All practice questions

Question 1001Question →

If $2(x - 6) = 10$ , what is the value of $x + 4$ ?

12

3

15

11

Show answer & explanation

Answer: 15

Answer

15

The correct answer is 15. Isolating the variable in the equation

2(x - 6) = 10

can be done by first dividing both sides by 2, which gives

x - 6 = 5

. Adding 6 to both sides yields

x = 11

. Substituting this value into the expression

x + 4

results in

11 + 4 = 15

. Alternatively, distributing the 2 gives

2x - 12 = 10

, and adding 12 to both sides gives

2x = 22

, which leads to

x = 11

and thus

x + 4 = 15

.

Step-by-Step Solution

1

Distribute the 2 to the terms inside the parentheses on the left side of the equation.

2x - 12 = 10

To remove the parentheses and prepare to isolate the variable term.

2

Add 12 to both sides of the equation.

2x = 22

To isolate the variable term

2x

on the left side of the equation.

3

Divide both sides of the equation by 2.

x = 11

To find the value of

x

.

4

Substitute the value of

x

into the expression

x + 4

.

11 + 4 = 15

To find the final value requested by the question.

Key Concept

Solving linear equations in one variable by applying the distributive property, isolating the variable, and evaluating an expression.

Alternative Method

Instead of distributing first, divide both sides of the equation

2(x - 6) = 10

by 2 directly to get

x - 6 = 5

. Then, add 10 to both sides of the equation

x - 6 = 5

to get the value of

x + 4

directly:

(x - 6) + 10 = 5 + 10 \Rightarrow x + 4 = 15

.

Estimated Time:45s

Question 1002Question →

Which form of the verb 'confirm' correctly completes the sentence to conform to the conventions of Standard English?

Fill in the blanks below

In his 1915 paper on general relativity, Albert Einstein proposed that gravity is not an attractive force between masses but rather a structural curvature of spacetime. Although many physicists initially doubted this radical new model, British astronomer Arthur Eddington the theory in 1919 by measuring the deflection of starlight passing near the Sun during a total solar eclipse, providing the first empirical evidence for Einstein's ideas.

Show answer & explanation

Answer

The correct answer is 'confirmed' because the sentence refers to a completed historical action that took place at a specific time in the past.

The simple past tense form 'confirmed' is correct because the sentence describes a completed historical event that occurred at a specific point in the past, indicated by the time marker 'in 1919.' This form also maintains grammatical consistency with other past tense verbs in the passage, such as 'proposed' and 'doubted.'

Step-by-Step Solution

1

Identify the temporal context of the sentence containing the blank.

The action occurred in a specific, completed past year (1919).

This helps determine whether the verb should be in the past, present, or future tense.

2

Check the tenses of other verbs in the passage describing the same sequence of events.

The verbs 'proposed' and 'doubted' are in the simple past tense.

Ensuring tense consistency is necessary unless there is a clear reason to shift tenses.

3

Select the appropriate form of the verb 'confirm'.

The simple past tense form 'confirmed' correctly completes the sentence.

It matches the historical timeline and maintains tense consistency with the surrounding past tense verbs.

Key Concept

Verb Tense, Aspect, and Mood

Estimated Time:1m 0s

Question 1003Question →

A linear function $f$ has a slope of $\frac{2}{3}$ . The function $g$ is defined by $g(x) = f(x + 4) - 3$ . If the graph of $y = g(x)$ in the $xy$ -plane has an $x$ -intercept at $(5, 0)$ , what is the $x$ -intercept of the graph of $y = f(x)$ ?

Show answer & explanation

Answer: 4.5

Answer

4.5

The correct answer is

4.5

(or

\frac{9}{2}

). By using the function transformation relation, we find

g(5) = f(9) - 3 = 0

, which yields

f(9) = 3

. With a slope of

\frac{2}{3}

, the equation of the line is

f(x) = \frac{2}{3}x - 3

. Setting

f(x) = 0

yields the

x

-intercept at

x = 4.5

.

Step-by-Step Solution

1

Use the definition of function

g

and its given

x

-intercept to find a point on the graph of

f

.

f(9) = 3

Since the graph of

y = g(x)

has an

x

-intercept at

(5, 0)

, we know

g(5) = 0

. Substituting

x = 5

into the definition

g(x) = f(x + 4) - 3

gives

g(5) = f(5 + 4) - 3 = f(9) - 3

. Since

g(5) = 0

, it follows that

f(9) - 3 = 0

, or

f(9) = 3

.

2

Determine the equation of the linear function

f

using its slope and the point identified in Step 1.

f(x) = \frac{2}{3}x - 3

The function

f

is linear with a slope of

\frac{2}{3}

and passes through the point

(9, 3)

. Using the point-slope formula, we get

f(x) - 3 = \frac{2}{3}(x - 9)

, which simplifies to

f(x) = \frac{2}{3}x - 3

.

3

Find the

x

-intercept of the graph of

y = f(x)

by setting

f(x) = 0

.

x = 4.5

(or

\frac{9}{2}

)

To find the

x

-intercept, set

f(x) = 0

. This gives

\frac{2}{3}x - 3 = 0

. Adding

3

to both sides and multiplying by

\frac{3}{2}

yields

x = \frac{9}{2}

, which is equal to

4.5

.

Key Concept

Linear Functions and Graphs

Alternative Method

Alternatively, we can write the equation of

f

in slope-intercept form as

f(x) = \frac{2}{3}x + b

. Then the definition of

g(x)

becomes

g(x) = \frac{2}{3}(x + 4) + b - 3 = \frac{2}{3}x + \frac{8}{3} + b - 3 = \frac{2}{3}x + b - \frac{1}{3}

. Since the graph of

g

has an

x

-intercept at

(5, 0)

, we substitute

x = 5

and

g(5) = 0

to get

\frac{2}{3}(5) + b - \frac{1}{3} = 0

, which simplifies to

3 + b = 0

, so

b = -3

. This gives

f(x) = \frac{2}{3}x - 3

. Finally, the

x

-intercept is found by solving

\frac{2}{3}x - 3 = 0

, resulting in

x = 4.5

.

Estimated Time:2m 0s

Question 1004Question →

Although she did not live to see the final decipherment of the ancient Aegean script known as Linear B, classicist Alice Kober played a vital role in its decoding. Having painstakingly recorded and analyzed grammatical patterns on thousands of index cards, _______

Which choice completes the text so that it conforms to the conventions of Standard English?

Kober's grammatical grid was established, which eventually enabled Michael Ventris to decipher the language.

Michael Ventris was eventually enabled to decipher the language using a grammatical grid established by Kober.

Kober established the grammatical grid that eventually enabled Michael Ventris to decipher the language.

the final decipherment of the language by Michael Ventris was built upon a grammatical grid that she established.

Show answer & explanation

Answer: Kober established the grammatical grid that eventually enabled Michael Ventris to decipher the language.

Answer

Kober established the grammatical grid that eventually enabled Michael Ventris to decipher the language.

The correct option correctly places 'Kober' as the subject of the main clause immediately following the introductory modifying phrase. Since she is the person who recorded and analyzed the grammatical patterns, this structure conforms to Standard English rules of subject-modifier agreement.

Step-by-Step Solution

1

Identify the introductory modifying phrase and determine its logical subject.

The introductory phrase is 'Having painstakingly recorded and analyzed grammatical patterns on thousands of index cards.' The logical subject who performed this action is classicist Alice Kober.

An introductory modifier must immediately precede the noun it logically describes to avoid a dangling or misplaced modifier.

2

Evaluate the options to find the one where the subject immediately following the comma is the logical subject ('Kober').

Only the option beginning with 'Kober' correctly aligns the subject with the modifier. The options beginning with 'Kober's grammatical grid,' 'Michael Ventris,' or 'the final decipherment' create dangling or misplaced modifiers because these nouns cannot logically record and analyze cards.

Ensuring standard English modifier agreement rules requires the subject of the main clause to be the agent of the action described in the modifying phrase.

Key Concept

Subject-Modifier Agreement

Question 1005Question →

Read the passage about marine biology research. Which word, when inserted into the blank, maintains a parallel structure with the other items in the list?

Fill in the blanks below

During their month-long research expedition in the northern Pacific Ocean, the marine biologists focused on three primary tasks. They spent their days mapping the deep ocean floor, collecting water samples at various depths, and the migration patterns of local whale populations. Through these efforts, the team gathered invaluable data that will help protect the region's marine habitats for future generations.

Show answer & explanation

Answer

A present participle verb ending in -ing (such as 'tracking', 'observing', 'monitoring', or 'studying') that matches the grammatical form of the preceding gerunds ('mapping' and 'collecting').

The sentence contains a list of activities that the biologists performed: 'mapping the deep ocean floor,' 'collecting water samples at various depths,' and the final item in the series. The first two items are gerunds/present participles ('mapping' and 'collecting'). To maintain parallel structure, the blank must be filled with a present participle verb ending in -ing (such as 'tracking', 'observing', 'monitoring', 'studying', 'recording', or 'analyzing') to complete the series in the same grammatical form.

Step-by-Step Solution

1

Identify the list of items in the sentence that must be parallel.

The items are 'mapping the deep ocean floor', 'collecting water samples at various depths', and the final action in the blank.

The coordinating conjunction 'and' signals a list of elements that must share the same grammatical form.

2

Analyze the grammatical structure of the first two items in the list.

Both 'mapping' and 'collecting' are present participles ending in -ing.

To maintain parallel structure, all items in the series must have the same grammatical construction.

3

Determine the required form for the blank and select an appropriate verb.

The blank requires a present participle ending in -ing (such as 'tracking', 'observing', 'monitoring', or 'studying') that fits the context of whale migration.

Using any other verb form (like the infinitive 'to track' or simple present 'track') violates parallel structure.

Key Concept

Parallel structure requires that items in a list or series follow the same grammatical pattern.

Estimated Time:45s

Question 1006Question →

In the $xy$ -plane, line $L$ passes through the points $(0, 4)$ and $(6, 0)$ . A second line, $M$ , is perpendicular to line $L$ and intersects line $L$ at a point on the line $y = x$ . If line $M$ is represented by the equation $y = px + q$ , where $p$ and $q$ are constants, what is the value of $p + q$ ?

\frac{6}{5}

\frac{10}{3}

\frac{3}{10}

-\frac{9}{2}

Show answer & explanation

Answer: $\frac{3}{10}$

Answer

The correct answer is the value stating three-tenths

To find the sum of the slope

p

and

y

-intercept

q

of line

M

, we first determine the equation of line

L

. Line

L

passes through

(0, 4)

and

(6, 0)

, so its slope is

\frac{0 - 4}{6 - 0} = -\frac{2}{3}

. The equation of line

L

is therefore

y = -\frac{2}{3}x + 4

. Setting

y = x

gives

x = -\frac{2}{3}x + 4

, which yields

x = 2.4

. The intersection point is

(2.4, 2.4)

. Because line

M

is perpendicular to line

L

, its slope is the negative reciprocal of

-\frac{2}{3}

, which is

p = 1.5

. Using point-slope form with the intersection point

(2.4, 2.4)

, the equation of line

M

is

y - 2.4 = 1.5(x - 2.4)

, which simplifies to

y = 1.5x - 1.2

. Thus,

q = -1.2

. The sum

p + q

is

1.5 - 1.2 = 0.3

, or

\frac{3}{10}

.

Step-by-Step Solution

1

Find the equation of line

L

using the given points

(0, 4)

and

(6, 0)

.

The slope of line

L

is

m_L = \frac{0 - 4}{6 - 0} = -\frac{2}{3}

. Since the

y

-intercept is

(0, 4)

, the equation of line

L

is

y = -\frac{2}{3}x + 4

.

To find where line

L

intersects another line, we first need to establish its linear equation.

2

Find the intersection point of line

L

and the line

y = x

.

Substitute

y = x

into the equation for line

L

:

x = -\frac{2}{3}x + 4 \implies \frac{5}{3}x = 4 \implies x = 2.4

. Thus, the intersection point is

(2.4, 2.4)

.

The problem states that line

M

intersects line

L

at a point on the line

y = x

.

3

Determine the equation of line

M

using the intersection point and the perpendicular relationship.

Since line

M

is perpendicular to line

L

, its slope

p

is the negative reciprocal of

-\frac{2}{3}

, which is

p = \frac{3}{2} = 1.5

. Using point-slope form at

(2.4, 2.4)

:

y - 2.4 = 1.5(x - 2.4) \implies y = 1.5x - 1.2

. This gives

q = -1.2

.

Perpendicular lines in the coordinate plane have slopes that multiply to negative one. We then use the point-slope formula to define the equation of line

M

.

4

Calculate the sum of

p

and

q

.

p + q = 1.5 + (-1.2) = 0.3 = \frac{3}{10}

.

The question asks for the value of the expression

p + q

.

Key Concept

Perpendicular lines and linear graph intersections

Estimated Time:3m 0s

Question 1007Question →

The following passage is about the development of a mathematical theorem. Which punctuation mark must be inserted into the blank to ensure that the nonessential element in the sentence is punctuated correctly?

Fill in the blanks below

Although her contributions were initially undervalued by contemporaries due to institutional gender bias, her formulation—which posits that every differentiable symmetry of a physical system corresponds to a specific conservation lawfundamentally reshaped the theoretical framework of modern physics.

Show answer & explanation

Answer

The correct punctuation mark is an em dash (—).

The correct punctuation mark is an em dash (—). The sentence contains a nonessential relative clause ('which posits that every differentiable symmetry of a physical system corresponds to a specific conservation law') that modifies the noun 'formulation'. This nonessential clause is opened with an em dash, so it must be closed with an em dash to maintain punctuation symmetry before continuing to the main verb 'fundamentally'.

Step-by-Step Solution

1

Identify the nonessential element in the sentence.

The clause 'which posits that every differentiable symmetry of a physical system corresponds to a specific conservation law' is a nonessential relative clause modifying the noun 'formulation'.

Nonessential elements provide extra information that can be removed without altering the core grammatical structure of the sentence.

2

Check the punctuation mark used to introduce the nonessential element.

The nonessential clause is introduced with an em dash (—) right after 'formulation'.

Standard English grammar requires that nonessential elements be set off by matching pairs of punctuation.

3

Determine the closing punctuation mark.

Since the nonessential element begins with an em dash, it must also end with an em dash (—) before the main verb 'fundamentally'.

Using asymmetrical punctuation, such as opening with an em dash and closing with a comma or parenthesis, violates standard grammatical rules.

Key Concept

Punctuation of Nonessential Elements

Question 1008Question →

The following passage contains a grammatical blank. Which physicist's name correctly fills the blank to ensure that the sentence maintains subject-modifier agreement?

Fill in the blanks below

Recognized for her crucial contributions to the Manhattan Project and her experimental proof of parity violation, challenged a widely accepted law of physics in 1956.

Show answer & explanation

Answer

Chien-Shiung Wu

The introductory modifying phrase describes a female scientist's achievements. To ensure subject-modifier agreement, the subject of the sentence immediately following the comma must be the person who matches this description. Placing the name Chien-Shiung Wu directly after the modifier establishes a clear and grammatically correct connection.

Step-by-Step Solution

1

Identify the introductory modifying phrase at the start of the sentence.

The modifying phrase is 'Recognized for her crucial contributions to the Manhattan Project and her experimental proof of parity violation.'

This phrase contains the possessive pronoun 'her', indicating that it must modify a specific female scientist.

2

Determine who or what must immediately follow the modifying phrase to avoid a dangling modifier.

The name of the physicist, Chien-Shiung Wu, must be placed directly after the comma as the subject of the main clause.

Subject-modifier agreement requires that the noun being described by the introductory modifier immediately follow it.

Key Concept

Subject-modifier agreement

Question 1009Question →

The passage below is missing a punctuation mark. Write the single punctuation mark that should fill the blank to correctly set off the nonessential element.

Fill in the blanks below

The red panda—a small mammal native to the eastern Himalayasshares a name but not a close genetic relationship with the giant panda. While both species feed primarily on bamboo, recent genetic analysis has revealed that they belong to entirely different families. The red panda is actually the sole living member of the family Ailuridae, whereas the giant panda is classified under Ursidae.

Show answer & explanation

Answer

An em dash (—)

The sentence contains a nonessential appositive phrase ('a small mammal native to the eastern Himalayas') that provides extra information about the subject ('The red panda'). Since the phrase is introduced with an em dash, it must be closed with a matching em dash to maintain grammatical symmetry.

Step-by-Step Solution

1

Identify the nonessential element in the sentence.

The phrase 'a small mammal native to the eastern Himalayas' is an appositive that can be removed without changing the core grammatical structure of the sentence.

This determines where punctuation is needed to set off the nonessential element.

2

Examine the punctuation mark used to introduce the nonessential element.

The nonessential element is preceded by an em dash ('—').

Matching pairs of punctuation (comma-comma, dash-dash, or parenthesis-parenthesis) must be used to set off nonessential elements.

3

Determine the correct punctuation mark to close the nonessential element.

An em dash ('—') must be placed in the blank.

Using an em dash maintains structural symmetry with the em dash that introduces the phrase.

Key Concept

Punctuation of Nonessential Elements

Estimated Time:45s

Question 1010Question →

In the $xy$ -plane, the graphs of the linear equations $y = 3x - 5$ and $y = -x + 7$ intersect at the point $(x, y)$ . What is the value of $x$ ?

1

3

4

6

Show answer & explanation

Answer: 3

Answer

3

To find the

x

-value of the intersection point of the two graphs, set the two equations equal to each other because both represent the same

y

-value at that point:

3x - 5 = -x + 7

. Adding

x

to both sides of the equation gives

4x - 5 = 7

. Adding

5

to both sides gives

4x = 12

. Dividing both sides by

4

yields

x = 3

.

Step-by-Step Solution

1

Set the two expressions for

y

equal to each other to find the

x

-coordinate of the intersection point.

3x - 5 = -x + 7

Since both equations are solved for

y

, their right-hand sides must be equal at the point of intersection.

2

Add

x

to both sides of the equation to collect the variable terms on one side.

4x - 5 = 7

Moving the variable terms together allows us to isolate

x

.

3

Add

5

to both sides of the equation to isolate the term with

x

.

4x = 12

Grouping the constant terms on the opposite side prepares the equation for final division.

4

Divide both sides by

4

to find the value of

x

.

x = 3

This isolates

x

and gives the final solution.

Key Concept

Solving a system of linear equations by setting the equations equal to find the point of intersection in the coordinate plane.

Question 1011Question →

Architects in ancient Rome achieved remarkable structural durability by mixing volcanic ash with slaked lime. This specific volcanic ash _______ was sourced primarily from the region around Mount Vesuvius and contained reactive silica and alumina. When mixed with water, the combination triggered a chemical reaction that formed a mineral called stratlingite, which prevented the spread of microscopic cracks throughout the concrete.

Which choice completes the text so that it conforms to the conventions of Standard English?

—known as pozzolana,

, known as pozzolana—

—known as pozzolana—

, known as pozzolana

Show answer & explanation

Answer: —known as pozzolana—

Answer

The correct answer is the option that sets off the nonessential phrase with matching em dashes ('—known as pozzolana—').

The phrase 'known as pozzolana' is a nonessential appositive that provides extra, nonessential information about the volcanic ash. To conform to the conventions of Standard English, nonessential elements must be set off from the rest of the sentence using a matching pair of punctuation marks. The correct punctuation uses matching em dashes at both the beginning and the end of the phrase, properly isolating it.

Step-by-Step Solution

1

Identify the nonessential element in the sentence.

The phrase 'known as pozzolana' is an appositive phrase that provides extra details about the 'volcanic ash' but is not grammatically necessary for the sentence to function.

Recognizing that the phrase is nonessential determines that it must be set off by matching punctuation on both sides.

2

Examine the punctuation options to find a matching pair.

Only the choice utilizing matching em dashes on both sides correctly isolates the nonessential phrase.

Standard English requires that nonessential elements be set off with matching punctuation (either two commas, two em dashes, or two parentheses); mixing punctuation marks or leaving one side unpunctuated is incorrect.

Key Concept

Punctuation of Nonessential Elements

Question 1012Question →

In the equation below,

a

and

b

are constants.

a(3x - 5) - b(2x + 2) = 4x - 12

If the equation has infinitely many solutions for

x

, what is the value of

a + b

?

3

7

8

13

Show answer & explanation

Answer: 3

Answer

3

To find the value of

a + b

that results in infinitely many solutions, we first distribute the constants

a

and

b

to rewrite the equation as

(3a - 2b)x - (5a + 2b) = 4x - 12

. For a linear equation to have infinitely many solutions, the coefficient of

x

on both sides must be equal, and the constant terms on both sides must be equal. This gives the system of equations

3a - 2b = 4

and

5a + 2b = 12

. Adding these two equations yields

8a = 16

, which gives

a = 2

. Substituting

a = 2

into

3a - 2b = 4

yields

6 - 2b = 4

, which gives

b = 1

. Therefore, the value of

a + b

is

2 + 1 = 3

.

Step-by-Step Solution

1

Distribute the constants

a

and

b

on the left side of the equation and group the terms.

(3a - 2b)x - (5a + 2b) = 4x - 12

To write the linear equation in the standard form

Ax + B = Cx + D

so that coefficients can be compared.

2

Set up a system of equations by equating the coefficient of

x

and the constant term on both sides of the equation.

3a - 2b = 4

and

-(5a + 2b) = -12

(which simplifies to

5a + 2b = 12

)

For a linear equation in one variable to have infinitely many solutions, the coefficient of

x

on both sides must be equal, and the constant terms on both sides must be equal.

3

Solve the system of equations by adding them to eliminate

b

.

8a = 16 \implies a = 2

Adding

3a - 2b = 4

and

5a + 2b = 12

eliminates

b

, allowing us to solve directly for

a

.

4

Substitute

a = 2

back into

3a - 2b = 4

to solve for

b

.

3(2) - 2b = 4 \implies 6 - 2b = 4 \implies -2b = -2 \implies b = 1

Substituting the value of

a

allows us to determine

b

.

5

Calculate the value of

a + b

.

a + b = 2 + 1 = 3

To find the sum of the two constants as requested.

Key Concept

Linear Equations in One Variable (Infinitely Many Solutions)

Question 1013Question →

In her analysis of the agricultural transition in the Neolithic Near East, archaeologist Sarah Stein posits that the shift to sedentary farming was not a sudden revolution. Instead, she argues that this prolonged transition was driven not only by communities adapting to localized climatic fluctuations but also by _______

Which choice completes the text so that it conforms to the conventions of Standard English?

the development of complex regional trade networks by societies.

developing complex regional trade networks among societies.

societies developing complex regional trade networks.

to develop complex regional trade networks within societies.

Show answer & explanation

Answer: societies developing complex regional trade networks.

Answer

societies developing complex regional trade networks.

The correct answer maintains parallel structure by matching the grammatical form of the first element in the 'not only... but also...' construction. The first element, 'communities adapting to localized climatic fluctuations,' consists of a plural noun followed by a present participle phrase. The correct choice, 'societies developing complex regional trade networks,' follows this exact pattern (plural noun + present participle phrase), providing grammatical symmetry.

Step-by-Step Solution

1

Identify the parallel trigger and the structure of the first element in the comparison.

The sentence uses the correlative conjunction 'not only... but also...'. The first parallel element following the preposition 'by' is 'communities adapting to localized climatic fluctuations', which consists of a plural noun ('communities') modified by an -ing participle phrase ('adapting...').

Parallel structure requires that elements linked by correlative conjunctions share the same grammatical form.

2

Determine the grammatical structure required for the second element following the preposition 'by'.

The second element must also consist of a plural noun followed by an -ing participle phrase.

This maintains the precise grammatical symmetry established by the first element.

3

Evaluate the answer choices to find the one that matches this grammatical pattern.

The option 'societies developing complex regional trade networks' consists of a plural noun ('societies') modified by the -ing participle phrase ('developing...'), making it the only parallel choice.

Other choices use different grammatical forms such as noun phrases, bare gerunds, or infinitive phrases, which break the parallel pattern.

Key Concept

Parallel Structure

Question 1014Question →

If $\frac{2}{3}(x - 4) = 6$ , what is the value of $x$ ?

Show answer & explanation

Answer: 13

Answer

13

To solve the linear equation

\frac{2}{3}(x - 4) = 6

, multiply both sides of the equation by

\frac{3}{2}

to isolate

x - 4

. This yields

x - 4 = 9

. Adding

4

to both sides of the equation isolates

x

and gives the value

x = 13

.

Step-by-Step Solution

1

Multiply both sides of the equation

\frac{2}{3}(x - 4) = 6

by the reciprocal of the fraction, which is

\frac{3}{2}

.

x - 4 = 9

Multiplying a fraction by its reciprocal simplifies it to 1, leaving the term in parentheses isolated on the left side.

2

Add

4

to both sides of the equation

x - 4 = 9

.

x = 13

Adding

4

isolates the variable

x

on the left side.

Key Concept

Solving a one-variable linear equation using inverse operations.

Question 1015Question →

In the $xy$ -plane, line $l$ passes through the point $(3, 4)$ and has a slope of $m$ , where $m > 0$ . Line $k$ is perpendicular to line $l$ and passes through the point $(2, 2)$ . If the sum of the $y$ -intercepts of line $l$ and line $k$ is $1$ , what is the value of $m$ ?

\frac{1}{3}

\frac{2}{3}

1

2

Show answer & explanation

Answer: $2$

Answer

The correct value of

m

is

2

.

The correct answer is the value

2

. By finding the equations of both lines in slope-intercept form, we express their

y

-intercepts in terms of

m

: the

y

-intercept of line

l

is

4 - 3m

and the

y

-intercept of line

k

is

\frac{2}{m} + 2

. Setting their sum to

1

gives the equation

3m^2 - 5m - 2 = 0

. Factoring this quadratic yields the solutions

m = -\frac{1}{3}

and

m = 2

. Since

m

must be positive,

m = 2

is the only valid solution.

Step-by-Step Solution

1

Find the equation and

y

-intercept of line

l

.

Using the point-slope form

y - y_1 = m(x - x_1)

with the point

(3, 4)

and slope

m

, the equation of line

l

is

y - 4 = m(x - 3)

, which simplifies to

y = mx + 4 - 3m

. Thus, the

y

-intercept of line

l

is

4 - 3m

.

Expressing the

y

-intercept of line

l

in terms of

m

allows us to use it in the sum equation.

2

Find the equation and

y

-intercept of line

k

.

Since line

k

is perpendicular to line

l

, its slope is the negative reciprocal of

m

, which is

-\frac{1}{m}

. Using the point-slope form with the point

(2, 2)

, the equation of line

k

is

y - 2 = -\frac{1}{m}(x - 2)

, which simplifies to

y = -\frac{1}{m}x + \frac{2}{m} + 2

. Thus, the

y

-intercept of line

k

is

\frac{2}{m} + 2

.

Expressing the

y

-intercept of line

k

in terms of

m

allows us to use it in the sum equation.

3

Set up the equation for the sum of the

y

-intercepts and solve for

m

.

The sum of the

y

-intercepts is

(4 - 3m) + (\frac{2}{m} + 2) = 1

. Simplifying this equation gives

6 - 3m + \frac{2}{m} = 1 \implies 5 - 3m + \frac{2}{m} = 0

. Multiplying by

m

yields

3m^2 - 5m - 2 = 0

. Factoring the quadratic gives

(3m + 1)(m - 2) = 0

, which has solutions

m = -\frac{1}{3}

and

m = 2

. Since

m > 0

, we have

m = 2

.

To determine the unique positive slope that satisfies the given conditions.

Key Concept

Writing equations of perpendicular lines and finding their intercepts using slopes and points.

Question 1016Question →

A food truck sells tacos and burritos. On Tuesday, the food truck sold a total of 120 tacos and burritos. The number of tacos sold, $t$ , was 20 more than the number of burritos sold, $b$ . This situation is represented by the system of equations below:

t + b = 120

t = b + 20

How many burritos did the food truck sell on Tuesday?

Show answer & explanation

Answer: 50

Answer

The food truck sold 50 burritos on Tuesday.

To find the number of burritos sold,

b

, we substitute

t = b + 20

into the first equation

t + b = 120

, giving

(b + 20) + b = 120

. Combining like terms yields

2b + 20 = 120

. Subtracting 20 from both sides results in

2b = 100

. Finally, dividing both sides by 2 gives

b = 50

. Thus, the food truck sold 50 burritos on Tuesday.

Step-by-Step Solution

1

Substitute the expression for

t

from the second equation into the first equation

(b + 20) + b = 120

To eliminate the variable

t

and set up an equation with only one variable,

b

.

2

Combine like terms on the left side of the equation

2b + 20 = 120

To simplify the equation for solving.

3

Subtract 20 from both sides of the equation

2b = 100

To isolate the variable term on one side of the equation.

4

Divide both sides of the equation by 2

b = 50

To solve for the variable

b

representing the number of burritos.

Key Concept

Solving systems of linear equations using the substitution method

Question 1017Question →

Complete the passage with the form of the verb in parentheses that correctly completes the sentence to conform to the conventions of Standard English.

Fill in the blanks below

By the time the European Space Agency’s JUICE spacecraft reaches the Jovian system in 2031, planetary scientists (develop) advanced simulation models based on preliminary telescopic observations. These sophisticated models will subsequently assist researchers in analyzing data collected during close flybys of the icy moons Ganymede, Callisto, and Europa.

Show answer & explanation

Answer

will have developed

The correct answer is 'will have developed' because the sentence establishes a future reference point ('By the time the European Space Agency’s JUICE spacecraft reaches the Jovian system in 2031') and describes an action that will be completed prior to that point. The future perfect tense ('will have' followed by the past participle of 'develop') is the standard grammatical structure to convey this sequence.

Step-by-Step Solution

1

Analyze the temporal clues and clauses in the passage.

The passage begins with the dependent clause 'By the time the European Space Agency’s JUICE spacecraft reaches the Jovian system in 2031', referencing a future event using the present-tense verb 'reaches'.

This establishes a future deadline by which another action must be completed.

2

Select the correct tense to show completion prior to a future time.

The main clause requires the future perfect tense ('will have' plus the past participle of 'develop') to indicate that the simulation models will be completed before the spacecraft arrives.

The future perfect tense ('will have developed') correctly expresses an action that will be completed before a specified future reference point.

Key Concept

Future Perfect Tense

Question 1018Question →

During the late Renaissance, the Flemish cartographer Gerardus Mercator revolutionized maritime navigation with the introduction of his 1569 world map. Designed primarily to assist sailors, this projection _______ allowed mariners to steer ships across vast oceans without constantly adjusting their bearings. Although the map distorted the relative size of landmasses near the poles, its utility for navigation made it an enduring standard in cartography.

Which choice completes the text so that it conforms to the conventions of Standard English?

, which represented sailing courses of constant angle as straight lines—

—which represented sailing courses of constant angle as straight lines,

—which represented sailing courses of constant angle as straight lines—

which represented sailing courses of constant angle as straight lines

Show answer & explanation

Answer: —which represented sailing courses of constant angle as straight lines—

Answer

The correct option is the one that sets off the nonessential relative clause with matching em dashes: '—which represented sailing courses of constant angle as straight lines—'.

The correct answer properly sets off the nonessential relative clause 'which represented sailing courses of constant angle as straight lines' with matching em dashes. In English grammar, nonessential elements—which provide supplementary information that can be removed without altering the sentence's basic meaning—must be set off with symmetrical punctuation (either two commas, two parentheses, or two em dashes).

Step-by-Step Solution

1

Identify the nonessential element in the sentence.

The clause 'which represented sailing courses of constant angle as straight lines' is a nonessential relative clause because it provides extra explanatory detail about 'this projection' but is not necessary to maintain the core grammatical structure or meaning of the main sentence.

Recognizing whether a modifier is nonessential determines whether it requires punctuation to set it off from the rest of the sentence.

2

Determine the punctuation requirements for nonessential elements.

Nonessential elements must be set off from the main clause by matching punctuation marks: either two commas, two em dashes, or two parentheses.

Standard English conventions require punctuation surrounding nonessential elements to be symmetrical.

3

Evaluate the choices for matching punctuation pairs.

The option beginning with an em dash and ending with an em dash is the only one that uses a matching pair of punctuation marks to isolate the nonessential clause.

Options using mixed punctuation (a comma with an em dash, or an em dash with a comma) or no punctuation at all violate the symmetry rule for nonessential elements.

Key Concept

Punctuation of Nonessential Elements

Question 1019Question →

In 2024, a specialized commission of international environmental lawyers and policy experts convened to draft a legally binding treaty aimed at reducing global carbon emissions. Although this prominent group initially faced intense skepticism from industrial representatives who questioned the feasibility of the new regulations, _______ final report was highly praised by the UN delegation for its scientific rigor.

Which choice completes the text so that it conforms to the conventions of Standard English?

their

its

it's

they

Show answer & explanation

Answer: its

Answer

its

The singular possessive pronoun 'its' is correct because it agrees in number with the singular collective noun antecedent 'group' and properly modifies 'final report' to show possession.

Step-by-Step Solution

1

Identify the antecedent that the pronoun in the blank must replace or refer to.

The pronoun refers to the noun phrase 'this prominent group' (which stands for 'commission'). This is a singular collective noun.

Establishing the correct antecedent determines the singular or plural number of the pronoun.

2

Determine the grammatical function and case of the pronoun in the sentence.

The pronoun precedes the noun phrase 'final report' and must indicate ownership, meaning a possessive case pronoun is required.

The sentence structure dictates the case of the pronoun needed to modify the noun.

3

Evaluate the choices to find the singular possessive pronoun.

'its' is singular and possessive, agreeing with 'group' and correctly modifying 'final report.'

Selecting the option that matches both the required number and case completes the sentence grammatically.

Key Concept

Pronoun-antecedent agreement requires a pronoun to match its antecedent in both number (singular or plural) and gender. Singular collective nouns (such as 'group' or 'commission') take singular pronouns, even when followed by intervening plural nouns (like 'lawyers' or 'experts') in prepositional phrases. Additionally, writers must distinguish possessive pronouns like 'its' from contractions like 'it's'.

Question 1020Question →

During a chemistry experiment, the temperature of a liquid sample is decreased at a constant rate. At the start of the experiment, the temperature of the liquid is $80^\circ\text{C}$ . After $4$ minutes, the temperature of the liquid is $68^\circ\text{C}$ . If $T(t)$ represents the temperature of the liquid, in degrees Celsius, $t$ minutes after the experiment starts, which of the following equations defines $T$ ?

T(t) = -3t + 80

T(t) = 3t + 80

T(t) = -\frac{1}{3}t + 80

T(t) = 80t - 3

Show answer & explanation

Answer: $T(t) = -3t + 80$

Answer

The equation

T(t) = -3t + 80

correctly defines the temperature function.

The correct equation shows a starting value of

80

and a rate of change of

-3

, which corresponds to the initial temperature of

80^\circ\text{C}

and a decrease of

3^\circ\text{C}

per minute.

Step-by-Step Solution

1

Identify the

y

-intercept (initial value) of the linear function.

The initial temperature at

t = 0

is

80^\circ\text{C}

, so the

y

-intercept

b

is

80

.

The initial value of a linear relationship in context corresponds to the

y

-intercept.

2

Calculate the slope (constant rate of change) using the coordinate points

(0, 80)

and

(4, 68)

.

The slope

m

is

\frac{68 - 80}{4 - 0} = \frac{-12}{4} = -3

.

The slope is defined as the change in the dependent variable (temperature) divided by the change in the independent variable (time).

3

Substitute the slope

m

and

y

-intercept

b

into the slope-intercept form

T(t) = mt + b

.

T(t) = -3t + 80

Substituting the specific slope and intercept values into the general form defines the particular function.

Key Concept

Determining a linear equation from a real-world scenario by identifying the initial value and constant rate of change.

Answer

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