All practice questions

Question 961Question →

In 539 BCE, the Persian king Cyrus the Great conquered Babylon, establishing one of the largest empires in ancient history. A baked-clay artifact discovered in the ruins of Babylon in 1879, _______

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

the king’s conquests and policies of religious tolerance are detailed in a cuneiform declaration on the Cyrus Cylinder.

the cuneiform declaration on the Cyrus Cylinder details the king’s conquests and outlines his policies of religious tolerance.

the Cyrus Cylinder features a cuneiform declaration that details the king’s conquests and outlines his policies of religious tolerance.

scholars have translated the cuneiform declaration on the Cyrus Cylinder, which details the king’s conquests and policies of religious tolerance.

Show answer & explanation

Answer: the Cyrus Cylinder features a cuneiform declaration that details the king’s conquests and outlines his policies of religious tolerance.

Answer

the Cyrus Cylinder features a cuneiform declaration that details the king’s conquests and outlines his policies of religious tolerance.

The introductory phrase 'A baked-clay artifact discovered in the ruins of Babylon in 1879' describes the Cyrus Cylinder. Standard English conventions require that an introductory modifying phrase be followed immediately by the noun it logically modifies. Therefore, the subject of the main clause must be 'the Cyrus Cylinder'. The option stating that 'the Cyrus Cylinder features a cuneiform declaration' correctly places the logical subject immediately after the modifier.

Step-by-Step Solution

1

Identify the introductory modifying phrase and its logical subject.

The introductory phrase is 'A baked-clay artifact discovered in the ruins of Babylon in 1879'. This modifier describes a physical object, which is the Cyrus Cylinder.

Determining the logical subject of the modifier tells us which noun must immediately follow the comma.

2

Evaluate the grammatical subjects at the beginning of each option.

Only the option starting with 'the Cyrus Cylinder' correctly aligns the physical artifact modifier with the physical artifact itself. Options starting with 'the king's conquests', 'the cuneiform declaration', or 'scholars' create illogical modifier relationships.

Standard English conventions require that a modifying phrase be placed immediately adjacent to the noun it modifies to prevent dangling or misplaced modifiers.

Key Concept

Subject-Modifier Agreement

Question 962Question →

In the $xy$ -plane, the graph of a linear function $f$ has a $y$ -intercept of $(0, r)$ and an $x$ -intercept of $(s, 0)$ , where $r$ and $s$ are nonzero constants. A second line, $g$ , is perpendicular to the graph of $f$ and passes through the point $(s, r)$ . Which of the following expressions represents the $x$ -intercept of line $g$ ?

\frac{s^2 - r^2}{r}

\frac{s^2 + r^2}{s}

\frac{s^2 - r^2}{s}

\frac{r^2 - s^2}{r}

Show answer & explanation

Answer: $\frac{s^2 - r^2}{s}$

Answer

The expression

\frac{s^2 - r^2}{s}

The slope of line

f

is found using its intercepts

(0, r)

and

(s, 0)

, which gives a slope of

m_f = -\frac{r}{s}

. Since line

g

is perpendicular to line

f

, its slope is the negative reciprocal,

m_g = \frac{s}{r}

. Using the point-slope form with the point

(s, r)

, the equation for line

g

is

y - r = \frac{s}{r}(x - s)

. Setting

y = 0

to find the

x

-intercept yields

x = \frac{s^2 - r^2}{s}

.

Step-by-Step Solution

1

Determine the slope of the linear function

f

using its intercepts.

The slope of

f

is

m_f = \frac{0 - r}{s - 0} = -\frac{r}{s}

.

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

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

.

2

Find the slope of line

g

, which is perpendicular to the graph of

f

.

The slope of line

g

is

m_g = -\frac{1}{m_f} = -\frac{1}{-r/s} = \frac{s}{r}

.

Perpendicular lines have slopes that are negative reciprocals of each other.

3

Write the equation of line

g

in point-slope form using the point

(s, r)

.

The equation of line

g

is

y - r = \frac{s}{r}(x - s)

.

The point-slope form of a linear equation is

y - y_1 = m(x - x_1)

.

4

Solve for the

x

-intercept of line

g

by setting

y = 0

.

Setting

y = 0

gives

-r = \frac{s}{r}(x - s) \implies -\frac{r^2}{s} = x - s \implies x = s - \frac{r^2}{s} = \frac{s^2 - r^2}{s}

.

The

x

-intercept of a line is the value of

x

when

y = 0

.

Key Concept

Finding the equation of a perpendicular line and its intercepts using algebraic representations.

Question 963Question →

In 1964, radio astronomers Arno Penzias and Robert Wilson ______ a persistent background noise in their horn antenna that they could not eliminate. Although they initially attributed the static to nesting pigeons, subsequent measurements by a Princeton research group confirmed that the signal was actually the ancient thermal radiation left over from the early universe.

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

detect

have detected

detected

are detecting

Show answer & explanation

Answer: detected

Answer

detected

The correct option is the simple past tense verb 'detected.' The passage describes events that took place in a specific past year (1964), and the surrounding verbs ('could not,' 'attributed,' 'confirmed,' and 'was') are all in the simple past tense. Therefore, the simple past tense is required to maintain verb tense consistency.

Step-by-Step Solution

1

Analyze the temporal markers and surrounding verbs in the passage to establish the correct time frame.

The phrase 'In 1964' and verbs like 'could not,' 'attributed,' 'confirmed,' and 'was' indicate that the passage describes past events.

Verb tenses within a passage must remain consistent unless there is a logical reason to shift time frames.

2

Evaluate the choices to find the form of 'detect' that matches this past time frame.

The simple past tense 'detected' matches the established past context, whereas 'detect,' 'have detected,' and 'are detecting' introduce inappropriate present-tense elements.

Using the simple past tense maintains grammatical consistency and correctly positions the action in the specified year (1964).

Key Concept

Verb Tense Consistency

Estimated Time:1m 0s

Question 964Question →

If $3(2n - 5) = 4n + 9$ , what is the value of $n$ ?

Show answer & explanation

Answer: 12

Answer

12

To solve the equation

3(2n - 5) = 4n + 9

, we first distribute the 3 to the terms inside the parentheses to get

6n - 15 = 4n + 9

. Next, we subtract

4n

from both sides to group the variable terms, giving

2n - 15 = 9

. We then add 15 to both sides to isolate the variable term, resulting in

2n = 24

. Finally, dividing both sides by 2 gives the solution

n = 12

.

Step-by-Step Solution

1

Distribute 3 to the terms inside the parentheses.

6n - 15 = 4n + 9

To simplify the left side of the equation and remove parentheses.

2

Subtract

4n

from both sides of the equation.

2n - 15 = 9

To group the variable terms on one side of the equation.

3

Add 15 to both sides of the equation.

2n = 24

To isolate the term with the variable

n

.

4

Divide both sides of the equation by 2.

n = 12

To solve for

n

.

Key Concept

Solving a linear equation in one variable by applying the distributive property and isolating the variable.

Question 965Question →

Complete the sentence by filling in the blank with the name of the scientist described by the introductory modifier.

Fill in the blanks below

Having studied the genetic inheritance of maize for decades, discovered transposable elements in 1948.

Show answer & explanation

Answer

Barbara McClintock

The introductory modifier 'Having studied the genetic inheritance of maize for decades' describes an action performed by a person. Therefore, the subject immediately following the comma must be the person who did the studying, which is Barbara McClintock. Placing her name in the blank ensures proper subject-modifier agreement.

Step-by-Step Solution

1

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

The phrase is 'Having studied the genetic inheritance of maize for decades.'

This modifier describes an action performed by a logical subject.

2

Determine who logically performed the action of studying maize.

The scientist Barbara McClintock is the logical subject who performed this study.

To avoid a dangling modifier, the subject immediately following the comma must be the person who performed the action.

Key Concept

Subject-Modifier Agreement

Question 966Question →

Constructed as the flagship of the Swedish navy, the warship *Vasa* was intended to project military power in the Baltic region. However, had the vessel not sunk on its maiden voyage in 1628, its timber ________ within decades due to wood-boring organisms. Instead, the brackish, cold waters of the Stockholm harbor preserved the ship nearly intact, allowing archaeologists to salvage it in 1961.

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

decayed

would decay

would have decayed

will have decayed

Show answer & explanation

Answer: would have decayed

Answer

would have decayed

The text establishes a counterfactual scenario in the past, beginning with the inverted conditional clause 'had the vessel not sunk.' To complete this past conditional structure, the main clause requires the past conditional verb phrase 'would have decayed' to describe what would have happened to the timber under those counterfactual conditions.

Step-by-Step Solution

1

Analyze the sentence structure to identify the conditional relationship and temporal framework.

The sentence begins with an inverted conditional clause 'had the vessel not sunk,' which is equivalent to 'if the vessel had not sunk,' establishing a past counterfactual scenario.

Understanding the grammar of the conditional clause is necessary to select the correct mood and tense for the main verb.

2

Determine the correct verb form for the main clause of a past counterfactual conditional sentence.

A past counterfactual (third conditional) sentence requires the past conditional form 'would have' followed by the past participle ('would have decayed') in the main clause.

This form accurately expresses a hypothetical past outcome that did not actually occur.

Key Concept

Past Counterfactual Conditionals

Estimated Time:45s

Question 967Question →

In the $xy$ -plane, the graph of the linear function $f$ passes through the points $(-2, c)$ and $(c, 7)$ , where $c$ is a constant. If the slope of the graph of $f$ is $\frac{1}{2}$ , what is the value of $c$ ?

1

4

6

16

Show answer & explanation

Answer: 4

Answer

The value of

c

is 4.

Using the slope formula

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

with the points

(-2, c)

and

(c, 7)

and the slope

m = \frac{1}{2}

gives

\frac{7 - c}{c - (-2)} = \frac{1}{2}

. Simplifying the denominator yields

\frac{7 - c}{c + 2} = \frac{1}{2}

. Cross-multiplying gives

2(7 - c) = c + 2

, which expands to

14 - 2c = c + 2

. Adding

2c

to both sides and subtracting

2

from both sides gives

12 = 3c

. Dividing both sides by

3

gives

c = 4

.

Step-by-Step Solution

1

Set up the slope formula for the line passing through

(-2, c)

and

(c, 7)

.

m = \frac{7 - c}{c - (-2)}

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

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

.

2

Substitute the given slope

m = \frac{1}{2}

into the formula and simplify the denominator.

\frac{7 - c}{c + 2} = \frac{1}{2}

We are given that the slope of the line is

\frac{1}{2}

.

3

Solve the equation for

c

by cross-multiplying and isolating the variable.

2(7 - c) = c + 2 \implies 14 - 2c = c + 2 \implies 12 = 3c \implies c = 4

Cross-multiplying eliminates the fractions, allowing us to solve the linear equation for

c

.

Key Concept

Finding the slope of a line given two points containing an unknown coordinate

Question 968Question →

The passage below contains a nonessential element set off by punctuation. Which punctuation mark correctly completes the passage by setting off the nonessential element?

Fill in the blanks below

For decades, cognitive scientists have debated the validity of linguistic relativity. Proposed in the mid-twentieth century, the Sapir-Whorf hypothesis—the controversial theory that the structure of a language influences its speakers' cognition and world viewcontinues to spark debate among contemporary scholars who investigate how grammatical categories might gently nudge human perception in subtle, non-linguistic ways.

Show answer & explanation

Answer

An em dash (—)

The correct punctuation mark is an em dash (—). In the sentence, the parenthetical appositive phrase 'the controversial theory that the structure of a language influences its speakers' cognition and world view' is a nonessential element. Nonessential elements must be set off from the rest of the sentence using matching punctuation marks on both sides. Since the phrase is opened with an em dash after 'hypothesis', it must be closed with a matching em dash before 'continues'.

Step-by-Step Solution

1

Identify the nonessential element in the sentence.

The phrase 'the controversial theory that the structure of a language influences its speakers' cognition and world view' is an appositive phrase that provides nonessential explanatory information about the Sapir-Whorf hypothesis.

Nonessential elements must be set off from the rest of the sentence by matching punctuation marks.

2

Examine the punctuation mark that introduces the nonessential element.

The nonessential element begins with an em dash (—) immediately following 'hypothesis'.

Punctuation setting off a nonessential element must be symmetrical (e.g., comma-comma, dash-dash, or parenthesis-parenthesis).

3

Determine the punctuation mark needed to close the nonessential element.

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

An em dash is required to maintain grammatical symmetry and avoid punctuation asymmetry.

Key Concept

Punctuation of Nonessential Elements

Estimated Time:1m 0s

Question 969Question →

Consider the system of equations below.

x = 2y

3x - y = 10

If $(x, y)$ is the solution to the system of equations above, what is the value of $x$ ?

Show answer & explanation

Answer: 4

Answer

The value of

x

is

4

.

The system of equations can be solved by substituting

x = 2y

from the first equation into the second equation, which gives

3(2y) - y = 10

. Simplifying this equation yields

6y - y = 10

, or

5y = 10

. Dividing by

5

gives

y = 2

. Substituting

y = 2

back into the first equation yields

x = 2(2) = 4

. Therefore, the value of

x

is

4

.

Step-by-Step Solution

1

Substitute the expression for

x

from the first equation into the second equation.

3(2y) - y = 10

Since the first equation gives

x

in terms of

y

, substituting it into the second equation reduces the system to a single linear equation in one variable.

2

Solve the resulting equation for

y

.

y = 2

Simplifying

3(2y) - y = 10

yields

6y - y = 10

, which simplifies further to

5y = 10

. Dividing both sides by

5

gives

y = 2

.

3

Substitute the value of

y

back into the first equation to solve for

x

.

x = 4

Using

x = 2y

and substituting

y = 2

gives

x = 2(2) = 4

.

Key Concept

Solving systems of linear equations using the substitution method.

Question 970Question →

The following passage is from a text about the discovery of a rare tree species. Read the passage and determine which punctuation mark best completes the sentence.

In 1994, field officer David Noble discovered a small grove of unusual trees in a remote canyon within Australia's Wollemi National Park. These trees—later named Wollemi pines (Wollemia nobilis){{blank_1}} represent one of the greatest botanical discoveries of the twentieth century. Previously known only through fossil records dating back millions of years, the species offers scientists a rare window into prehistoric ecosystems. Because fewer than a hundred mature trees exist in the wild, the exact location of the grove remains a closely guarded secret to protect them from disease and damage.

Which punctuation mark must be inserted into the blank to correctly set off the nonessential element?

Fill in the blanks below

In 1994, field officer David Noble discovered a small grove of unusual trees in a remote canyon within Australia's Wollemi National Park. These trees—later named Wollemi pines (Wollemia nobilis) represent one of the greatest botanical discoveries of the twentieth century. Previously known only through fossil records dating back millions of years, the species offers scientists a rare window into prehistoric ecosystems. Because fewer than a hundred mature trees exist in the wild, the exact location of the grove remains a closely guarded secret to protect them from disease and damage.

Show answer & explanation

Answer

An em dash (—)

The correct punctuation mark is an em dash (—). The phrase 'later named Wollemi pines (Wollemia nobilis)' acts as a nonessential appositive phrase that provides additional information about 'These trees'. Since this nonessential element is opened with an em dash (—), it must be closed with an em dash to maintain grammatical symmetry and avoid mixing different types of punctuation.

Step-by-Step Solution

1

Identify the nonessential element in the target sentence.

The phrase 'later named Wollemi pines (Wollemia nobilis)' is a nonessential element modifying 'These trees'.

A nonessential element provides extra information that is not essential to the core grammatical structure of the sentence.

2

Examine the punctuation that begins the nonessential element.

The nonessential element starts with an em dash (—) after the word 'trees'.

To maintain punctuation symmetry, a nonessential element that begins with an em dash must also end with an em dash.

3

Determine the appropriate punctuation mark to close the nonessential element.

An em dash (—) must be inserted at the blank position to properly close the parenthetical information.

Using any other punctuation mark, such as a comma or parenthesis, would result in punctuation asymmetry.

Key Concept

Punctuation of Nonessential Elements

Question 971Question →

In the equation below,

a

and

b

are positive constants.

\frac{2}{3} \left( \frac{3}{2}x - a \right) - \frac{3}{4} \left( 8 - bx \right) = \frac{5}{2}(x - 3) - \frac{1}{2}

If the equation has infinitely many solutions, what is the value of

a^b

?

Show answer & explanation

Answer: 9

Answer

The correct answer is 9.

The correct answer is 9. Expanding the left side of the equation yields

\left(1 + \frac{3}{4}b\right)x - \left(6 + \frac{2}{3}a\right)

, and simplifying the right side yields

\frac{5}{2}x - 8

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

x

on the left,

1 + \frac{3}{4}b

, must equal the coefficient of

x

on the right,

\frac{5}{2}

, which gives

b = 2

. Similarly, the constant term on the left,

-\left(6 + \frac{2}{3}a\right)

, must equal the constant term on the right,

-8

, which simplifies to

6 + \frac{2}{3}a = 8

and gives

a = 3

. Evaluating

a^b

with these values yields

3^2 = 9

.

Step-by-Step Solution

1

Expand both sides of the equation to collect like terms.

\left( 1 + \frac{3}{4}b \right)x - \left( 6 + \frac{2}{3}a \right) = \frac{5}{2}x - 8

Expanding allows us to compare the coefficient of

x

and the constant term on each side of the equation.

2

Equate the coefficients of

x

on both sides of the equation.

1 + \frac{3}{4}b = \frac{5}{2} \implies b = 2

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

x

must be identical on both sides.

3

Equate the constant terms on both sides of the equation.

-\left( 6 + \frac{2}{3}a \right) = -8 \implies a = 3

For a linear equation to have infinitely many solutions, the constant terms must also be identical on both sides.

4

Calculate the value of

a^b

using the solved values of

a

and

b

.

3^2 = 9

The question asks for the value of the expression

a^b

where

a = 3

and

b = 2

.

Key Concept

A linear equation in one variable of the form

Ax + B = Cx + D

has infinitely many solutions if and only if

A = C

and

B = D

.

Alternative Method

Since the equation must hold for all values of

x

if it has infinitely many solutions, you can substitute convenient values for

x

to solve for

a

and

b

directly. Substituting

x = 0

simplifies the equation to

-\frac{2}{3}a - 6 = -8

, which quickly yields

a = 3

. Then, substituting

x = 2

and

a = 3

simplifies the equation to

-\frac{3}{4}(8 - 2b) = -3

, which yields

b = 2

. Calculating

a^b

gives

3^2 = 9

.

Estimated Time:3m 0s

Question 972Question →

Complete the text by filling in the blank with the option that conforms to the conventions of Standard English. Choose from the following options:
- the faint light source detected in 1994 was eventually recognized
- astronomers eventually recognized the faint light source detected in 1994
- the recognition of the faint light source detected in 1994 eventually occurred
- detecting the faint light source in 1994 eventually led to its recognition

Fill in the blanks below

Although initially dismissed as a minor instrumental glitch by contemporary astronomers, as Gliese 229B, the first confirmed brown dwarf.

Show answer & explanation

Answer

the faint light source detected in 1994 was eventually recognized

The introductory modifying phrase 'Although initially dismissed as a minor instrumental glitch by contemporary astronomers' describes a physical observation or light source. The correct option begins with the noun phrase 'the faint light source,' ensuring that the modifier correctly describes the subject of the main clause.

Step-by-Step Solution

1

Identify the introductory modifying phrase and determine its logical subject.

The modifying phrase is 'Although initially dismissed as a minor instrumental glitch by contemporary astronomers.' The entity being 'dismissed as a minor instrumental glitch' must be the physical light source or astronomical observation itself.

A modifying phrase must logically describe the noun or pronoun that immediately follows it.

2

Evaluate the first word or phrase of each option to find the one that represents the logical subject.

The option beginning with 'the faint light source' correctly places the logical subject immediately after the modifier. The options beginning with 'astronomers,' 'the recognition,' or 'detecting' do not represent the entity dismissed as a glitch.

Placing any other word immediately after the modifier results in a dangling or misplaced modifier.

Key Concept

Subject-Modifier Agreement

Question 973Question →

If $\frac{5}{3}(3x - 6) - \frac{1}{2}(4x + 8) = 2$ , what is the value of $3x - 5$ ?

Show answer & explanation

Answer: 11

Answer

11

Distributing the fraction

\frac{5}{3}

to

(3x - 6)

yields

5x - 10

. Distributing the term

-\frac{1}{2}

to

(4x + 8)

yields

-2x - 4

. Combining these results gives

(5x - 2x) + (-10 - 4) = 2

, which simplifies to

3x - 14 = 2

. Adding 14 to both sides yields

3x = 16

. Finally, subtracting 5 from both sides of this equation gives

3x - 5 = 11

.

Step-by-Step Solution

1

Distribute

\frac{5}{3}

and

-\frac{1}{2}

to the terms within their respective parentheses.

5x - 10 - 2x - 4 = 2

This removes the parentheses and allows us to group terms.

2

Combine the variable terms and the constant terms on the left side of the equation.

3x - 14 = 2

Simplifying the expression on the left side makes it easier to solve.

3

Add 14 to both sides of the equation to isolate the term

3x

.

3x = 16

Since the question asks for the value of

3x - 5

, isolating

3x

allows direct evaluation.

4

Subtract 5 from both sides of the equation

3x = 16

.

3x - 5 = 11

This directly yields the required value without needing to compute the fractional value of

x

first.

Key Concept

Solving linear equations in one variable by distribution, combining like terms, and evaluating expressions.

Estimated Time:1m 30s

Question 974Question →

Read the passage and determine the correct form of the verb 'share' to complete the sentence. To ensure the passage conforms to the conventions of Standard English, what form of the verb 'share' should be written in the blank?

Fill in the blanks below

In 1944, German chemist Otto Hahn was solely awarded the Nobel Prize in Chemistry for the discovery of nuclear fission. Many modern historians of science argue that if the Swedish Academy had fully recognized Austrian physicist Lise Meitner’s critical theoretical contributions to the research, she the prestigious prize alongside him.

Show answer & explanation

Answer

would have shared

The correct verb form 'would have shared' is required to complete the conditional construction. The passage sets up a past counterfactual scenario using the past perfect tense in the conditional clause ('if the Swedish Academy had fully recognized...'). To maintain grammatical mood and tense consistency, the main clause must use the conditional perfect form ('would have shared') to indicate an action that would have occurred in the past if the condition had been met.

Step-by-Step Solution

1

Identify the conditional structure of the sentence.

The sentence contains an 'if' clause stating a hypothetical condition: 'if the Swedish Academy had fully recognized...'. This condition is expressed in the past perfect tense ('had recognized'), indicating a past counterfactual scenario.

Determining the tense of the condition is essential for selecting the correct verb form in the main clause.

2

Apply the appropriate conditional mood construction.

A past counterfactual condition ('if' followed by past perfect) requires the main clause to use the conditional perfect tense ('would have' followed by the past participle) to describe the hypothetical past outcome.

This maintains grammatical consistency and correct mood alignment for a past hypothesis.

3

Conjugate the verb 'share' to match this conditional perfect structure.

The base verb 'share' becomes 'would have shared'.

This completes the main clause of the conditional sentence correctly.

Key Concept

Past counterfactual conditional structures (third conditional) require the past perfect tense in the conditional clause and the conditional perfect tense in the main clause.

Question 975Question →

In the $xy$ -plane, the graph of a linear function $f$ has a positive $y$ -intercept and a positive $x$ -intercept. The area of the triangular region in the first quadrant bounded by the graph of $f$ and the coordinate axes is $36$ . If the graph of $f$ passes through the point $(4, 4)$ and has a slope less than $-1$ , what is the $y$ -coordinate of the $y$ -intercept of the graph of $f$ ?

Show answer & explanation

Answer: 12

Answer

The

y

-coordinate of the

y

-intercept of the graph of

f

is

12

.

The correct answer is

12

because the system of equations derived from the area constraint (

qr = 72

) and the point constraint (

q+r=18

) yields two possible values for the

y

-intercept:

6

or

12

. The condition that the slope must be less than

-1

means that the line must be steeper than a slope of

-1

, which requires the

y

-intercept to be larger than the

x

-intercept (

r > q

). Thus, the

y

-intercept is

12

.

Step-by-Step Solution

1

Express the equation of the line using intercept form.

\frac{x}{q} + \frac{y}{r} = 1

, where

q > 0

is the

x

-intercept and

r > 0

is the

y

-intercept.

Since the line intersects the positive axes, this form directly relates the intercepts to the coordinates of points on the line.

2

Use the area of the triangle to find a relationship between

q

and

r

.

qr = 72

The area of the right triangle formed by the axes and the intercepts is given by

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2}qr = 36

.

3

Substitute the point

(4, 4)

into the equation of the line and simplify using the area relationship.

q + r = 18

Plugging in

x = 4

and

y = 4

yields

\frac{4}{q} + \frac{4}{r} = 1 \implies 4(q + r) = qr

. Substituting

qr = 72

gives

4(q + r) = 72 \implies q + r = 18

.

4

Solve the system of equations

q + r = 18

and

qr = 72

.

(q, r) = (6, 12)

or

(q, r) = (12, 6)

Substituting

r = 18 - q

into

qr = 72

yields

q(18 - q) = 72 \implies q^2 - 18q + 72 = 0 \implies (q - 6)(q - 12) = 0

.

5

Apply the slope condition to determine the unique value of

r

.

r = 12

The slope of the line is

m = -\frac{r}{q}

. If

q = 12

and

r = 6

, then

m = -0.5

, which is not less than

-1

. If

q = 6

and

r = 12

, then

m = -2

, which is less than

-1

. Thus,

r = 12

is the correct

y

-intercept.

Key Concept

Using intercepts and area to determine the equation of a linear function under constraints.

Question 976Question →

In the equation $3(2x - a) - 4(x + 1) = 2x + 5$ , $a$ is a constant. If the equation has infinitely many solutions, what is the value of $a$ ?

-\frac{1}{3}

-3

3

-9

Show answer & explanation

Answer: $-3$

Answer

The value of

a

must be

-3

.

The value of

-3

is correct. Expanding and simplifying the left side of the equation yields

2x - 3a - 4 = 2x + 5

. For a linear equation to have infinitely many solutions, the constant terms on both sides must be identical. Setting

-3a - 4 = 5

and solving for

a

gives

-3a = 9

, which simplifies to

a = -3

.

Step-by-Step Solution

1

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

6x - 3a - 4x - 4 = 2x + 5

Distributing eliminates the parentheses, allowing like terms to be combined.

2

Combine the variable terms on the left side of the equation.

2x - 3a - 4 = 2x + 5

Grouping the

x

terms (

6x - 4x = 2x

) simplifies the structure of the equation.

3

Set up the condition for the equation to have infinitely many solutions.

-3a - 4 = 5

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

x

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

x

coefficients are already equal (

2 = 2

), so we equate the constants.

4

Isolate the constant

a

by adding

4

to both sides and then dividing by

-3

.

a = -3

Adding

4

yields

-3a = 9

, and dividing by

-3

isolates the variable to find the solution.

Key Concept

Linear Equations in One Variable

Alternative Method

Instead of fully simplifying the equation, substitute

x = 0

directly into the original equation, since a statement with infinitely many solutions must hold true for all values of

x

. Substituting

x = 0

gives

3(-a) - 4(1) = 5

, which simplifies directly to

-3a - 4 = 5

, yielding

a = -3

.

Estimated Time:1m 15s

Question 977Question →

The table below shows some values of $x$ and the corresponding values of $f(x)$ for a linear function $f$ .

$x$	$f(x)$
3	7
6	12
9	17

What is the value of $f(15)$ ?

Show answer & explanation

Answer: 27

Answer

The value of the linear function evaluated at 15 is 27.

To find

f(15)

for the linear function, we determine its constant rate of change. Using points

(3, 7)

and

(6, 12)

, the change in

f(x)

is

12 - 7 = 5

for a change in

x

of

6 - 3 = 3

. This gives a slope of

\frac{5}{3}

. Using point-slope form with

(3, 7)

gives the equation

f(x) - 7 = \frac{5}{3}(x - 3)

, which simplifies to

f(x) = \frac{5}{3}x + 2

. Substituting

x = 15

yields

f(15) = \frac{5}{3}(15) + 2 = 25 + 2 = 27

. Alternatively, since

x

increases by 6 from 9 to 15, which is twice the step size of 3, the value of

f(x)

must increase by

2 \times 5 = 10

from

17

, resulting in

17 + 10 = 27

.

Step-by-Step Solution

1

Calculate the slope of the linear function

slope

m = \frac{5}{3}

Linear functions have a constant rate of change, which is the slope.

2

Determine the equation of the function

f(x) = \frac{5}{3}x + 2

Using the slope and one point from the table helps define the function for all inputs.

3

Evaluate the function at

x = 15

f(15) = 27

Substitute the given input value into the function equation to find the corresponding output value.

Key Concept

Linear Functions and Graphs

Question 978Question →

The following passage is missing a punctuation mark. Read the passage and write the single punctuation mark that correctly completes the sentence by setting off the nonessential element.

Fill in the blanks below

In 1895, Harvard physics professor Wallace Clement Sabine was tasked with improving the abysmal acoustics of the newly constructed Fogg Art Museum lecture hall. Sabine realized that the duration of sound persistence a metric he later defined as reverberation time, was directly proportional to the room's volume and inversely proportional to its absorptive surfaces.

Show answer & explanation

Answer

A comma (,)

The correct answer is a comma. The nonessential appositive phrase 'a metric he later defined as reverberation time' is closed with a comma after 'time'. To maintain punctuation symmetry, the phrase must also be opened with a comma.

Step-by-Step Solution

1

Identify the nonessential appositive phrase in the target sentence.

The phrase 'a metric he later defined as reverberation time' is a nonessential appositive that renames and describes the noun phrase 'duration of sound persistence'.

Recognizing the boundaries of the nonessential element is necessary to determine where punctuation is required.

2

Locate the punctuation mark used at the closing boundary of the nonessential phrase.

A comma is used after the word 'time' to close the nonessential phrase.

Nonessential elements must be set off with matching punctuation on both sides to maintain structural symmetry.

3

Select the punctuation mark that matches the closing punctuation.

A comma must be placed in the blank to open the nonessential phrase.

Using any other punctuation mark, such as a dash or parenthesis, would result in mismatched punctuation (asymmetry), which is grammatically incorrect.

Key Concept

Punctuation of Nonessential Elements

Estimated Time:1m 30s

Question 979Question →

Although the nineteenth-century poet Emily Dickinson published fewer than a dozen poems during her lifetime, her posthumously recovered manuscript books ______ contain nearly 1,800 closely guarded works that challenge the period’s conventional poetic forms. Today, literary scholars analyze these handmade booklets to understand Dickinson's intentional thematic ordering and punctuation choices.

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

—which she referred to as fascicles—

, which she referred to as fascicles—

—which she referred to as fascicles,

which she referred to as fascicles—

Show answer & explanation

Answer: —which she referred to as fascicles—

Answer

—which she referred to as fascicles—

The sentence contains a nonessential relative clause ('which she referred to as fascicles') that provides extra information about Dickinson's manuscript books. To conform to Standard English conventions, nonessential elements must be set off from the rest of the sentence by matching punctuation marks: either two commas, two parentheses, or two em dashes. The option that begins and ends the clause with em dashes correctly isolates the nonessential clause. The options that pair a comma with an em dash, pair an em dash with a comma, or omit the opening punctuation mark are incorrect because they create asymmetrical punctuation.

Step-by-Step Solution

1

Identify the nonessential element in the sentence.

The clause 'which she referred to as fascicles' is nonessential because it provides supplementary information about the 'manuscript books' without restricting the identity of the noun it modifies.

Nonessential elements must be identified so that they can be set off from the main clause with appropriate punctuation.

2

Determine the required punctuation pattern to set off the nonessential clause.

The punctuation marks setting off a nonessential element must be symmetrical—either a pair of commas, a pair of em dashes, or a pair of parentheses.

Standard English conventions require that the opening and closing punctuation marks for parenthetical or nonessential information match.

3

Evaluate the choices to find the one that maintains symmetrical punctuation.

Only the option using an em dash at both the beginning and the end of the clause provides symmetrical punctuation. The other options combine different punctuation marks or omit the opening mark, creating asymmetrical constructions.

Ensuring symmetry in punctuation is necessary to follow standard grammatical conventions.

Key Concept

Symmetrical punctuation of nonessential elements using matching punctuation marks.

Estimated Time:1m 0s

Question 980Question →

The variables $x$ and $y$ satisfy the system of equations below.

y = 10 - 2x

3x - y = 5

What is the value of $x$ ?

15

4

3

1

Show answer & explanation

Answer: 3

Answer

3

Substituting the expression for

y

from the first equation,

10 - 2x

, into the second equation yields

3x - (10 - 2x) = 5

. Distributing the negative sign results in

3x - 10 + 2x = 5

. Combining the like terms of

x

gives

5x - 10 = 5

. Adding 10 to both sides of the equation results in

5x = 15

. Finally, dividing both sides by 5 gives

x = 3

.

Step-by-Step Solution

1

Substitute the expression for

y

from the first equation into the second equation.

3x - (10 - 2x) = 5

To eliminate the variable

y

and obtain a single linear equation in terms of

x

.

2

Distribute the negative sign through the parentheses and combine like terms.

3x - 10 + 2x = 5

, which simplifies to

5x - 10 = 5

To simplify the linear equation before isolating the variable.

3

Isolate the variable

x

by adding 10 to both sides and then dividing by 5.

5x = 15

, which gives

x = 3

To determine the value of

x

.

Key Concept

Solving systems of linear equations using substitution

Estimated Time:45s

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

Alternative Method

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