All practice questions

Question 901Question →

The following passage is an excerpt from an art history essay. Complete the passage by typing the punctuation mark that correctly fills the blank to ensure standard English conventions.

Fill in the blanks below

The American artist Mary Cassatt was celebrated for her intimate portraits of mothers and children, which departed from traditional European portraiture. Her famous painting *The Child's Bath*, which she completed in 1893 showcases her mastery of Japanese woodblock print techniques through its bold lines and unique perspective. By emphasizing domestic scenes, Cassatt helped introduce Impressionist ideas to a broader American audience.

Show answer & explanation

Answer

A comma (,) is required to close the nonessential relative clause.

The clause 'which she completed in 1893' is a nonessential relative clause describing the painting *The Child's Bath*. To correctly set off this nonessential element, the punctuation must be symmetrical. Since the clause is introduced by a comma after the painting's title, it must also be closed with a comma before the main verb 'showcases'.

Step-by-Step Solution

1

Identify the nonessential element in the sentence.

The clause 'which she completed in 1893' is a nonessential relative clause because removing it does not change the core meaning of the main sentence.

Nonessential elements must be set off from the main clause with punctuation on both sides.

2

Examine the punctuation that opens the nonessential element.

The clause begins after a comma: 'Her famous painting *The Child's Bath*, which...'

This establishes that commas are the chosen punctuation style for setting off the nonessential element.

3

Select the matching punctuation mark to close the nonessential element.

A matching comma (,) must be placed at the end of the clause before the main verb 'showcases'.

Punctuation marks setting off nonessential elements must be symmetrical (comma-comma).

Key Concept

Punctuation of Nonessential Elements

Question 902Question →

In the equation $\frac{3}{4}(8x - 12) + kx = 10x - 9$ , $k$ is a constant. If the equation has infinitely many solutions, what is the value of $k$ ?

Show answer & explanation

Answer: 4

Answer

The value of

k

is

4

.

Distributing the fraction on the left side of the equation yields

\frac{3}{4}(8x) - \frac{3}{4}(12) = 6x - 9

. Substituting this back gives

6x - 9 + kx = 10x - 9

. Factoring out

x

on the left side gives

(6 + k)x - 9 = 10x - 9

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

x

on both sides must be identical, and the constants must be identical. Since the constants on both sides are already

-9

, we set the coefficients equal:

6 + k = 10

. Subtracting 6 from both sides gives the correct value

k = 4

.

Step-by-Step Solution

1

Distribute the fraction

\frac{3}{4}

to the terms inside the parentheses.

6x - 9 + kx = 10x - 9

To simplify the expression and eliminate the parentheses.

2

Factor out

x

from the terms on the left side of the equation.

(6 + k)x - 9 = 10x - 9

To group the

x

terms together to easily compare coefficients.

3

Set the coefficient of

x

on the left side equal to the coefficient of

x

on the right side.

6 + k = 10

For the equation to have infinitely many solutions, the coefficients of the variable on both sides must be equal when the constant terms are equal.

4

Solve for

k

by subtracting 6 from both sides of the equation.

k = 4

To isolate the constant

k

.

Key Concept

A linear equation in one variable has infinitely many solutions when it can be simplified to an identity of the form

Ax + B = Ax + B

, meaning both the coefficients of

x

and the constant terms on both sides of the equation are equal.

Question 903Question →

The passage below is from an academic text about linguistic history. Which form of the verb 'struggle' correctly completes the sentence in accordance with the rules of Standard English conventions?

Fill in the blanks below

Before the basalt slab known as the Rosetta Stone was discovered in Egypt in 1799, European scholars for centuries to decipher ancient hieroglyphic inscriptions. Because these researchers had long operated under the false assumption that the characters were entirely symbolic rather than phonetic, they made no progress until the slab provided a bilingual key.

Show answer & explanation

Answer

had struggled (or had been struggling)

The correct answer is 'had struggled' or 'had been struggling'. The sentence describes an action that occurred over a duration ('for centuries') before another event in the past ('was discovered in 1799'). To indicate that an action was ongoing or completed prior to a specific point in the past, the past perfect tense ('had struggled') or past perfect continuous tense ('had been struggling') must be used.

Step-by-Step Solution

1

Identify the time frame established by the context.

The phrase 'was discovered in Egypt in 1799' establishes a specific event in the past.

Verbs must align with the temporal relationship defined by the text.

2

Determine the relationship between the missing verb and the established past event.

The action of trying to decipher the inscriptions occurred over a period ('for centuries') prior to the discovery in 1799.

Actions that were completed or ongoing prior to a specific point in the past require the past perfect tense.

3

Select the correct past perfect form of the verb 'struggle'.

'had struggled' (past perfect) or 'had been struggling' (past perfect continuous) correctly indicates this sequence.

These forms correctly show that the action occurred before the discovery.

Key Concept

Verb Tense, Aspect, and Mood

Question 904Question →

Complete the sentence with the proper subject so that it conforms to the conventions of Standard English, ensuring that the introductory modifying phrase logically describes the subject.

Fill in the blanks below

Designed by the architect Antoni Gaudí, has been under construction in Barcelona since 1882.

Show answer & explanation

Answer

the Sagrada Família

The introductory participial phrase 'Designed by the architect Antoni Gaudí' must logically modify the subject of the main clause. The basilica 'the Sagrada Família' is the entity designed by Gaudí, making it the grammatically correct subject to place immediately after the modifier.

Step-by-Step Solution

1

Identify the introductory modifying phrase and its logical target.

The phrase 'Designed by the architect Antoni Gaudí' describes a specific structure or building designed by Gaudí.

An introductory modifier must logically describe the subject that immediately follows it.

2

Determine the correct subject that fits both the modifier and the context of the sentence.

The iconic basilica 'the Sagrada Família' is the famous building designed by Gaudí that has been under construction in Barcelona since 1882.

Placing 'the Sagrada Família' immediately after the comma ensures that the modifier correctly describes the subject of the main clause, avoiding a dangling modifier.

Key Concept

Subject-Modifier Agreement

Question 905Question →

In the $xy$ -plane, the graph of a linear function $f$ passes through the points $(0, 1)$ and $(2, k)$ . The graph of another linear function $g$ is perpendicular to the graph of $f$ and passes through the points $(-4, 8)$ and $(2, k)$ . Which of the following could be the value of $k$ ?

2

8

5

9

Show answer & explanation

Answer: 5

Answer

5

The slope of the line representing function

f

is calculated as

m_f = \frac{k - 1}{2}

, and the slope of the line representing function

g

is

m_g = \frac{k - 8}{6}

. Since the lines are perpendicular, the product of their slopes must be

-1

. This gives the equation

\frac{k - 1}{2} \cdot \frac{k - 8}{6} = -1

, which simplifies to

(k - 1)(k - 8) = -12

. Expanding and setting the quadratic equation to zero gives

k^2 - 9k + 20 = 0

. Factoring this expression yields

(k - 4)(k - 5) = 0

. Thus,

k

can be either

4

or

5

. Since 5 is the only option listed, the option containing 5 is the correct answer.

Step-by-Step Solution

1

Express the slope of the linear function

f

in terms of

k

.

m_f = \frac{k - 1}{2 - 0} = \frac{k - 1}{2}

The slope formula is

m = \frac{y_2 - y_1}{x_2 - x_1}

.

2

Express the slope of the linear function

g

in terms of

k

.

m_g = \frac{k - 8}{2 - (-4)} = \frac{k - 8}{6}

The slope formula is applied to the points

(-4, 8)

and

(2, k)

.

3

Apply the perpendicular lines condition to set up an equation for

k

.

m_f \cdot m_g = -1 \implies \left(\frac{k - 1}{2}\right)\left(\frac{k - 8}{6}\right) = -1

Perpendicular lines have slopes that are negative reciprocals of each other, meaning their product is

-1

.

4

Solve the quadratic equation for

k

.

(k - 1)(k - 8) = -12 \implies k^2 - 9k + 8 = -12 \implies k^2 - 9k + 20 = 0 \implies (k - 4)(k - 5) = 0

Multiply both sides by 12, expand the product, move all terms to one side, and factor the quadratic expression.

5

Identify the possible values of

k

and match with the options.

k = 4

or

k = 5

. Since 5 is the only value present in the options, the correct value is 5.

Both values satisfy the condition, but only one is listed among the multiple-choice options.

Key Concept

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other (their product is

-1

).

Question 906Question →

In the equation below, $a$ and $b$ are constants.

\frac{1}{3}(2a - 5x) - \frac{3}{4}(x - b) = -\frac{29}{12}x + 5

If the equation has infinitely many solutions for $x$ , what is the value of $8a + 9b$ ?

Show answer & explanation

Answer: 60

Answer

The value of

8a + 9b

is

60

.

To find the value of

8a + 9b

that makes the equation have infinitely many solutions, we first expand and simplify the left side of the equation:

\frac{2}{3}a - \frac{5}{3}x - \frac{3}{4}x + \frac{3}{4}b = -\frac{29}{12}x + 5

. Combining the

x

terms gives

-\frac{29}{12}x + \(\frac{2}{3}a + \frac{3}{4}b\) = -\frac{29}{12}x + 5

. Since the coefficients of

x

on both sides are equal (

-\frac{29}{12}

), the equation will have infinitely many solutions if the constant terms on both sides are also equal. This requires

\frac{2}{3}a + \frac{3}{4}b = 5

. Multiplying this entire equation by the least common multiple of the denominators, which is 12, yields

8a + 9b = 60

.

Step-by-Step Solution

1

Distribute the constants through the parentheses on the left side of the equation.

\frac{2}{3}a - \frac{5}{3}x - \frac{3}{4}x + \frac{3}{4}b = -\frac{29}{12}x + 5

This separates the variable terms from the constant terms so the equation can be simplified.

2

Combine the coefficients of the

x

terms on the left side using 12 as the common denominator.

-\frac{29}{12}x + \frac{2}{3}a + \frac{3}{4}b = -\frac{29}{12}x + 5

Simplifying the variable terms allows us to compare the coefficients on both sides of the equation.

3

Equate the constant terms from the left and right sides of the equation.

\frac{2}{3}a + \frac{3}{4}b = 5

A linear equation has infinitely many solutions when the coefficients of the variable on both sides are equal and the constant terms on both sides are also equal.

4

Multiply both sides of the equation by 12 to eliminate the fractional denominators.

8a + 9b = 60

Multiplying the equation by the common denominator directly evaluates the target expression

8a + 9b

.

Key Concept

For a linear equation in one variable to have infinitely many solutions, it must be reducible to an identity of the form

cx + d = cx + d

, where both the variable coefficients and the constant terms on both sides of the equation are equal.

Question 907Question →

The following passage is from an academic text about South American archaeology. Complete the passage by entering the single punctuation mark that should go in the blank to correctly set off the nonessential element.

Fill in the blanks below

In 1992, archaeologist Maria Gutierrez uncovered a collection of ancient stone tools at the Arroyo Seco site in Argentina. The artifacts, which were found in close association with the remains of extinct Pleistocene megafaunastrongly suggest that early humans inhabited South America thousands of years earlier than previously assumed by the scientific community.

Show answer & explanation

Answer

A comma is the correct punctuation mark to set off the nonessential element.

The sentence contains the nonessential relative clause 'which were found in close association with the remains of extinct Pleistocene megafauna.' Because this clause is introduced with a comma after the noun 'artifacts', it must also be closed with a comma before the verb 'strongly suggest' to maintain punctuation symmetry.

Step-by-Step Solution

1

Identify the nonessential element within the sentence.

The phrase 'which were found in close association with the remains of extinct Pleistocene megafauna' is a nonessential relative clause that provides extra description but is not grammatically essential to the main clause.

This helps determine which part of the sentence needs to be set off by punctuation.

2

Locate the punctuation mark used to open the nonessential clause.

The clause is introduced with a comma after the word 'artifacts'.

Symmetrical punctuation rules require that the opening and closing punctuation marks for a nonessential element must match.

3

Determine the matching punctuation mark needed to close the nonessential clause.

Since the clause begins with a comma, it must close with a comma right before the main verb 'strongly suggest'.

Using a comma maintains punctuation symmetry and grammatically isolates the nonessential clause.

Key Concept

Nonessential elements (such as appositives, parenthetical phrases, or relative clauses) must be set off from the rest of the sentence using matching punctuation marks: two commas, two em dashes, or parentheses. Mixing punctuation marks (such as starting with a comma and ending with a dash) is incorrect.

Question 908Question →

The following passage is an excerpt from an academic text on astrobiology. The second sentence contains a nonessential appositive phrase. What punctuation mark should be inserted into the blank to correctly set off this phrase?

Fill in the blanks below

In the hyper-arid McMurdo Dry Valleys of Antarctica, astrobiologists have discovered endolithic communities, which are microbial ecosystems that survive inside the microscopic pores of translucent rocks. These resilient organisms—primarily lichens and cyanobacteria obtain their moisture from transient snowmelt and photosynthesize using light that filters through the mineral layers. This habitat demonstrates that life can persist under the most extreme terrestrial conditions, offering a potential model for past life on Mars.

Show answer & explanation

Answer

An em dash (—)

The sentence includes the nonessential appositive phrase 'primarily lichens and cyanobacteria' to describe 'These resilient organisms.' In Standard English, nonessential elements must be set off from the rest of the sentence using a matching pair of punctuation marks: two commas, two em dashes, or two parentheses. Because the appositive phrase is introduced with an em dash (—), it must be closed with a matching em dash (—) before the main verb 'obtain' to maintain grammatical symmetry.

Step-by-Step Solution

1

Identify the nonessential element in the sentence.

The phrase 'primarily lichens and cyanobacteria' is an appositive phrase that provides nonessential details about the subject 'These resilient organisms.'

Identifying the nonessential element is necessary to determine which part of the sentence must be set off with punctuation.

2

Examine the punctuation mark used to introduce the nonessential element.

The phrase is introduced with an em dash (—) immediately following the word 'organisms.'

Standard English conventions require that nonessential elements be set off using matching pairs of punctuation (such as two commas, two em dashes, or two parentheses).

3

Determine the matching punctuation mark for the blank.

To match the opening em dash, the blank must be filled with an em dash (—) before the main verb 'obtain.'

Using any other punctuation mark, such as a comma, would create an asymmetrical punctuation pattern, which violates Standard English conventions.

Key Concept

Punctuation of nonessential elements requires matching pairs of commas, em dashes, or parentheses to set off the parenthetical information.

Question 909Question →

If $20 - 2(x + 3) = 8$ , what is the value of $x$ ?

-11

-3

3

9

Show answer & explanation

Answer: 3

Answer

3

The correct answer is 3. Distributing

-2

to the terms inside the parentheses gives

20 - 2x - 6 = 8

. Simplifying the left side yields

14 - 2x = 8

. Subtracting 14 from both sides results in

-2x = -6

. Dividing both sides by

-2

gives

x = 3

.

Step-by-Step Solution

1

Distribute the coefficient

-2

to all terms inside the parentheses

(x + 3)

.

20 - 2x - 6 = 8

Applying the distributive property requires multiplying both

x

and

3

by

-2

.

2

Combine the constant terms

20

and

-6

on the left side of the equation.

14 - 2x = 8

Simplifying the expression makes it easier to isolate the variable term.

3

Subtract

14

from both sides of the equation to isolate the variable term

-2x

.

-2x = -6

This moves the constant term to the opposite side of the equation.

4

Divide both sides of the equation by

-2

to solve for

x

.

x = 3

Dividing by the coefficient of the variable isolates the variable completely.

Key Concept

Solving linear equations in one variable with parentheses by distributing coefficients, combining like terms, and isolating the variable.

Question 910Question →

Which verb form correctly completes the blank in the academic passage below?

Fill in the blanks below

Although modern historians often view the 1919 Treaty of Versailles as a deeply flawed accord that paved the way for subsequent global conflict, many of the allied diplomats who actually attended the peace negotiations in Paris firmly that the final agreement would establish a stable and lasting peace throughout Europe.

Show answer & explanation

Answer

believed

The simple past tense verb 'believed' is correct because it maintains tense consistency with the other past tense verbs in the sentence, specifically 'paved' and 'attended', which establish a historical narrative about the 1919 Treaty of Versailles.

Step-by-Step Solution

1

Identify the time frame established by the context of the sentence.

The sentence discusses a historical event in the past (the 1919 Treaty of Versailles and negotiations in Paris).

Temporal indicators such as '1919' and the past tense verbs 'paved' and 'attended' establish a past-tense narrative framework.

2

Select the verb form that maintains consistency with this past-tense framework.

The simple past tense verb 'believed' matches the established past tense.

The action of believing occurred at the same time as the negotiations described by 'attended', so the simple past tense is required to maintain consistency.

Key Concept

Verb Tense Consistency

Estimated Time:1m 0s

Question 911Question →

A factory manufactures solar panels at a constant rate. At the start of a morning shift, the factory has already manufactured $80$ solar panels. The total number of solar panels manufactured $t$ hours after the shift begins is modeled by a linear function. If the factory has manufactured a total of $200$ solar panels $3$ hours after the shift begins, how many hours after the shift begins will the factory have manufactured a total of $440$ solar panels?

Show answer & explanation

Answer: 9

Answer

9

To find the time when the total number of solar panels reaches

440

, we first determine the constant rate of production. The change in solar panels over the first

3

hours is

200 - 80 = 120

panels. Dividing by the

3

hours gives a constant rate of

40

panels per hour. The linear equation representing the total panels is

N(t) = 40t + 80

. Setting

N(t) = 440

gives

440 = 40t + 80

, which simplifies to

360 = 40t

. Solving for

t

yields

t = 9

.

Step-by-Step Solution

1

Calculate the constant rate of production (slope) of the linear function.

The rate is

40

solar panels per hour.

The factory starts with

80

solar panels and reaches

200

solar panels in

3

hours. The rate of change is the change in the number of panels divided by the change in time:

\frac{200 - 80}{3 - 0} = \frac{120}{3} = 40

panels per hour.

2

Write the linear equation representing the total number of solar panels manufactured,

N(t)

, after

t

hours.

N(t) = 40t + 80

Since the initial quantity is

80

and the rate of production is

40

panels per hour, the linear function is

N(t) = 40t + 80

.

3

Solve for the time

t

when the total number of solar panels is

440

.

t = 9

hours

Set

N(t) = 440

in the equation:

440 = 40t + 80

. Subtract

80

from both sides to get

360 = 40t

. Divide both sides by

40

to find

t = 9

.

Key Concept

Linear Functions and Graphs

Estimated Time:1m 30s

Question 912Question →

In the equation $\frac{1}{2}(kx - 4) - \frac{2}{3}(x - k) = \frac{5}{6}x - 1$ , where $k$ is a constant, the equation has no solution for $x$ . What is the value of $k$ ?

Show answer & explanation

Answer: 3

Answer

The value of the constant

k

for which the equation has no solution is

3

.

For a linear equation of the form

Ax + B = Cx + D

to have no solution, the coefficients of the variable terms must be equal (

A = C

) but the constant terms must be unequal (

B \neq D

). Distributing the terms on the left side of the given equation and grouping them yields

\left(\frac{1}{2}k - \frac{2}{3}\right)x + \left(\frac{2}{3}k - 2\right) = \frac{5}{6}x - 1

. Setting the coefficients of

x

equal to each other gives

\frac{1}{2}k - \frac{2}{3} = \frac{5}{6}

. Adding

\frac{2}{3}

to both sides yields

\frac{1}{2}k = \frac{9}{6}

, or

\frac{1}{2}k = \frac{3}{2}

, which simplifies to

k = 3

. Evaluating the constant terms when

k = 3

gives

\frac{2}{3}(3) - 2 = 0

on the left side and

-1

on the right side. Since

0 \neq -1

, the equation has no solution when

k = 3

.

Step-by-Step Solution

1

Distribute the fractional coefficients on the left side of the equation.

\frac{1}{2}kx - 2 - \frac{2}{3}x + \frac{2}{3}k = \frac{5}{6}x - 1

To separate the variable terms from the constants for further algebraic manipulation.

2

Group the terms on the left side into a coefficient for

x

and a single constant term.

\left(\frac{1}{2}k - \frac{2}{3}\right)x + \left(\frac{2}{3}k - 2\right) = \frac{5}{6}x - 1

To represent the equation in the standard linear format

Ax + B = Cx + D

.

3

Set the coefficients of

x

from both sides equal to each other.

\frac{1}{2}k - \frac{2}{3} = \frac{5}{6}

A linear equation has no solution when the variable terms on both sides cancel out, requiring their coefficients to be identical.

4

Solve the linear equation to isolate the constant

k

.

k = 3

Add

\frac{2}{3}

to both sides to get

\frac{1}{2}k = \frac{9}{6}

, which simplifies to

\frac{1}{2}k = \frac{3}{2}

. Multiplying both sides by

2

yields

k = 3

.

5

Verify that the constant terms are not equal when substituting

k = 3

.

The left-side constant is

0

and the right-side constant is

-1

. Since

0 \neq -1

, the condition is satisfied.

To ensure the equation does not simplify to an identity with infinitely many solutions (which occurs when both the variable coefficients and the constant terms are equal).

Key Concept

Determining the conditions under which a linear equation in one variable has no solution.

Alternative Method

To avoid working with fractions, multiply every term in the equation by

6

(the least common multiple of

2, 3, 6

) at the start:

3(kx - 4) - 4(x - k) = 5x - 6

. Expand the parentheses to get

3kx - 12 - 4x + 4k = 5x - 6

, and group the terms:

(3k - 4)x + (4k - 12) = 5x - 6

. For there to be no solution, set the coefficients of

x

equal to each other:

3k - 4 = 5 \implies 3k = 9 \implies k = 3

. Check the constant terms with

k = 3

:

4(3) - 12 = 0

, which is unequal to

-6

. This confirms

k = 3

is the correct answer.

Estimated Time:2m 0s

Question 913Question →

If $\frac{2}{3}(x - 6) - \frac{1}{2}(x - 4) = 4$ , what is the value of $3x - 12$ ?

168

96

36

24

Show answer & explanation

Answer: 96

Answer

96

The correct answer is 96. Multiplying the entire equation by the common denominator 6 clears the fractions to yield

4(x - 6) - 3(x - 4) = 24

. Distributing the factors gives

4x - 24 - 3x + 12 = 24

, which simplifies to

x - 12 = 24

. Adding 12 to both sides determines that

x = 36

. Evaluating the expression

3x - 12

with

x = 36

yields

3(36) - 12 = 96

.

Step-by-Step Solution

1

Multiply the entire equation by 6 (the least common multiple of the denominators 3 and 2) to eliminate the fractions.

4(x - 6) - 3(x - 4) = 24

Clearing the denominators simplifies the algebraic manipulation and reduces the risk of fractional arithmetic errors.

2

Distribute the coefficients to the terms inside the parentheses, taking care to distribute the negative sign for the second term.

4x - 24 - 3x + 12 = 24

Applying the distributive property expands the expression so that like terms can be grouped.

3

Combine like terms on the left side of the equation.

x - 12 = 24

Grouping

4x - 3x

to get

x

and

-24 + 12

to get

-12

simplifies the equation to a single variable and constant.

4

Isolate the variable

x

by adding 12 to both sides of the equation.

x = 36

Adding the opposite of the constant term isolates the variable on one side.

5

Substitute

x = 36

into the requested expression

3x - 12

to find its value.

3(36) - 12 = 96

The question asks for the value of the expression

3x - 12

, not just the value of

x

.

Key Concept

Solving multi-step linear equations in one variable and evaluating algebraic expressions.

Estimated Time:1m 30s

Question 914Question →

Complete the text by filling in the blank with a noun phrase that conforms to the conventions of Standard English (specifically, subject-modifier agreement).

Fill in the blanks below

Using sticky, glistening droplets to trap unsuspecting insects, survives in nutrient-poor bogs by extracting nitrogen directly from its prey.

Show answer & explanation

Answer

the sundew plant (or a similar noun phrase identifying the plant itself as the subject)

The introductory phrase 'Using sticky, glistening droplets to trap unsuspecting insects' is a participial modifier that must describe the grammatical subject that follows it. Since the sundew plant is the entity that uses these droplets, the main clause must begin with a noun phrase representing the plant (such as 'the sundew plant' or 'the plant') to maintain subject-modifier agreement.

Step-by-Step Solution

1

Analyze the introductory modifying phrase.

The phrase 'Using sticky, glistening droplets to trap unsuspecting insects' describes an action.

To identify what entity must serve as the subject of the sentence.

2

Determine the logical subject of the action.

The sundew plant is the entity that uses sticky droplets to trap insects.

To ensure the modifier logically and grammatically aligns with the subject of the main clause.

3

Formulate the subject noun phrase to fill the blank.

The noun phrase 'the sundew plant' (or 'the plant') is placed directly after the modifier.

This establishes correct subject-modifier agreement and avoids a dangling or misplaced modifier.

Key Concept

Subject-Modifier Agreement

Question 915Question →

In the $xy$ -plane, the graph of the linear function $f(x) = px + q$ , where $p$ and $q$ are constants, passes through the point $(2, 7)$ . Line $L$ is parallel to the graph of $f$ and has a $y$ -intercept that is $3$ units below the $y$ -intercept of the graph of $f$ . If the $x$ -intercept of line $L$ is $(-6, 0)$ , what is the value of $p$ ?

Show answer & explanation

Answer: 0.5

Answer

The value of

p

is

0.5

(or the fraction

1/2

).

The correct answer is

0.5

(or

1/2

). The graph of

f(x) = px + q

passes through

(2, 7)

, which means

7 = 2p + q

, or

q = 7 - 2p

. Line

L

is parallel to

f

, so its slope is

p

, and its

y

-intercept is

q - 3

. Thus, the equation of line

L

is

y = px + q - 3

. Since line

L

has an

x

-intercept at

(-6, 0)

, we can substitute

x = -6

and

y = 0

into its equation, yielding

0 = -6p + q - 3

. Substituting

q = 7 - 2p

into this equation gives

0 = -6p + (7 - 2p) - 3

, which simplifies to

-8p + 4 = 0

. Solving for

p

gives

8p = 4

, or

p = 0.5

.

Step-by-Step Solution

1

Express the relationship between

p

and

q

using the given point

(2, 7)

on the graph of

f

.

q = 7 - 2p

The point

(2, 7)

must satisfy the equation

f(x) = px + q

.

2

Formulate the equation of line

L

using the parallel slope and the shifted

y

-intercept.

y = px + q - 3

Parallel lines have equal slopes, and the

y

-intercept of

L

is

3

units below the

y

-intercept of

f

, which is

q

.

3

Substitute the

x

-intercept

(-6, 0)

into the equation of line

L

.

-6p + q - 3 = 0

The

x

-intercept is a point on the line where

y = 0

.

4

Substitute the expression for

q

from Step 1 into the equation from Step 3 and solve for

p

.

p = 0.5

(or

\frac{1}{2}

)

Solving the linear equation

-6p + (7 - 2p) - 3 = 0

simplifies to

-8p + 4 = 0

, giving

p = 0.5

.

Key Concept

Understanding linear functions, their graphs, slopes of parallel lines, and intercepts.

Alternative Method

Instead of solving for

q

first, you can use the point-slope form. Line

L

passes through

(-6, 0)

and has slope

p

, so its equation is

y = p(x + 6)

, or

y = px + 6p

. The

y

-intercept of

L

is

6p

. The

y

-intercept of the graph of

f

is

q

. We are given that the

y

-intercept of

L

is

3

units below the

y

-intercept of

f

, so

6p = q - 3

. Since

f

passes through

(2, 7)

, we have

7 = 2p + q

, which means

q = 7 - 2p

. Substituting this into

6p = q - 3

gives

6p = (7 - 2p) - 3

, or

6p = 4 - 2p

. Adding

2p

to both sides gives

8p = 4

, which results in

p = 0.5

.

Estimated Time:2m 30s

Question 916Question →

Originally cultivated in the Andean highlands for its nutritious seeds, _______

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

quinoa is now grown by farmers worldwide due to its ability to tolerate diverse soil conditions.

farmers worldwide now grow quinoa due to its ability to tolerate diverse soil conditions.

the global popularity of quinoa has led farmers worldwide to cultivate it in diverse soil conditions.

cultivation of quinoa has spread to farmers worldwide who value its ability to tolerate diverse soil conditions.

Show answer & explanation

Answer: quinoa is now grown by farmers worldwide due to its ability to tolerate diverse soil conditions.

Answer

quinoa is now grown by farmers worldwide due to its ability to tolerate diverse soil conditions.

The introductory modifying phrase 'Originally cultivated in the Andean highlands for its nutritious seeds' describes a crop. The option beginning with 'quinoa' correctly places the logical subject immediately after the comma, satisfying the subject-modifier agreement rule.

Step-by-Step Solution

1

Identify the introductory modifying phrase and the noun it logically describes.

The modifying phrase 'Originally cultivated in the Andean highlands for its nutritious seeds' must describe a crop or plant (quinoa), not farmers, popularity, or cultivation.

An introductory modifying phrase must be immediately followed by the noun it logically modifies to avoid a dangling or misplaced modifier.

2

Evaluate the choices to find the one where the logical subject immediately follows the modifier.

The option beginning with 'quinoa' is the only choice that correctly places the logical subject immediately after the comma.

Placing 'quinoa' at the beginning of the main clause ensures that the modifier correctly and logically describes the subject.

Key Concept

Subject-Modifier Agreement

Estimated Time:1m 0s

Question 917Question →

A rental company charges a flat fee of $35$ dollars plus $12$ dollars per hour to rent a bicycle. If a customer was charged a total of $95$ dollars, for how many hours did the customer rent the bicycle?

Show answer & explanation

Answer: 5

Answer

5

To find the number of hours the bicycle was rented, we can set up the equation

12h + 35 = 95

, where

h

is the number of hours. Subtracting

35

from both sides gives

12h = 60

. Dividing both sides by

12

gives

h = 5

. Thus, the customer rented the bicycle for

5

hours.

Step-by-Step Solution

1

Set up a linear equation representing the total cost.

12h + 35 = 95

, where

h

represents the number of hours rented.

The total cost consists of a flat fee of

35

dollars plus an hourly charge of

12

dollars multiplied by the number of hours

h

.

2

Subtract the flat fee from both sides of the equation to isolate the variable term.

12h = 60

Subtracting

35

from both sides simplifies the equation to find the total hourly charge portion of the cost.

3

Divide by the hourly rate to solve for the number of hours

h

.

h = 5

Dividing both sides by

12

isolates

h

to find the number of rental hours.

Key Concept

Linear Equations in One Variable

Question 918Question →

In the $xy$ -plane, the graph of the linear function $f$ has a slope of $-3$ and passes through the point $(4, 18)$ . What is the $y$ -coordinate of the $y$ -intercept of the graph of $f$ ?

Show answer & explanation

Answer: 30

Answer

The correct answer is 30.

The equation of a linear function can be written in slope-intercept form as

f(x) = mx + b

, where

m

is the slope and

b

is the

y

-coordinate of the

y

-intercept. Substituting the given slope

m = -3

yields

f(x) = -3x + b

. Since the point

(4, 18)

lies on the graph of

f

, we substitute

x = 4

and

f(x) = 18

into the equation to get

18 = -3(4) + b

, which simplifies to

18 = -12 + b

. Adding

12

to both sides of the equation yields

b = 30

. Thus, the

y

-coordinate of the

y

-intercept of the graph of

f

is

30

.

Step-by-Step Solution

1

Write the general slope-intercept form of a linear equation.

f(x) = -3x + b

The slope of the line is given as

-3

, so we substitute

m = -3

into

f(x) = mx + b

.

2

Substitute the point

(4, 18)

into the equation.

18 = -3(4) + b

Since the graph of

f

passes through

(4, 18)

, these coordinates must satisfy the function equation.

3

Solve for the

y

-intercept

b

.

b = 30

Simplify to

18 = -12 + b

, and then add

12

to both sides to isolate

b

.

Key Concept

Finding the equation of a line using its slope and a point.

Question 919Question →

Although many people associate lavender primarily with modern aromatherapy, its cultivation dates back over two thousand years. Known for its sweet scent and vibrant purple flowers, _______

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

lavender is cultivated by gardeners around the world.

gardeners around the world cultivate lavender for various uses.

the cultivation of lavender is common among gardeners worldwide.

cultivating lavender is a common practice for gardeners worldwide.

Show answer & explanation

Answer: lavender is cultivated by gardeners around the world.

Answer

lavender is cultivated by gardeners around the world.

The introductory phrase 'Known for its sweet scent and vibrant purple flowers' is a modifier that must logically describe the noun 'lavender'. The correct option is the only choice that places 'lavender' immediately after the comma, ensuring correct subject-modifier agreement.

Step-by-Step Solution

1

Identify the introductory modifying phrase and its logical subject.

The modifying phrase is 'Known for its sweet scent and vibrant purple flowers'. The logical subject being described is the plant 'lavender'.

According to English grammar rules, an introductory modifying phrase must be immediately followed by the noun it modifies.

2

Evaluate the choices to see which one begins with the logical subject.

Only the option beginning with 'lavender' places the correct subject immediately after the modifier.

Other choices place 'gardeners', 'the cultivation of lavender', or 'cultivating lavender' directly after the comma, creating dangling or misplaced modifiers.

Key Concept

Subject-Modifier Agreement

Question 920Question →

The table below shows some values of the linear function $g$ .

$x$	$g(x)$
$2$	$5$
$4$	$11$
$6$	$17$

What is the slope of the graph of $g$ in the $xy$ -plane?

\frac{1}{3}

6

3

\frac{5}{2}

Show answer & explanation

Answer: $3$

Answer

The slope of the graph is

3

.

The slope of a linear function is the constant rate of change, calculated as the change in the function values divided by the change in the input values. Choosing the points

(2, 5)

and

(4, 11)

, the slope is

\frac{11 - 5}{4 - 2} = \frac{6}{2} = 3

. This same slope is obtained using any other pair of points from the table, such as

(4, 11)

and

(6, 17)

, where the slope is

\frac{17 - 11}{6 - 4} = \frac{6}{2} = 3

.

Step-by-Step Solution

1

Select two points from the table to find the change in the input and output values.

Using the points

(2, 5)

and

(4, 11)

, the change in the output

g(x)

is

11 - 5 = 6

, and the change in the input

x

is

4 - 2 = 2

.

The slope is defined as the change in the vertical coordinate divided by the change in the horizontal coordinate.

2

Divide the change in

g(x)

by the change in

x

to find the slope.

The slope is

\frac{6}{2} = 3

.

Dividing the vertical change by the horizontal change yields the constant rate of change of the linear function.

Key Concept

Finding the slope of a linear function from a table of values

Estimated Time:45s

Answer

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