \\ x=\square \\ y=\square \end{array}
−
4
x
+
3
y
=
−
2
y
=
x
−
1
x
=
□
y
=
□
Get tutor help
Solve the system of equations.
\newline
5
x
−
7
y
=
58
y
=
−
x
+
2
x
=
□
y
=
□
\begin{array}{l} 5 x-7 y=58 \\ y=-x+2 \\ x=\square \\ y=\square \end{array}
5
x
−
7
y
=
58
y
=
−
x
+
2
x
=
□
y
=
□
Get tutor help
Solve the system of equations.
\newline
2
x
+
7
y
=
3
x
=
−
4
y
x
=
□
y
=
□
\begin{array}{l} 2 x+7 y=3 \\ x=-4 y \\ x=\square \\ y=\square \end{array}
2
x
+
7
y
=
3
x
=
−
4
y
x
=
□
y
=
□
Get tutor help
Solve the system of equations.
\newline
−
7
x
−
6
y
=
4
x
=
−
3
y
+
8
x
=
□
y
=
□
\begin{array}{l} -7 x-6 y=4 \\ x=-3 y+8 \\ x=\square \\ y=\square \end{array}
−
7
x
−
6
y
=
4
x
=
−
3
y
+
8
x
=
□
y
=
□
Get tutor help
Solve the system of equations.
\newline
13
x
−
6
y
=
22
x
=
y
+
6
x
=
□
y
=
□
\begin{array}{l} 13 x-6 y=22 \\ x=y+6 \\ x=\square \\ y=\square \end{array}
13
x
−
6
y
=
22
x
=
y
+
6
x
=
□
y
=
□
Get tutor help
Solve the system of equations.
\newline
5
x
−
4
y
=
−
10
y
=
2
x
−
5
x
=
□
y
=
□
\begin{array}{l} 5 x-4 y=-10 \\ y=2 x-5 \\ x=\square \\ y=\square \end{array}
5
x
−
4
y
=
−
10
y
=
2
x
−
5
x
=
□
y
=
□
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