Resources

Resources

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

Two groups of volunteers are cleaning up the football stadium after the Homecoming game. Volunteers from the Band Booster Club have already cleaned

2

rows of bleachers and will continue to clean at a rate of

7

rows per minute. The leadership class has completed

9

rows and will continue working at

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

\_

minutes, the groups will each cleaned

\_

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

points, and his little brother starts with

27

points. Wyatt gets

2

points for every successful shot, and his brother, since he is younger, gets

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?

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Wyatt and his brother will have each made ___ shots, for a tied score of ___.

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

, and paint and canvas for each painting costs

\$79

; he will sell each painting for

\$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?

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Once Kurt has sold _____ paintings, his expenses and receipts will both equal

\$

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

for showing up, plus

\$73

per hour. Levi charges

\$169

\$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

\$

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

to purchase soap-making equipment, and the materials for each pound of soap cost

\$7

. Kiara sells the soap for

\$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?

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Once Kiara sells _____ pounds of soap, her sales and costs will both be

\$

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

for showing up, plus

\$73

per hour. Vince charges

\$186

\$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

\$

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

per charm, plus

\$64

for the bracelet. Patton Jewelers, in contrast, charges

\$12

\$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.

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

to purchase soap-making equipment, and the materials for each pound of soap cost

\$8

. Denise sells the soap for

\$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

\$

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

setup fee and then buy the shirts for

\$7

each. Another option is to go to City Printing, paying

\$46

for a setup fee and an additional

\$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

\$

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

\$4

per mile. The second costs

\$13

\$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

\$

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

for showing up, plus

\$70

per hour. Troy charges

\$178

\$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

\$

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

as a booking fee and an additional

\$33

per hour. Stanley charges

\$34

per hour, plus a booking fee of

\$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

\$

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

to purchase soap-making equipment, and the materials for each pound of soap cost

\$4

. Nicole sells the soap for

\$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

\$

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

baskets with apples and is filling them at a rate of

4

baskets per hour. Carly has

11

full baskets and will continue picking at

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?

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In hours, the cousins will each have filled baskets with apples.

Isabel tried to evaluate an expression step by step.

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\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}

\newline

Find Isabel's first mistake.

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1

\newline

(A) Changing the groups changes the sum's value.

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8

8

16

0

\newline

2

0

0

2

Amar tried to evaluate an expression step by step.

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\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}

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Find Amar's first mistake.

\newline

1

\newline

(A) Amar was supposed to add

3

3

\newline

5

3

-2

2

\newline

-2

2

-4

0

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

seconds, even though the video had already played

190

frames. The video plays

25

frames per second.

\newline

How many frames had the video already played when the time was equal to

-3 \frac{2}{5}

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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

\frac{2}{3}

of a minute to prepare one bag.

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How many hours will it take Ashley and

4

of her friends to prepare

1575

bags of samples?

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Sophia has a bag with

3

purple marbles and

4

blue marbles in it. She is going to randomly draw a marble from the bag

280

times, putting the marble back in the bag after each draw.

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Complete the following statement with the best prediction.

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Sophia will draw a purple marble...

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1

\newline

120

\newline

120

times but probably not exactly

120

\newline

160

\newline

160

times but probably not exactly

160

Scarlett is playing a video game. She spends

900

minerals to create

18

workers. Each worker costs the same number of minerals.

\newline

Write an equation to describe the relationship between

m

, the amount of minerals, and

w

, the number of workers.

A unicorn daycare center requires there to be

2

supervisors for every

18

\newline

Write an equation that shows the relationship between

n

, the number of supervisors, and

u

, the number of baby unicorns.

\newline

Please note that this is a magical daycare center, so fractional supervisors are allowed.

Esther the Clown does face painting at the city carnival. She paints

7

21

minutes and spends the same amount of time painting each face.

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Write an equation that relates

f

, the number of faces she paints, and

m

, the time she spends painting in minutes.

Tara is taller than Jeff. Jeff is

120

centimeters tall.

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Write an inequality that compares Tara's height in centimeters,

h

, to Jeff's height.

Jake is younger than Sophie. Sophie is

14

\newline

Write an inequality that compares Jake's age in years,

j

, to Sophie's age.

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

, in hours, since Dominic sent the letter, and the number of people,

P_{\text {hour }}(t)

, who receive the email is modeled by the following function:

\newline

P_{\text {hour }}(t)=18 \cdot(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

Luke and Owen have

\$ 100

each. Their friend offered to invest their money, promising to return a sum

r

times as great as what they invested. Luke was suspicious, so he invested

\$ 10

only, but Owen invested his entire

\$ 100

. Fortunately, the friend did indeed return a sum

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

. Again, they got a sum

r

times as great as what they invested. In the end, Owen had

\$ 337.50

more than Luke.

\newline

Write an equation in terms of

r

that models the situation.

n=\left(\frac{1.96 \sigma}{E}\right)^{2}

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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

, that he needs to include in his sample given a standard deviation of

\sigma

hours and a maximum allowable error of

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

1

\newline

\sigma=\sqrt{\frac{E n}{1.96}}

\newline

\sigma=\frac{E \sqrt{n}}{1.96}

\newline

\sigma=\sqrt{\frac{1.96 n}{\pi}}

Kudzu is a vine that was introduced to the United States from Japan in

1876

as an ornamental plant. Starting in the year

1935

, kudzu was planted throughout the Southeast to combat soil erosion. By

1946

, kudzu covered approximately

3

million acres of land in the Southeast. Sixty years later, about

7

million acres were covered by kudzu. Which of the following functions best models the amount of kudzu, in millions of acres,

t

1946 ?

\newline

1

\newline

K(t)=3 \cdot(1.014)^{t}

\newline

K(t)=3 \cdot(0.014)^{t}

\newline

K(t)=3+(1.014) t

\newline

K(t)=3+(0.014) t

During last year's basketball season, Jackson attempted

150

free throws of which he made

120

. So far this season, he's attempted

24

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

1

\newline

67 \%

\newline

73 \%

\newline

78 \%

\newline

80 \%

One cookbook recommends that a person can substitute

1

\mathrm{Tbsp}

) of dried mint leaves for

\frac{1}{4}

cup (c) of fresh mint leaves. The salad recipe calls for

2

\mathrm{Tbsp}

of fresh mint leaves. How many

\mathrm{Tbsp}

of dried mint leaves could a person substitute into the recipe?

\newline

16

\mathrm{Tbsp}

1 \mathrm{c}

A train traveling at a speed of

s

miles per hour applies its brakes before a buffer stop. Assuming

d \geq 0

s \geq 0

, the distance,

d

, in yards from the train to the buffer stop once the train comes to rest is:

\newline

d=0.5\left(-s^{2}-1.2 s+184\right.

\newline

Which of the following equivalent expressions for

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

1

\newline

-0.5 s^{2}-0.6 s+92.3

\newline

-0.5(s-13)(s+14.2)

\newline

-0.5(s+0.6)^{2}+92.48

\newline

(6.5-0.5 s)(s+14.2)

For a particular delivery, a courier travels by bicycle at a speed of

v

miles per hour for a distance of

1

3

miles. After making the delivery, the courier travels the same distance back, but travels

2

miles per hour faster than on the way to the delivery. If the courier spent

0

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

1

\newline

0.2 v^{2}-2.2 v-2.6=0

\newline

1.3 v^{2}-2 v-0.2=0

\newline

0.2\left(v^{2}-3\right)+2.6=0

\newline

3(0.2-v)(2.6-v)=0

Chang is measuring the speeds of stars as they travel around a black hole. She notices that the speed,

s

, in kilometers per second

t

days after March

1^{\text {st }}

\newline

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

1

\newline

96+34 t-t^{2}

\newline

-(t-17)^{2}+385

\newline

-(t-34)^{2}+385

\newline

104+30 t-t^{2}

The width of a rectangular box is

2 \mathrm{~cm}

less than its length. The height of the box is

8 \mathrm{~cm}

. If the box has a length of

x

centimeters, which of the following functions represents the surface area,

S

, in square centimeters?

\newline

1

\newline

S=x^{2}+14 x-16

\newline

S=2 x^{2}+28 x-32

\newline

S=16+2 x-x^{2}

\newline

S=32+4 x-2 x^{2}

The denarius was a unit of currency in ancient Rome. Suppose it costs the Roman government

10

denarii per day to support

4

legionaries and

4

archers. It only costs

5

denarii per day to support

2

legionaries and

2

archers. Use a system of linear equations in two variables.

\newline

Can we solve for a unique cost for each soldier?

\newline

1

\newline

(A) Yes; a legionary costs

1

denarius per day to support, and an archer costs

2

denarii per day to support.

\newline

(B) Yes; a legionary costs

2

denarii per day to support, and an archer costs

\frac{4}{3}

denarii per day to support.

\newline

(C) No; the system has many solutions.

\newline

(D) No; the system has no solution.

The denarius was a unit of currency in ancient Rome. Suppose it costs the Roman government

10

denarius per day to support

3

legionaries and

3

archers. It only costs

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

1

\newline

(A) Yes; a legionary costs

1

denarius per day to support, and an archer costs

2

denarius per day to support.

\newline

(B) Yes; a legionary costs

2

denarius per day to support, and an archer costs

\frac{4}{3}

denarius per day to support.

\newline

(C) No; the system has many solutions.

\newline

(D) No; the system has no solution.

Quinn has a large family. She has

4

cousins who live in Texas,

3

cousins who live in Nebraska, and the other

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

1

\newline

3

16

\newline

4

16

\newline

7

16

\newline

9

16

A washing machine is being redesigned to handle a greater volume of water. One part is a pipe with a radius of

3

(\mathrm{cm})

and a length of

11 \mathrm{~cm}

. It gets replaced with a pipe of radius

4 \mathrm{~cm}

, and the same length. The new pipe can hold

w \pi

more cubic centimeters

\left(\mathrm{cm}^{3}\right)

of water than the old pipe, where

w

is a constant. What is the value of

w

Anya has two different sized cylindrical coffee mugs. The larger mug has an internal height of

15

(\mathrm{cm})

, and the smaller mug has an internal height of

10 \mathrm{~cm}

. Both mugs have an internal diameter of

8 \mathrm{~cm}

. Which of the following is closest to the difference in cubic centimeters

\left(\mathrm{cm}^{3}\right)

between the internal volume of the larger mug and the internal volume of the smaller mug?

\newline

1

\newline

63 \mathrm{~cm}^{3}

\newline

251 \mathrm{~cm}^{3}

\newline

754 \mathrm{~cm}^{3}

\newline

1,005 \mathrm{~cm}^{3}

Akoni, a painter, painted a circular helicopter landing pad one color. Next she painted over a

3

meter radius circle in the center, in a different color. If the second coat of paint covered an area half the size of the first coat, which of the following is most nearly the radius of the landing pad?

\newline

1

\newline

2

\newline

3

\newline

4

\newline

6

An architect has created a scale drawing for a residential townhouse in the shape of a rectangular prism. In the drawing, the internal width of the townhouse is labeled as

18

(\mathrm{ft})

, the internal length as

20 \mathrm{ft}

, and the internal height as

30 \mathrm{ft}

. The local building department tells the architect that if built, the building would be too tall according to local zoning laws and that its height must be reduced by

10 \%

. If the architect creates a second scale drawing where the townhouse's height is reduced by

10 \%

, what will be its new proposed internal volume in cubic feet?

468-6 g

\newline

The volume of a rectangular birthday cake is

468

cubic inches. The expression above models the amount of cake remaining, in cubic inches, after serving

g

guests. How much cake, in cubic inches, would remain after serving

50

300-1.5 p

\newline

To promote his new business, Claudio made

300

tamales to give to pedestrians. The expression above describes the number of tamales remaining after passing out samples to

p

pedestrians. If Claudio passed out samples to

160

pedestrians, how many tamales would remain?

5,500 P+4,500 S>50,000

\newline

Huang Yi works as a paleontologist and a sculptor. She earns

\$ 5,500

per month as a paleontologist and

\$ 4,500

per month as a sculptor. Huang Yi wants to make more than

\$ 50,000

this year. In the given inequality,

P

represents the number of months Huang Yi can work as a paleontologist and

S

represents the number of months she can work as a sculptor to reach her goal. If Huang Yi worked for

6

months as a paleontologist this year, what is the minimum whole number of months that she must work as a sculptor to reach her goal?

\begin{array}{l} m+p=23 \\ m-7=p \end{array}

\newline

The given system of equations relates

m

, the number of shirts Mara owns, and

p

, the number of shirts Phil owns. Which of the following is the best interpretation of the number

7

\newline

1

\newline

7

more shirts than Phil does.

\newline

7

more shirts than Mara does.

\newline

7

\newline

7

A cable company with a reputation for poor customer service is losing subscribers at a rate of approximately

3 \%

per year. The company had

2

million subscribers at the start of

2014

. Assume that the company continues to lose subscribers at the same rate, and that there are no new subscribers. Which of the following functions,

S

, models the number of subscribers (in millions) remaining

t

years after the start of

2014

\newline

1

\newline

S(t)=2(1.03)^{t}

\newline

S(t)=2(0.97)^{t}

\newline

S(t)=2(0.70)^{t}

\newline

S(t)=2(0.97) t

Two climate scientists are creating models for global rising sea levels. One scientist proposes a

10

-year doubling model, which predicts that the change in sea level,

R(t)

, will grow according to the following function:

\newline

R(t)=\frac{125}{256} \cdot(2)^{\frac{t}{10}}

\newline

t

is the number of years after

2005

. The other scientist proposes a steady increase model where the change in sea level will increase by

5

(\mathrm{cm})

per year. Both models predict that in the year

2105

, the sea level will increase by

500 \mathrm{~cm}

. To the nearest centimeter, how much greater is the predictions of the steady increase model than the

10

-year doubling model for the change in sea level in the year

2055

h(t)=-4.9 t^{2}+18 t+0.5

\newline

The equation models

h

, the height, in meters, of a soccer ball

t

seconds after the goalkeeper kicks it. Which of the following statements is the best interpretation of the ordered pair

(3,10.4)

\newline

1

\newline

3

seconds after the kick, the distance between the soccer ball and the goalkeeper is

10

4

\newline

3

seconds after the kick, the soccer ball is traveling at a speed of

10

4

meters per second.

\newline

3

seconds after the kick, the height of the soccer ball is

10

4

\newline

10

4

seconds after the kick, the height of the soccer ball is

3

3

years older than Daniel. Two years ago, Kevin was

4

times as old as Daniel.

\newline

k

be Kevin's age and let

d

be Daniel's age.

\newline

Which system of equations represents this situation?

\newline

1

\newline

\left\{\begin{array}{l}k=d+3 \\ k-2=4(d-2)\end{array}\right.

\newline

\left\{\begin{array}{l}k-2=(d-2)+3 \\ k=4 d\end{array}\right.

\newline

\left\{\begin{array}{l}d-2=(k-2)+3 \\ d=4 k\end{array}\right.

\newline

\left\{\begin{array}{l}d=k+3 \\ d-2=4(k-2)\end{array}\right.

P=\frac{19}{2}+\frac{4}{25} d

\newline

The amount of paint,

P

, in pints, that an art studio can buy with

d

dollars is given by the equation. What is the best interpretation of

\frac{4}{25}

as shown in the equation?

\newline

1

\newline

(A) The paint costs

\frac{4}{25}

of a dollar per pint.

\newline

(B) One dollar can buy

\frac{4}{25}

of a pint of paint.

\newline

(C) The art studio started with

\frac{4}{25}

of a pint of paint.

\newline

(D) The art studio can purchase at most

\frac{4}{25}

of a pint of paint.