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Solve a system of equations using any method

Dimitri is solving the following equation for

x

\newline

2 x^{2}-7=33

\newline

His first few steps are given below.

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2 x^{2}=40

\newline

x^{2}=20

\newline

x= \pm \sqrt{20}

\newline

Is it necessary for Dimitri to check his answers for extraneous solutions?

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1

\newline

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Emily is solving the following equation for

z

\newline

z-8=\sqrt{3 z-5}

\newline

Her first few steps are given below.

\newline

\begin{aligned} (z-8)^{2} & =(\sqrt{3 z-5})^{2} \\ z^{2}-16 z+64 & =3 z-5 \end{aligned}

\newline

Is it necessary for Emily to check her answers for extraneous solutions?

\newline

1

\newline

\newline

Ahmed is solving the following equation for

x

\newline

2 \sqrt[3]{x-7}+11=3

\newline

His first few steps are given below.

\newline

\begin{aligned} 2 \sqrt[3]{x-7} & =-8 \\ \sqrt[3]{x-7} & =-4 \\ (\sqrt[3]{x-7})^{3} & =(-4)^{3} \\ x-7 & =-64 \end{aligned}

\newline

Is it necessary for Ahmed to check his answers for extraneous solutions?

\newline

1

\newline

\newline

A chameleon is looking for prey. Let positive numbers represent the elevation of prey above the chameleon and negative numbers represent the elevation of prey below the chameleon.

\newline

The chameleon spots a fly at

4 \mathrm{~m}

and a grasshopper at

-6 \mathrm{~m}

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What does an elevation of

0 \mathrm{~m}

represent in this situation?

\newline

1

\newline

(A) The elevation of the fly

\newline

(B) The elevation of the chameleon

\newline

(C) The elevation of the grasshopper

An arithmetic sequence is defined as follows:

\newline

\left\{\begin{array}{l} a_{1}=-138 \\ a_{i}=a_{i-1}+6 \end{array}\right.

\newline

Find the sum of the first

35

terms in the sequence.

José is on a gameshow that involves a taste test. He gets to see a list of

15

different secret ingredients before tasting

3

dishes one after the other. The

3

dishes are all identical to each other except that each dish contains

1

15

secret ingredients. He gets

60

seconds to identify what secret ingredient was in each dish.

\newline

The permutation formula

n \mathrm{P} r

can be used to find the number of unique ways to arrange the ingredients in the dishes.

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What are the appropriate values of

n

r

\newline

\begin{array}{l} n=\square \\ r=\square \end{array}

A small college with

1

200

total students has a student government of

40

members. From its members, the student government will elect a president, vice president, secretary, and treasurer. No single member can hold more than

1

4

\newline

The permutation formula

n \mathrm{Pr}

can be used to find the number of unique ways the student government can arrange its members into these positions.

\newline

What are the appropriate values of

n

r

\newline

\begin{array}{l} n=\square \\ r=\square \end{array}

Priya's favorite singer has made

6

albums containing

75

songs in total. Priya wants to make a playlist of

10

of those songs, and she won't repeat

1

75

\newline

The permutation formula

n \mathrm{Pr}

can be used to find the number of unique ways Priya can pick and arrange the songs for the playlist.

\newline

What are the appropriate values of

n

r

\newline

\begin{array}{l} n= \\ r= \end{array}

A biology teacher has

5

different pets they keep in their classroom. For an upcoming holiday break, the teacher will send the pets home with students. Suppose that

16

of the teacher's

75

students volunteer to take a pet home, and the teacher will randomly select

5

of those volunteers to each take

1

\newline

The permutation formula

n \mathrm{P} r

can be used to find the number of unique ways the teacher can distribute pets to the volunteers.

\newline

What are the appropriate values of

n

r

\newline

\begin{array}{l} n=\square \\ r=\square \end{array}

Gottfried wanted to see how contagious yawning can be. To better understand this, he conducted a social experiment by yawning in front of a random large crowd and observing how many people yawned as a result.

\newline

The relationship between the elapsed time

t

, in minutes, since Gottfried yawned, and the number of people in the crowd,

P_{\text {minute }}(t)

, who yawned as a result is modeled by the following function:

\newline

P_{\text {minute }}(t)=5 \cdot(1.03)^{t}

\newline

Complete the following sentence about the hourly rate of change in the number of people who yawn in Gottfried's experiment.

\newline

Round your answer to two decimal places.

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Every hour, the number of people who yawn in Gottfried's experiment grows by a factor of

The population of Springville is on the rise. Each year, on January

1

, the town holds its own census and the population is recorded. In

2010

, the town recorded a population of

12

000

2011

census reported a

5 \%

increase from the

2010

population, and the

2012

census reported a

7 \%

increase from the

2011

population. By how many people did the population of Springville increase over the

2

\newline

1

\newline

240

\newline

882

\newline

\mathbf{1 4 4 0}

\newline

\mathbf{1 4 8 2}

In School District X,

6.9 \%

29

students in Ms. Walker's sixthgrade class identify as Asian. The average sixth-grade class size in the school district is

29

. If the students in Ms. Walker's class are representative of students in the school district's sixth-grade classes and there are

112

sixth-grade classes in the district, which of the following best estimates the number of sixth-grade students in the school district who do not identify as Asian?

\newline

1

\newline

3

248

\newline

3

024

\newline

1

007

\newline

224

In a random sample of

200

152

have read at least one book in the past

12

months. Based on this information, approximately how many of the

255

million U.S. adults have not read a book in the past

12

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1

\newline

48

\newline

61

\newline

\mathbf{1 0 3}

\newline

\mathbf{1 9 4}

Takada is hosting an event for

100

of her fans. For the event, she meets each of her fans one at a time at a constant rate. After

90

minutes, she has met

45 \%

of her fans at the event. Which of the following equations models the number of fans,

F

, remaining for Takada to meet

m

minutes after the event started?

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1

\newline

F=100-0.5 m

\newline

F=100-0.45 m

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F=100(0.45)^{\frac{m}{90}}

\newline

F=100(0.55)^{\frac{m}{90}}

The nutrition facts on a container of mini pretzels state that each serving of the mini pretzels contains

310

milligrams of sodium, which is

13 \%

of the recommended daily value for adults. If

x

servings of mini pretzels contain

p

percent of an adult's recommended daily value of sodium, which of the following expresses

p

x

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1

\newline

p=(1.13)^{x}

\newline

p=310(0.13)^{x}

\newline

p=13 x

\newline

p=310(0.13 x)

A new shopping mall records

150

total shoppers on their first day of business. Each day after that, the number of shoppers is

15 \%

more than the number of shoppers the day before.

\newline

Which expression gives the total number of shoppers in the first

n

days of business?

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1

\newline

1.15\left(\frac{1-150^{n}}{1-150}\right)

\newline

0.85\left(\frac{1-150^{n}}{1-150}\right)

\newline

150\left(\frac{1-1.15^{n}}{1-1.15}\right)

\newline

150\left(\frac{1-0.85^{n}}{1-0.85}\right)

A high school's graduation rate is defined to be the percentage of the senior class that graduates. Last year

406

of Sagamore High School's

452

seniors graduated. This year the school expects the previous year's graduation rate to increase by approximately

2

percentage points. If there are

436

students in this year's senior class, which of the following best approximates the number of seniors that Sagamore High School expects to graduate this year?

\newline

1

\newline

390

\newline

400

\newline

410

\newline

420

Ivy is downloading a computer program from the Internet. After

8

minutes, the computer program is

35 \%

downloaded. If the computer program continues to download at the current rate, about how much longer will it take for Ivy's computer to finish downloading the program?

\newline

1

\newline

12

\newline

\mathbf{1 5}

\newline

18

\newline

23

A grocery store normally sells lemonade for

\$ 3.50

per bottle. The grocery store is currently having a sale on lemonade which advertises

6

bottles for only

\$ 13.50

. How much cheaper is the lemonade on sale per bottle compared to the normal price per bottle?

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1

\newline

\$ 0.75

\newline

\$ 1.25

\newline

\$ 1.75

\newline

\$ 2.25

An umbrella sprinkler is positioned on a ceiling at a point whose

x

0

. Negative values of

x

indicate distances, in meters, to the left of the position of the sprinkler, and positive values indicate distances to the right.

\newline

The path of water from the sprinkler can be modeled by the quadratic function

\newline

w(x)=-\frac{1}{4}\left(x^{2}-12\right)

\newline

w(x)

is the height of the water, in meters, at position

x

\newline

Which of the following equivalent expressions displays the height of the ceiling as a constant or coefficient?

\newline

1

A forest ranger travels up a river by boat to collect soil samples. The boat travels at a speed of

v

miles per hour for a distance of

2

miles. After collecting the samples, the ranger travels the same distance back, but the boat travels

5

miles per hour slower than it did on the way up the river. If the ranger spent

3

hours travelling up and back, which of the following equations could be used to determine the speed of the boat on the way up the river?

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1

\newline

2 v^{2}-13 v+17=0

\newline

3 v^{2}-15 v-4=0

\newline

3 v^{2}-11 v+10=0

\newline

3 v^{2}-19 v+10=0

In game theory, the average branching factor refers to the average number of legal moves available to a player at each turn. In chess, the average branching factor is about

35

\newline

If Jules wants to write a chess simulation program that represents every possible sequence of moves in a game that lasts

n

moves, the following function approximates the number of distinct sequences

s(n)

he would simulate.

\newline

s(n)=35^{n}

\newline

How many times greater does the number of sequences become each time the game lasts

2

The Capulet and Montague families love writing.

\newline

Last year, each Capulet wrote

4

essays, each Montague wrote

6

essays, and both families wrote

100

essays in total.

\newline

This year, each Capulet wrote

8

essays, each Montague wrote

12

essays, and both families wrote

200

essays in total.

\newline

How many Capulets and Montagues are there?

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1

\newline

(A) There is not enough information to determine the exact number of Capulets and Montagues.

\newline

(B) The given information describes an impossible situation.

\newline

16

6

\newline

6

16

When Famke charges her phone, it gains

2

percentage points each minute. When Famke uses her phone, it loses

5

percentage points each minute.

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Once, Famke used her phone for some time, and then charged it for twice the amount of time she used it. In the end, her phone lost

10

percentage points.

\newline

How long did Famke use her phone and how long did she charge it?

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1

\newline

(A) There is not enough information to determine the exact time Famke used and charged her phone.

\newline

(B) The given information describes an impossible situation.

\newline

(C) Famke used her phone for

10

minutes, and charged her phone for

20

\newline

(D) Famke used her phone for

20

minutes, and charged her phone for

\mathbf{1 0}

Liam's bookstore sold

40

20

newspapers for a total of

\$ 130

\newline

A day later, the bookstore sold

8

4

newspapers at the same prices, for a total of

\$ 28

\newline

How much does a notebook and a newspaper cost at Liam's bookstore?

\newline

1

\newline

(A) There is not enough information to determine the exact cost of a notebook and a newspaper.

\newline

(B) The given information describes an impossible situation.

\newline

(C) A notebook costs

\$ 2.50

and a newspaper costs

\$ 1.50

\newline

(D) A notebook costs

\$ 1.50

and a newspaper costs

\$ 2.50

6

tubes of red paint,

2

tubes of yellow paint,

13

tubes of white paint, and

9

tubes of blue paint for a mural.

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What is the ratio of tubes of blue paint to all of the tubes of paint used?

\newline

1

\newline

30

9

\newline

30

21

\newline

9

30

\newline

21

30

Erin loves to play sports. In all, she has earned

3

tennis trophies,

4

basketball trophies,

7

soccer trophies, and

1

volleyball trophy.

\newline

What is the ratio of Erin's tennis trophies to all of her trophies?

\newline

1

\newline

3: 4

\newline

3: 7

\newline

3: 12

\newline

3: 15

F=P+P r t

\newline

For an investment that bears simple interest, one can predict future value

F

t

years if the principal or initial amount invested

P

and annual interest rate

r

are known, by using the given equation. Which of the following correctly expresses the principal amount in terms of the annual rate, number of years invested, and future value?

\newline

1

\newline

P=\frac{F}{1+r t}

\newline

P=\frac{F}{1-r t}

\newline

P=\frac{F+P}{r t}

\newline

P=\frac{F-P}{r t}

\frac{P_{1} V_{1}}{T_{1}}=\frac{P_{2} V_{2}}{T_{2}}

\newline

Commonly known as the Combined Gas Law, the given equation relates the pressure,

P

V

, and temperature,

T

, of an ideal gas between its initial (

1

2

) states. Which of the following shows the final volume in terms of the initial and final pressure, the initial and final temperatures, and the initial volume?

\newline

1

\newline

V_{1}=V_{2}

\newline

V_{2}=\frac{P_{1} T_{2} V_{1}}{\left(T_{1} / P_{2}\right)}

\newline

V_{2}=\frac{P_{1} T_{2} V_{1}}{P_{2} T_{1}}

\newline

V_{2}=\frac{P_{1} V_{1}}{T_{1}}-\frac{T_{2}}{P_{2}}

\newline

For safety, each climber has

8

hooks when climbing the mountain. A large group of climbers climbs up a mountain and puts all their supplies in a heap in a cabin. One of the climbers wishes to know the number of climbers in the group, so he counts the number of hooks in the supply heap. If the climber counts

h

hooks, which of the following functions can be used to find the number of climbers,

c(h)

, in the group?

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1

\newline

c(h)=\frac{h}{64}

\newline

c(h)=\frac{h}{8}

\newline

c(h)=8 h

\newline

c(h)=64 h

Video games create a sense of motion by displaying still images, called frames, in rapid succession. A popular video game displays

60

frames per second. Which of the following equations shows the total number of frames,

f

, that the video game will display in

s

\newline

1

\newline

f=s+60

\newline

f=\frac{s}{60}

\newline

f=\frac{60}{s}

\newline

f=60 s

Miles's Internet service provider lets him use up to

1

229

gigabytes of data in a month and charges him an additional fee if he goes over the data limit. If Miles already used

984

gigabytes of data this month, which of the following represents

d

, the amount of data, in gigabytes, he can use in the rest of the month without incurring an additional fee?

\newline

1

\newline

d \leq 245

\newline

d \geq 245

\newline

d \leq 2,213

\newline

d \geq 2,213

3.5 c-1.5 d \geq 50

\newline

Esa's Pastries sells cupcakes for

\$ 3.50

each and donuts for

\$ 1.50

each. The given inequality represents the difference, in dollars, between cupcake sales and donut sales on a typical day based on

c

, the number of cupcakes sold and

d

, the number of donuts sold. If Esa sold

200

donuts on a typical day, what is the minimum number of cupcakes she sold on that day?

\newline

1

\newline

25

\newline

50

\newline

100

\newline

350

Jules is making a peanut butter sandwich with two slices of bread and chunky peanut butter. If the mass of two slices of bread is

60

grams and the mass of the entire sandwich is

92

grams, which of the following equations can be used to find

x

, the amount of peanut butter Jules used in grams?

\newline

1

\newline

30+x=92

\newline

60+x=92

\newline

120+x=92

\newline

60 x=92

Jasmine sleeps for an average of

6

hours each day. If Jasmine sleeps for

\frac{1}{3}

the amount of time that her cat sleeps each day, which of the following equations can be used to find

x

, the number of hours Jasmine's cat sleeps each day?

\newline

1

\newline

6 x=\frac{1}{3}

\newline

3 x=6

\newline

x-\frac{1}{3}=6

\newline

\frac{x}{3}=6

Aria has two pet snakes, Mian and Udon. The combined length of the two snakes is

34

inches, and Udon is

12

inches shorter than Mian. Which of the following systems of equations could be used to find

m

, the length of Mian in inches, and

u

, the length of Udon in inches?

\newline

1

\newline

m+u=12

\newline

m-u=34

\newline

m+u=12

\newline

u-m=34

\newline

m+u=34

\newline

m-u=12

\newline

m+u=34

\newline

u-m=12

\begin{array}{l} x+y=11 \\ x+3=y \end{array}

\newline

Bex and Yin are running errands together. The given system of equations relates

x

, the number of errands Bex has to run, and

y

, the number of errands Yin has to run. Based on the system, which of the following statements is true?

\newline

1

\newline

11

more errands to run than Yin does.

\newline

11

more errands to run than Bex does.

\newline

3

more errands to run than Yin does.

\newline

3

more errands to run than Bex does.

Which of the following statements about the graph of

y=12(0.75)^{x}

\newline

1

\newline

x

y

increases at an increasing rate.

\newline

x

y

increases at a decreasing rate.

\newline

x

y

decreases at an increasing rate.

\newline

x

y

decreases at a decreasing rate.

Anna would like to surround an

8

\times 10

in photograph with a border of uniform width. Let

x

be the width of the border, in inches. Which of the following functions could she use to determine the area,

A

, in square inches

\left(\mathrm{in}^{2}\right)

, of the picture and border combined?

\newline

1

\newline

A=(8-x)(10-x)

\newline

A=(8-2 x)(10-2 x)

\newline

A=(8+x)(10+x)

\newline

A=(8+2 x)(10+2 x)

Wang Lei would like to build a

144 \mathrm{ft}^{2}

rectangular garden. He plans to enclose this area with exactly

50 \mathrm{ft}

of fencing. Which of the following equations could be used to find the width,

x

, of Wang Lei's garden?

\newline

1

\newline

x(72-x)=50

\newline

x(25-x)=144

\newline

x(144-2 x)=50

\newline

x(50-2 x)=144

p(x)=(x-0.95)(500-100 x)

\newline

The equation models

p

, the daily profit, in dollars, a food truck makes by selling their signature tacos at a price of

x

dollars each. What is the daily profit if the tacos are sold at

\$ 3

\newline

1

\newline

\$ 410

\newline

\$ 600

\newline

\$ 820

\newline

\$ 2,460

h(t)=-16.1 t^{2}+100 t

\newline

The equation models

h

, the height of a firework shell in feet,

t

seconds after launch. What is the height, in feet, of the firework shell

2

seconds after launch?

\newline

1

\newline

135

6

\newline

167

8

\newline

232

2

\newline

264

4

P(x)=2,500(0.98)^{x}

\newline

The function models

P

, the population of the village of Kalm

x

1997

. Based on the model, what is the population of Kalm

18

1997

\newline

1

\newline

2

450

\newline

2

127

\newline

1

846

\newline

1

738

Josh is hiking Glacier National Park. He has now hiked a total of

17 \mathrm{~km}

2 \mathrm{~km}

\frac{1}{2}

of the way done with his hike. If

h

is the total length of Josh's hike, which of the following equations best describes the situation?

\newline

1

\newline

\frac{1}{2} h-2=17

\newline

\frac{1}{2} h+2=17

\newline

2 h-2=17

\newline

2 h+2=17

The rapid growth rate of Moso bamboo makes it an environmentally important plant, as it stores massive amounts of carbon dioxide

\left(\mathrm{CO}_{2}\right)

. Each year, one hectare of Moso bamboo in China absorbs

5

1

tons of carbon dioxide, which can compensate for the annual

\mathrm{CO}_{2}

3

people. Which of the following equations best models the number of hectares,

h

, of Moso bamboo that must be planted to compensate for the annual

\mathrm{CO}_{2}

100

people in China?

\newline

1

\newline

100=3 h

\newline

100=5.1 h

\newline

100=\frac{5.1}{3} h

\newline

100=\frac{3}{5.1} h

Felipe is saving money for a class trip. He already has saved

\$ 250

that he will put toward the trip. To save more money for the trip, Felipe gets a job where each month he can add

\$ 350

to his savings for the trip. Let

m

be the number of months that Felipe has worked at his new job. If Felipe needs to save

\$ 2700

to go on the trip, which equation best models the situation?

\newline

1

\newline

250 m-350=2700

\newline

250 m+350=2700

\newline

350 m-250=2700

\newline

350 m+250=2700

A reservoir can be filled to

\frac{1}{4}

1

day by Pump A, and to

\frac{1}{3}

1

day by Pump B. Both pumps have a constant rate of flow. Which of the following inequalities best describes the number of days,

d

, it will take for both pumps working at the same time to fill the reservoir to over

\frac{7}{8}

\newline

1

\newline

\frac{12}{7} d>\frac{7}{8}

\newline

\frac{4}{d}+\frac{3}{d} \leq \frac{7}{8}

\newline

\frac{7}{12} d>\frac{7}{8}

\newline

\frac{d}{4}+\frac{d}{3} \leq \frac{7}{8}

\newline

ar equations word problems | Lesson

At a particular college, a film club and a chess club were founded at the beginning of the same month. The film club began with

16

members and gained

1

new member per month and the chess club began with

4

members and gained

4

new members per month. At this rate, at the beginning of which month did the two clubs have the same membership?

\newline

1

\newline

2^{\text {nd }}

\newline

3^{\text {rd }}

\newline

4^{\text {th }}

\newline

5^{\text {th }}

John read the first

114

pages of a novel, which was

3

pages less than

\frac{1}{3}

of the novel. If

p

is the total number of pages in the novel, which of the following equations best describes the situation?

\newline

1

\newline

\frac{1}{3} p-3=114

\newline

\frac{1}{3} p+3=114

\newline

3 p-3=114

\newline

3 p+3=114

\begin{array}{l} -x+\frac{1}{2} y=-3 \\ 0=c x-2 y+10 \end{array}

\newline

In the system of equations,

c

is a constant. For what value of

c

does the system of linear equations have no solutions?

...