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Precalculus
Solve a system of equations using substitution: word problems
Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Aiden is going to ship some gifts to family members, and he is considering two shipping companies. The first shipping company charges a fee of
$
13
\$13
$13
to ship a medium box, plus an additional
$
6
\$6
$6
per kilogram. A second shipping company charges
$
15
\$15
$15
for the same size of box, plus an additional
$
5
\$5
$5
per kilogram. At a certain weight, the two shipping methods will cost the same amount. How much will it cost? What is that weight?
\newline
The two shipping methods both cost
$
\$
$
_____ at a weight of _____ kilograms.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
In a paint factory, an old conveyer line has filled
10
10
10
barrels of paint, and is filling more at a rate of
1
1
1
barrel per minute. A worker just switched on a newer line that can fill
6
6
6
barrels per minute. In a little while, the two lines will have filled an equal number of barrels. How many barrels will each line have filled? How long will that take?
\newline
Both lines will have filled
_
\_
_
barrels in
_
\_
_
minutes.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
The volleyball team and the wrestling team at Salem High School are having a joint car wash today, and they are splitting the revenues. The volleyball team gets
$
2
\$2
$2
per car. In addition, they have already brought in
$
100
\$100
$100
from past fundraisers. The wrestling team has raised
$
16
\$16
$16
in the past, and they are making
$
5
\$5
$5
per car today. After washing a certain number of cars together, each team will have raised the same amount in total. What will that total be? How many cars will that take?
\newline
Each team will have raised a total of
$
_
_
_
_
\$\_\_\_\_
$____
after washing
_
_
_
_
\_\_\_\_
____
cars.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
The volleyball team and the wrestling team at Newport High School are having a joint car wash today, and they are splitting the revenues. The volleyball team gets
$
1
\$1
$1
per car. In addition, they have already brought in
$
86
\$86
$86
from past fundraisers. The wrestling team has raised
$
14
\$14
$14
in the past, and they are making
$
3
\$3
$3
per car today. After washing a certain number of cars together, each team will have raised the same amount in total. What will that total be? How many cars will that take?
\newline
Each team will have raised a total of
$
\$
$
_____ after washing _____ cars.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
The volleyball team and the wrestling team at Danville High School are having a joint car wash today, and they are splitting the revenues. The volleyball team gets
$
1
\$1
$1
per car. In addition, they have already brought in
$
101
\$101
$101
from past fundraisers. The wrestling team has raised
$
10
\$10
$10
in the past, and they are making
$
2
\$2
$2
per car today. After washing a certain number of cars together, each team will have raised the same amount in total. How many cars will that take? What will that total be?
\newline
After washing
_
_
_
_
\_\_\_\_
____
cars, both teams will have raised a total of
$
_
_
_
_
\$\_\_\_\_
$____
.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Paula is starting a business selling handmade necklaces. She has decided to invest an initial amount of
$
41
\$41
$41
for advertising, and materials cost
$
8
\$8
$8
for each necklace she makes. Paula can sell her creations for
$
9
\$9
$9
per necklace. Once she makes and sells a certain number of necklaces, she will break even, with identical expenses and sales. What would the total expenses and sales be then? How many necklaces would that take?
\newline
Paula would have total expenses and sales
$
\$
$
_____ after selling _____ necklaces.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Sanjay has punch cards for his favorite tea house and his favorite coffee shop. He currently has
6
6
6
punches on the tea punch card and
8
8
8
punches on the coffee punch card. Given his regular routine, he consistently earns
5
5
5
new punches per week on the tea punch card and
4
4
4
on the coffee punch card. Before too long, Sanjay will have the same number of punches on each card. How many punches will Sanjay have on each card? How long will that take?
\newline
Sanjay will have
_
\_
_
punches on each card in
_
\_
_
weeks.
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Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks.
\newline
The drama club is selling gift baskets to raise money for new costumes. During the fall play, they sold a combined
9
9
9
regular gift baskets and
3
3
3
deluxe gift baskets, earning a total of
$
297
\$297
$297
. During the spring musical, they sold
1
1
1
deluxe gift basket, earning a total of
$
45
\$45
$45
. How much are they charging for the different sized gift baskets?
\newline
The drama club is charging
$
\$
$
_
_
_
_
\_\_\_\_
____
for a regular gift basket and
$
\$
$
_
_
_
_
\_\_\_\_
____
for a deluxe gift basket.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Jackie has
$
2363
\$2363
$2363
in her retirement account, and Edgar has
$
2359
\$2359
$2359
in his. Jackie is adding
$
9
\$9
$9
per day, whereas Edgar is contributing
$
11
\$11
$11
per day. Eventually, the two accounts will contain the same amount. How long will that take? What balance will each account have?
\newline
After
days
\text{days}
days
, Jackie and Edgar will each have a retirement account balance of
$
□
\$\square
$
□
.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Brian and Eve are making presentations for a class project. Brian's slideshow starts with a verbal introduction that is
13
13
13
seconds long, and then each slide is left up for
3
3
3
seconds. Eve leaves each slide onscreen for
4
4
4
seconds, and her introduction lasts
11
11
11
seconds. Brian and Eve notice that their presentations have both the same number of slides and the same duration. How many slides are in each presentation? How long is each presentation?
\newline
Each presentation has
_
\_
_
slides and is
_
\_
_
seconds long.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Fernando just got his commercial driver's license and is starting a new career as a truck driver. Getting trained and licensed involved a one-time cost of
$
437
\$437
$437
. Gas and insurance end up costing him
$
2
\$2
$2
per kilometer. For his first delivery, Fernando will get paid
$
403
\$403
$403
plus
$
3
\$3
$3
per kilometer that he drives. If he drives a certain distance on this delivery, Fernando will break even, making back all the money he had to spend. How much would both the costs and the earnings be? What distance would he have to drive?
\newline
The costs and the earnings would both be
$
_
_
_
_
\$\_\_\_\_
$____
if Fernando drives
_
_
_
_
\_\_\_\_
____
kilometers.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Sanjay is going to send some flowers to his wife. Richmond Florist charges
$
1
\$1
$1
per rose, plus
$
36
\$36
$36
for the vase. Kelly's Flowers, in contrast, charges
$
2
\$2
$2
per rose and
$
9
\$9
$9
for the vase. If Sanjay orders the bouquet with a certain number of roses, the cost will be the same with either flower shop. What would the total cost be? How many roses would there be?
\newline
The cost will be
$
\$
$
_____ if the bouquet contains _____ roses.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Cole's Bakery recently spent a total of
$
299
\$299
$299
on new equipment, and their average hourly operating costs are
$
13
\$13
$13
. Their average hourly receipts are
$
36
\$36
$36
. The bakery will soon make back the amount it invested in equipment. How many hours will that take? What would the total expenses and receipts both equal?
\newline
In
_
_
_
_
\_\_\_\_
____
hours, the bakery's total 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
Baldwin and Gavin, both teachers, are adding books to their class libraries. Baldwin's classroom started out with a collection of only
7
7
7
books, but he plans to purchase an additional
2
2
2
books per week. Gavin's library started out with
9
9
9
books, and he has enough money in his budget to purchase another
1
1
1
book per week. At some point, the two teachers' libraries will contain the same number of books. How many weeks will that take? How many books will each class have?
\newline
After
_
\_
_
weeks, the two teachers' libraries will each have
_
\_
_
books.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Kelsey works in the shipping department of a toy factory that makes radio-controlled helicopters. Small helicopters weigh
3
3
3
pounds each, and are shipped in a container that weighs
17
17
17
pounds. Large ones, on the other hand, weigh
4
4
4
pounds apiece, and are shipped in a container that weighs
11
11
11
pounds. If these boxes can hold a certain number of helicopters each, all of the packed containers will have the same shipping weight. What would the total weight be? How many helicopters would fit in either container?
\newline
The shipping weight of a full container of either size will be
_
\_
_
pounds if it holds
_
\_
_
helicopters.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Ernesto is going to ship some gifts to family members, and he is considering two shipping companies. The first shipping company charges a fee of
$
11
\$11
$11
to ship a medium box, plus an additional
$
2
\$2
$2
per pound. A second shipping company charges
$
9
\$9
$9
for the same size of box, plus an additional
$
3
\$3
$3
per pound. At a certain weight, the two shipping methods will cost the same amount. How much will it cost? What is that weight?
\newline
The two shipping methods both cost
$
\$
$
_____ at a weight of _____ pounds.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Dean works in the shipping department of a toy factory that makes radio-controlled helicopters. Small helicopters weigh
3
3
3
pounds each, and are shipped in a container that weighs
20
20
20
pounds. Large ones, on the other hand, weigh
7
7
7
pounds apiece, and are shipped in a container that weighs
16
16
16
pounds. If these boxes can hold a certain number of helicopters each, all of the packed containers will have the same shipping weight. How many helicopters would fit in either container? What would the total weight be?
\newline
If either container holds _____ helicopters, it will weigh a total of _____ pounds once it is packed for shipping.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Brendan and Kiera have summer jobs selling newspaper subscriptions door-to-door, but their compensation plans are different. Brendan earns a base wage of
$
15
\$15
$15
per hour, as well as
$
1
\$1
$1
for every subscription that he sells. Kiera gets
$
3
\$3
$3
per subscription sold, in addition to a base wage of
$
3
\$3
$3
per hour. If they each sell a certain number of subscriptions in an hour, they will end up earning the same amount. How much would each one earn? How many subscriptions would that be?
\newline
Brendan and Kiera will each earn
$
_
_
_
_
\$\_\_\_\_
$____
if they sell
_
_
_
_
\_\_\_\_
____
subscriptions in an hour.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Clare wants to buy a charm bracelet. Kingwood Fine Jewelry charges
$
8
\$8
$8
per charm, plus
$
39
\$39
$39
for the bracelet. Austin Jewelers, in contrast, charges
$
9
\$9
$9
per charm and
$
5
\$5
$5
for the bracelet. If Clare wants to add a certain number of charms to her bracelet, the cost will be the same at either jewelry shop. What would the total cost of the bracelet be? How many charms would that be?
\newline
The cost of the bracelet will be
$
\$
$
_____ if Clare adds _____ charms to it.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Derek wants to make scrapbooks with old family photos. An online scrapbooking company charges
$
10
\$10
$10
for a basic book and
$
4
\$4
$4
per page. Meanwhile, a family friend is willing to make a scrapbook for
$
43
\$43
$43
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
The volleyball team and the wrestling team at Newton High School are having a joint car wash today, and they are splitting the revenues. The volleyball team gets
$
5
\$5
$5
per car. In addition, they have already brought in
$
10
\$10
$10
from past fundraisers. The wrestling team has raised
$
101
\$101
$101
in the past, and they are making
$
4
\$4
$4
per car today. After washing a certain number of cars together, each team will have raised the same amount in total. How many cars will that take? What will that total be?
\newline
After washing
_
_
_
_
\_\_\_\_
____
cars, both teams will have raised a total of
$
_
_
_
_
\$\_\_\_\_
$____
.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Gabriel and Wyatt each want to run for president of their school's student body council. In order to do so, they must collect a certain number of signatures and get a nomination. So far, Gabriel has
15
15
15
signatures, and Wyatt has
13
13
13
. Gabriel is collecting signatures at an average rate of
8
8
8
per day, whereas Wyatt is averaging
9
9
9
signatures every day. Assuming that their rate of collection stays the same, eventually the two will have collected the same number of signatures. How long will that take? How many signatures will they both have?
\newline
In
_
\_
_
days, Gabriel and Wyatt will each have collected
_
\_
_
signatures.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
The volleyball team and the wrestling team at Arcadia High School are having a joint car wash today, and they are splitting the revenues. The volleyball team gets
$
3
\$3
$3
per car. In addition, they have already brought in
$
43
\$43
$43
from past fundraisers. The wrestling team has raised
$
93
\$93
$93
in the past, and they are making
$
1
\$1
$1
per car today. After washing a certain number of cars together, each team will have raised the same amount in total. What will that total be? How many cars will that take?
\newline
Each team will have raised a total of
$
_
_
_
_
\$\_\_\_\_
$____
after washing
_
_
_
_
\_\_\_\_
____
cars.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Marvin and Amelia each want to run for president of their school's student body council. In order to do so, they must collect a certain number of signatures and get a nomination. So far, Marvin has
14
14
14
signatures, and Amelia has
18
18
18
. Marvin is collecting signatures at an average rate of
7
7
7
per day, whereas Amelia is averaging
3
3
3
signatures every day. Assuming that their rate of collection stays the same, eventually the two will have collected the same number of signatures. How many signatures will they both have? How long will that take?
\newline
Marvin and Amelia will each have collected
_
\_
_
signatures in
_
\_
_
days.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Leslie works in the shipping department of a toy manufacturer. Toy cars weigh
5
5
5
pounds apiece and are shipped in a container that weighs
6
6
6
pounds when empty. Toy trucks, which weigh
1
1
1
pound apiece, are shipped in a container weighing
18
18
18
pounds. When packed with toys and ready for shipment, both kinds of containers have the same number of toys and the same weight. What is the weight of each container? What is the number of toys?
\newline
Each container weighs
_
\_
_
pounds and contains
_
\_
_
toys.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Janet and Kenny both plan to run for a spot on the school board in their city. They must each collect a certain number of signatures to get their name on the ballot. So far, Janet has
27
27
27
signatures, but Kenny just started and doesn't have any yet. Janet is collecting signatures at an average rate of
7
7
7
per hour, while Kenny can get
10
10
10
signatures every hour. Assuming that their rate of collection stays the same, eventually the two will have collected the same number of signatures. How many signatures will they both have? How many hours will have gone by?
\newline
Janet and Kenny will each have collected
_
\_
_
in
_
\_
_
hours.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Erin has been planting young trees in her garden. The maple tree that is
28
28
28
inches tall is growing
2
2
2
inches per month, whereas the oak tree that is
20
20
20
inches tall is growing
3
3
3
inches per month. In a few months, the two trees will be the same height. How long will that take? What will that height be?
\newline
In
_
\_
_
months, the two trees will both be
_
\_
_
inches tall.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Dr. Sloan just got her PhD, and she wants to print copies of her thesis in hardcover book format. She could use Silvergrove Printing, paying a setup fee of
$
43
\$43
$43
and
$
4
\$4
$4
for every book printed. Alternately, she could go through Weston University, paying an up-front fee of
$
28
\$28
$28
and
$
9
\$9
$9
per book. It turns out that, given the number of books Dr. Sloan wants to print, the two options cost the same amount. What is the amount? How many books is that?
\newline
The cost is
$
_
_
_
_
\$\_\_\_\_
$____
for
_
_
_
_
\_\_\_\_
____
books.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Janelle and Bridget both plan to run for a spot on the school board in their city. They must each collect a certain number of signatures to get their name on the ballot. So far, Janelle has
15
15
15
signatures, but Bridget just started and doesn't have any yet. Janelle is collecting signatures at an average rate of
5
5
5
per hour, while Bridget can get
20
20
20
signatures every hour. Assuming that their rate of collection stays the same, eventually the two will have collected the same number of signatures. How many hours will have gone by? How many signatures will they both have?
\newline
In
‾
\underline{\hspace{2cm}}
hours, Janelle and Bridget will each have collected
‾
\underline{\hspace{2cm}}
.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Two kids at a summer camp, Jonah and Mabel, are competing in a potato sack race. Jonah is younger, so he is given a head start of
30
30
30
meters. When the race starts, Jonah hops at a rate of
2
2
2
meters per second, and Mabel hops
3
3
3
meters per second. Eventually, Mabel will overtake Jonah. How long will that take? How far will Mabel have to hop?
\newline
It will take
_
\_
_
seconds for Mabel to hop
_
\_
_
meters and catch up to Jonah.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Kevin has punch cards for his favorite tea house and his favorite coffee shop. He currently has
6
6
6
punches on the tea punch card and
9
9
9
punches on the coffee punch card. Given his regular routine, he consistently earns
6
6
6
new punches per week on the tea punch card and
3
3
3
on the coffee punch card. Before too long, Kevin will have the same number of punches on each card. How many punches will Kevin have on each card? How long will that take?
\newline
Kevin will have
_
\_
_
punches on each card in
_
\_
_
weeks.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Leah and her sister Carly are making baby blankets to sell at a boutique. Leah has already completed
2
2
2
blankets and can finish
5
5
5
more blankets per day. Carly has already completed
12
12
12
blankets and can finish
4
4
4
more blankets per day. At some point, they will have completed the same number of blankets. How long will that take? How many blankets will each woman have made?
\newline
After
_
\_
_
days, each woman will have finished
_
\_
_
blankets.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Isaac 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. Isaac starts with
10
10
10
points, and his little brother starts with
30
30
30
points. Isaac gets
3
3
3
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
Isaac 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
Two classmates got together over the weekend to do their assigned History reading. Sophie can read
1
1
1
page per minute, while Quincy can read
4
4
4
pages per minute. When they met, Sophie had already read
83
83
83
pages, and Quincy had already gotten through
11
11
11
pages. After a while, they had both read the same number of pages. How many pages had each one read? How long did that take?
\newline
Sophie and Quincy had each read
_
\_
_
pages after
_
\_
_
minutes.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Two students in Mr. Mitchell's class, Matt and Ruth, have been assigned a workbook to complete at their own pace. They get together at Matt's house after school to complete as many pages as they can. Matt has already completed
81
81
81
pages and will continue working at a rate of
1
1
1
page per hour. Ruth has completed
59
59
59
pages and can work at a rate of
12
12
12
pages per hour. Eventually, the two students will be working on the same page. How long will that take? How many pages will each of them have completed?
\newline
After
_
\_
_
hours, Matt and Ruth will have each completed
_
\_
_
pages in their workbooks.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Porter and Matt both plan to run for a spot on the school board in their city. They must each collect a certain number of signatures to get their name on the ballot. So far, Porter has
16
16
16
signatures, but Matt just started and doesn't have any yet. Porter is collecting signatures at an average rate of
9
9
9
per hour, while Matt can get
17
17
17
signatures every hour. Assuming that their rate of collection stays the same, eventually the two will have collected the same number of signatures. How many hours will have gone by? How many signatures will they both have?
\newline
In
_
\_
_
hours, Porter and Matt will each have collected
_
\_
_
.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Brittany just became a personal trainer and is finalizing her pricing plans. One plan is to charge
$
41
\$41
$41
for the initial consultation and then
$
96
\$96
$96
per session. Another plan is to charge
$
13
\$13
$13
for the consultation and then
$
98
\$98
$98
per session. Brittany realizes that the two plans have the same cost for a certain number of sessions. How many sessions is that? What is that cost?
\newline
For _____ sessions, the cost is
$
\$
$
_____ on either plan.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Melissa is going to hire a makeup artist for a fashion show and is comparing prices. Sanjay charges
$
92
\$92
$92
as a booking fee and an additional
$
53
\$53
$53
per hour. Denise charges
$
52
\$52
$52
per hour, plus a booking fee of
$
95
\$95
$95
. 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
Jeanette just became a personal trainer and is finalizing her pricing plans. One plan is to charge
$
49
\$49
$49
for the initial consultation and then
$
68
\$68
$68
per session. Another plan is to charge
$
38
\$38
$38
for the consultation and
$
79
\$79
$79
per session. Jeanette realizes that the two plans have the same cost for a certain number of sessions. How many sessions is that? What is that cost?
\newline
For _____ sessions, the cost is
$
\$
$
_____ on either plan.
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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.
\newline
Ernest is a kickboxing instructor who will be teaching classes at a local gym. To get certified as an instructor, he spent a total of
$
99
\$99
$99
. Ernest will be earning a base salary of
$
91
\$91
$91
per month from the gym, plus an additional
$
4
\$4
$4
for every class he teaches. If Ernest teaches a certain number of classes during his first month as an instructor, he will earn back the amount he spent on certification. How many classes will that take? How much will Ernest's expenses and earnings be?
\newline
Once Ernest teaches _____ classes, his expenses and earnings 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
Lola just became a personal trainer and is finalizing her pricing plans. One plan is to charge
$
17
\$17
$17
for the initial consultation and then
$
38
\$38
$38
per session. Another plan is to charge
$
19
\$19
$19
for the consultation and
$
36
\$36
$36
per session. Lola realizes that the two plans have the same cost for a certain number of sessions. How many sessions is that? What is that cost?
\newline
For _____ sessions, the cost is
$
\$
$
_____ on either plan.
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