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Solve equations with variables on both sides

A manufacturer of pickles has a fixed monthly cost of

\$10,000

. Additionally, each jar of pickles that the manufacturer produces costs

\$0.75

. If these are the only monthly costs incurred by the manufacturer, which of the following functions best models the manufacturer's total monthly cost,

C

, when it produces

p

jars of pickles?

Gabriela opened a savings account with a

200

deposit. Each month, she deposited an additional

50

into the account. After a year, Gabriela stopped making deposits and upgraded her account to earn compounding interest. Which of the following graphs in the

tb

-plane could represent the balance,

b

, in dollars, of Gabriela's bank account after

t

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1

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1

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6

packs of marbles with a total mass of

629 \, g

. The packaging of each pack has a mass of

\newline

\frac{2}{3} \, g

and each marble has a mass of

\newline

4 \frac{1}{2} \, g

\newline

What equations can we use to determine

m

, the number of marbles per pack?

Each week, Nia takes a violin lesson and a dance lesson. The dance lesson costs

\dfrac{2}{3}

as much as the violin lesson, and the cost of both lessons combined is

\$75

. Which of the following systems of equations could be used to find

d

, the cost of the dance lesson in dollars, and

v

, the cost of the violin lesson in dollars?

Rearrange the equation so

A

is the independent variable.

3a-7=-4b+1

b=

Lily's car has a fuel efficiency of

8

100

kilometers. What is the fuel efficiency of Lily's car in kilometers per liter?

\dfrac{\text{km}}{\text{L}}

Which expressions are equivalent to

12r-5

? Choose all answers that apply: Choose all answers that apply: (Choice A)

6(2r+(-1))+1

6(2r+(-1))+1

-4(2+(-3r))+3

-4(2+(-3r))+3

(Choice C) None of the above C None of the above

The speed of sound in air is about \\(332\) meters per second \

\left(\dfrac{\text m}{\text s}\right)

at \\(0\) degrees Celsius \

(^\circ \text C)

. If the speed increases by \\(0\).\(6\)\,\dfrac{\text m}{\text{s}} for every increase in temperature of \\(1\)^\circ \text C, which inequality best represents the temperatures, \

T

, in degrees Celsius, for which the speed of sound in air exceeds \\(350\)\,\dfrac{\text m}{\text s} ?

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T > 30

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T > 25

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T > 20

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T > 15

During a timed test, Alexander typed

742

14

minutes. Assuming Alexander works at this rate for the next hour, which of the following best approximates the number of words he would type in that hour?

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1

\newline

53

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840

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3

180

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44

520

The formula for the distance traveled over time and at an average speed is . Amit ran for

t

minutes at a speed of about

v

kilometers per hour. What calculation will give us the estimated distance Amit ran in kilometers?

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1

\newline

1

\newline

A

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B

\newline

C

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D

The formula for the distance traveled over time and at an average speed is . Amit ran for

t

minutes at a speed of about

v

kilometers per hour. What calculation will give us the estimated distance Amit ran in kilometers?

\newline

1

\newline

1

\newline

A

\newline

B

\newline

C

\newline

D

The formula for the distance traveled over time and at an average speed is . Amit ran for

t

minutes at a speed of about

v

kilometers per hour. What calculation will give us the estimated distance Amit ran in kilometers?

\newline

1

\newline

1

\newline

A

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B

\newline

C

\newline

D

50

one-cent jcoins were stacked on top of each other in a column, the column would be approximately

3 \frac{7}{8}

inches tall. At this rate, which of the following is closest to the number of one-cent coins it would take to make an

8

-inch-tall column?

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75

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100

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200

\newline

390

Simplify the equations:

\newline

y=2x+3

\newline

y=-3x+3

\space

\space

\space

\space

\space

\space

\space

\space

\space

\space

\space

\space

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\space

\space

\space

\space

\space

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\{[6x-5y=1]\}

\space

\space

\space

\space

\space

\{[4x-3y=0]\}

\newline

\{[-2x+2y=-1]\}

\{[-2x+2y=-1]\}

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How can we get System B from System A ?

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1

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A) Replace one equation with the sum/difference of both equations

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B) Replace only the left-hand side of one equation with the sum/difference of the left-hand sides of both equations

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C) Replace one equation with a multiple of itself

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D) Replace one equation with a multiple of the other equation

Tanvi teaches art lessons at a day camp and needs to get supplies. She purchases

40

Bright Streak paint sets from the Color Splash art store. She uses a coupon she received in the mail for

\$10

off her purchase. Tanvi spends

\$310

in all. Which equation can you use to find

p

, the original price of each Bright Streak paint set? What was the original price of each Bright Streak paint set?

d

\newline

-5d = -8d - 6

\newline

d =

g

\newline

2 + 8g = 10g

\newline

g =

f

\newline

2 - \frac{1}{4}f = f - \frac{3}{4}f + 3

\newline

f =

z

\newline

\frac{5}{3}z + 1 = 3 - \frac{2}{3}z

\newline

z =

Giant kelp is one of the fastest-growing organisms on the planet. Desmond told his little brother that kelp grows about

1.38 \times 10^{-2}

feet per minute. What would be the most appropriate unit for Desmond to use instead of feet per minute?

\newline

\newline

(A) inches per minute

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(B) feet per second

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(C) feet per hour

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(D) feet per day

Vera studies the extinction of the bear population of Siberia over time. The following function gives the number of bears

t

years since Vera started tracking it:

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B(t)=2190 \cdot e^{-0.3 t}

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What is the instantaneous rate of change of the number of bears after

2

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1

\newline

1202

\newline

1202

\newline

-361

\newline

-361

A person stands

12

meters east of an intersection and watches a car driving away from the intersection to the north at

4

meters per second.

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At a certain instant, the car is

9

meters from the intersection.

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What is the rate of change of the distance between the car and the person at that instant (in meters per second)?

\newline

1

\newline

2

4

\newline

4 \sqrt{10}

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15

\newline

\frac{20}{3}

Two cars are driving away from an intersection in perpendicular directions.

\newline

The first car's velocity is

7

meters per second and the second car's velocity is

3

meters per second.

\newline

At a certain instant, the first car is

5

meters from the intersection and the second car is

12

meters from the intersection.

\newline

What is the rate of change of the distance between the cars at that instant (in meters per second)?

\newline

1

\newline

\frac{99}{13}

\newline

\mathbf{1 3}

\newline

\sqrt{58}

\newline

\frac{71}{13}

Two cars are driving away from an intersection in perpendicular directions.

\newline

The first car's velocity is

5

meters per second and the second car's velocity is

8

meters per second.

\newline

At a certain instant, the first car is

15

meters from the intersection and the second car is

20

meters from the intersection.

\newline

What is the rate of change of the distance between the cars at that instant (in meters per second)?

\newline

1

\newline

13

\newline

25

\newline

9

4

\newline

\sqrt{89}

29

-meter ladder is sliding down a vertical wall so the distance between the top of the ladder and the ground is decreasing at

7

meters per minute.

\newline

At a certain instant, the bottom of the ladder is

21

meters from the wall.

\newline

What is the rate of change of the distance between the bottom of the ladder and the wall at that instant (in meters per minute)?

\newline

1

\newline

\frac{20}{3}

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20

\newline

\frac{147}{20}

\newline

7

20

-meter ladder is sliding down a vertical wall so the distance between the top of the ladder and the ground is decreasing at

8

meters per minute.

\newline

At a certain instant, the top of the ladder is

12

meters from the ground.

\newline

What is the rate of change of the distance between the bottom of the ladder and the wall at that instant (in meters per minute)?

\newline

1

\newline

\frac{32}{3}

\newline

8

\newline

24

\newline

6

39

-meter ladder is sliding down a vertical wall so the distance between the bottom of the ladder and the wall is increasing at

10

meters per minute.

\newline

At a certain instant, the bottom of the ladder is

\mathbf{3 6}

meters from the wall.

\newline

What is the rate of change of the distance between the top of the ladder and the ground at that instant (in meters per minute)?

\newline

1

\newline

-\frac{25}{6}

\newline

-12

\newline

-\mathbf{1 0}

\newline

-24

A person stands

12

meters east of an intersection and watches a car driving away from the intersection to the north at

4

meters per second.

\newline

At a certain instant, the car is

9

meters from the intersection.

\newline

What is the rate of change of the distance between the car and the person at that instant (in meters per second)?

\newline

1

\newline

\frac{20}{3}

\newline

15

\newline

4 \sqrt{10}

\newline

2

4

Two cars are driving away from an intersection in perpendicular directions.

\newline

The first car's velocity is

7

meters per second and the second car's velocity is

3

meters per second.

\newline

At a certain instant, the first car is

5

meters from the intersection and the second car is

12

meters from the intersection.

\newline

What is the rate of change of the distance between the cars at that instant (in meters per second)?

\newline

1

\newline

\frac{99}{13}

\newline

\frac{71}{13}

\newline

\mathbf{1 3}

\newline

\sqrt{58}

Two cars are driving away from an intersection in perpendicular directions.

\newline

The first car's velocity is

5

meters per second and the second car's velocity is

8

meters per second.

\newline

At a certain instant, the first car is

15

meters from the intersection and the second car is

20

meters from the intersection.

\newline

What is the rate of change of the distance between the cars at that instant (in meters per second)?

\newline

1

\newline

13

\newline

9

4

\newline

25

\newline

\sqrt{89}

25

-meter ladder is sliding down a vertical wall so the distance between the bottom of the ladder and the wall is increasing at

3

5

meters per minute.

\newline

At a certain instant, the top of the ladder is

7

meters from the ground.

\newline

What is the rate of change of the distance between the top of the ladder and the ground at that instant (in meters per minute)?

\newline

1

\newline

-3

5

\newline

-12

\newline

-7

\newline

-24

29

-meter ladder is sliding down a vertical wall so the distance between the top of the ladder and the ground is decreasing at

7

meters per minute.

\newline

At a certain instant, the bottom of the ladder is

21

meters from the wall.

\newline

What is the rate of change of the distance between the bottom of the ladder and the wall at that instant (in meters per minute)?

\newline

1

\newline

7

\newline

\frac{147}{20}

\newline

20

\newline

\frac{20}{3}

20

-meter ladder is sliding down a vertical wall so the distance between the top of the ladder and the ground is decreasing at

8

meters per minute.

\newline

At a certain instant, the top of the ladder is

12

meters from the ground.

\newline

What is the rate of change of the distance between the bottom of the ladder and the wall at that instant (in meters per minute)?

\newline

1

\newline

8

\newline

6

\newline

\frac{32}{3}

\newline

24

A person stands

30

meters east of an intersection and watches a car driving away from the intersection to the north at

17

meters per second.

\newline

At a certain instant, the car is

16

meters from the intersection.

\newline

What is the rate of change of the distance between the car and the person at that instant (in meters per second)?

\newline

1

\newline

\sqrt{1189}

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34

\newline

36

125

\newline

8

x

\newline

\frac{1}{2x}+1=-1x+4

20

feet per second directly to a flowerbed from its hive. The bee stays at the flowerbed for

12

minutes, and then flies directly back to the hive at

12

feet per second. It is away from the hive for a total of

17

minutes. What equation can you use to find the distance of the flowerbed from the hive?

Stephanie is TWICE as old as her Sister Rosa. If Stephanie is

18

years old, How OLD is Rosa?

What is the area of the region between

2

consecutive polnts where the graphs of

f(x)=\cos (x)

g(x)=-\cos (x)+2

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1

\newline

4

\newline

4 \pi

\newline

2

\newline

2 \pi

Solve for the exact value of

x

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2 \ln (6 x+5)-11=-3

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Solve for the exact value of

x

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6 \ln (4 x+9)+4=-14

\newline

s

\newline

\frac{5}{8} - \frac{3}{8}s = 2 - s

\newline

s =

d

\newline

\frac{4}{5}d + 3 = -2 - \frac{1}{5}d

\newline

d =

A plane and a helicopter are both flying toward Clover City Airport. The plane is

1,200

miles from the airport, and it is flying at a constant speed of

510

miles per hour. The helicopter is

500

miles from the airport, and it is flying at a constant speed of

160

miles per hour.

\newline

Which equation can you use to find

h

, the number of hours it will take for the plane and the helicopter to be the same distance from the airport?

\newline

\newline

1,200 - 500h = 510 - 160h

\newline

1,200 - 510h = 500 - 160h

\newline

How many hours will it take for the plane and the helicopter to be the same distance from the airport?

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Simplify any fractions.

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\newline

Julia and Lena started reading a science fiction book at the same time. During the first week, Lena read

10

more pages than Julia. The following week, Lena read

14

pages, and Julia read the same number of pages she read the first week. Now they are on the same page.

\newline

Which equation can you use to find

j

, the number of pages Julia read during the first week?

\newline

\newline

j + 10 + 14 = j + j

\newline

j + 10 = j + j + 14

\newline

How many pages did Julia read during the first week?

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\_\_\_

A plane and a helicopter are both flying toward Clover City Airport. The plane is

1,200

miles from the airport, and it is flying at a constant speed of

510

miles per hour. The helicopter is

500

miles from the airport, and it is flying at a constant speed of

160

miles per hour.

\newline

Which equation can you use to find

h

, the number of hours it will take for the plane and the helicopter to be the same distance from the airport?

\newline

\newline

1,200 - 510h = 500 - 160h

\newline

1,200 - 500h = 510 - 160h

\newline

How many hours will it take for the plane and the helicopter to be the same distance from the airport?

\newline

Simplify any fractions.

\newline

\newline

k

\newline

2k = 3k + 4

p

\newline

6p = 7p + 10

\newline

p =

s

\newline

-8s = -7s + 10

\newline

s =

h

\newline

-6 + 6h = 9h

\newline

h =