All practice questions

In the $xy$ -plane, a line passes through the points $(2, 9)$ and $(5, 21)$ . What is the slope of this line?

Show answer & explanation

Answer: 4

Answer

The slope of the line is 4.

The slope

m

of a line passing through the points

(x_1, y_1)

and

(x_2, y_2)

is determined by the formula

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

. Substituting the given points

(2, 9)

and

(5, 21)

into the formula gives

m = \frac{21 - 9}{5 - 2} = \frac{12}{3} = 4

.

Step-by-Step Solution

1

Identify the coordinates of the two points on the line.

(x_1, y_1) = (2, 9)

and

(x_2, y_2) = (5, 21)

These points are used in the slope formula.

2

Apply the slope formula

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

to set up the calculation.

m = \frac{21 - 9}{5 - 2}

The slope is defined as the change in

y

divided by the change in

x

.

3

Simplify the fraction to calculate the final slope.

m = \frac{12}{3} = 4

Dividing the vertical change by the horizontal change yields the slope.

Key Concept

Calculating the slope of a line given two points in the coordinate plane.

Question 1222Question →

What form of the verb 'expand' correctly completes the sentence to conform to the conventions of Standard English?

Fill in the blanks below

In a 2000 study, researchers placed the single-celled organism *Physarum polycephalum* in a maze containing two food sources. Before they introduced the food, the organism throughout the entire maze in search of nutrients. However, once the food was positioned, the slime mold retracted its branches from the dead ends to form a single, efficient path connecting the two sources.

Show answer & explanation

Answer

had expanded

The correct answer is the past perfect form 'had expanded' because the context indicates that the organism's expansion occurred prior to the introduction of the food, which is already described in the past tense ('introduced').

Step-by-Step Solution

1

Identify the timeline established by the other verbs in the passage.

The narrative is set in the past, as shown by the verbs 'placed', 'introduced', 'was positioned', and 'retracted'.

Understanding the primary time frame is essential for choosing the correct tense and aspect.

2

Determine the temporal relationship between the action in the blank and the other actions.

The clause 'Before they introduced the food' establishes that the action of expanding happened prior to another past action ('introduced').

An action that is completed before another point in the past requires the past perfect tense.

3

Conjugate the verb 'expand' into the past perfect tense.

The past perfect form is 'had expanded'.

Past perfect is formed by combining the auxiliary verb 'had' with the past participle of the main verb ('expanded').

Key Concept

Verb Tense and Aspect (Past Perfect)

Question 1223Question →

In their comprehensive study of deep-sea *Bathymodiolus* mussels, marine biologists observed that these resilient organisms survive in extreme hydrothermal vent environments by hosting symbiotic bacteria in their gills, filtering organic matter from the surrounding water, and ________ metabolic waste through specialized physiological pathways. This allows the mussels to thrive where few other species can.

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

excreting

to excrete

they excrete

the excretion of

Show answer & explanation

Answer: excreting

Answer

excreting

The sentence outlines three parallel activities through which mussels survive, using a list connected by the coordinating conjunction 'and'. The first two items are gerund phrases ('hosting...' and 'filtering...'). To maintain parallel structure, the final item must also begin with a gerund. The option 'excreting' correctly provides this parallel form.

Step-by-Step Solution

1

Identify the grammatical structure of the sentence and locate the list of items connected by the conjunction 'and'.

The sentence lists three parallel methods by which the mussels survive: 'hosting...', 'filtering...', and the final action starting at the blank.

This establishes that the items in the list must share the same grammatical form.

2

Analyze the grammatical form of the existing list items.

The first item starts with the gerund 'hosting', and the second item starts with the gerund 'filtering'.

Determining the parts of speech of the established items tells us what form the final item must take to be parallel.

3

Select the choice that starts with a gerund (verb ending in -ing) to complete the parallel series.

The gerund 'excreting' matches the forms 'hosting' and 'filtering', completing the series correctly.

Conforming to parallel structure is required by the conventions of Standard English.

Key Concept

Parallel Structure

Estimated Time:1m 0s

Question 1224Question →

In the $xy$ -plane, the solution set to the system of inequalities below is a bounded region.

\begin{aligned} y &\leq -2x + 21 \\ y &\leq -\frac{1}{2}x + 9 \\ y &\geq -x + 5 \\ x &\geq 2 \\ y &\geq 1 \end{aligned}

What is the maximum possible value of $3x + 2y$ for a point $(x, y)$ in this region?

22

31

32

34

Show answer & explanation

Answer: 34

Answer

34

The correct value is 34. The solution set to the system of inequalities is a bounded pentagonal region. Evaluating the expression

3x + 2y

at each vertex of this region yields:

12

at

(2, 3)

,

22

at

(2, 8)

,

34

at

(8, 5)

,

32

at

(10, 1)

, and

14

at

(4, 1)

. Comparing these values shows that the maximum value is 34.

Step-by-Step Solution

1

Understand the behavior of the objective function over the region

The maximum value of a linear expression

3x + 2y

over a closed, bounded polygonal region must occur at one of the vertices (corner points) of the region.

This is a fundamental theorem of linear programming, which simplifies the search for the maximum value to only the boundary intersections of the system.

2

Determine the vertices of the feasible region

The boundary lines are

y = -2x + 21

,

y = -\frac{1}{2}x + 9

,

y = -x + 5

,

x = 2

, and

y = 1

. Finding their intersections that satisfy all inequalities yields five vertices:

(2, 3)

,

(2, 8)

,

(8, 5)

,

(10, 1)

, and

(4, 1)

.

Solving the pairwise equations of lines that bound the shaded region determines the exact coordinates of all corner points.

3

Evaluate the expression

3x + 2y

at each of the five vertices

At

(2, 3)

:

3(2) + 2(3) = 12

At

(2, 8)

:

3(2) + 2(8) = 22

At

(8, 5)

:

3(8) + 2(5) = 34

At

(10, 1)

:

3(10) + 2(1) = 32

At

(4, 1)

:

3(4) + 2(1) = 14

Calculating the value at each candidate vertex allows us to compare and find the absolute maximum.

4

Compare the evaluated values

The maximum value is

34

, occurring at the vertex

(8, 5)

.

Comparing all calculated values shows that 34 is the largest possible value.

Key Concept

Linear Programming and Bounded Systems of Inequalities

Estimated Time:3m 0s

Question 1225Question →

If $3(2y - 5) - 4(y - 2) = 11$ , what is the value of $3y + 2$ ?

8

9

29

53

Show answer & explanation

Answer: 29

Answer

29

The correct value is obtained by first distributing the terms, simplifying the equation to find the value of

y

, and then evaluating the expression

3y + 2

. Specifically, distributing yields

6y - 15 - 4y + 8 = 11

, which simplifies to

2y - 7 = 11

. Adding

7

to both sides gives

2y = 18

, so

y = 9

. Substituting

y = 9

into

3y + 2

gives

3(9) + 2 = 29

.

Step-by-Step Solution

1

Distribute the coefficients outside the parentheses.

6y - 15 - 4y + 8 = 11

To simplify the linear equation, we must first remove the parentheses by multiplying the terms inside by the factors outside.

2

Combine like terms on the left side of the equation.

2y - 7 = 11

Combining the variable terms (

6y - 4y = 2y

) and constant terms (

-15 + 8 = -7

) simplifies the equation.

3

Isolate the variable term by adding 7 to both sides, then divide by 2.

y = 9

Adding 7 gives

2y = 18

, and dividing by 2 isolates

y

.

4

Substitute the value of

y

into the expression

3y + 2

.

3(9) + 2 = 29

The question asks for the value of the expression

3y + 2

, not just the variable

y

.

Key Concept

Linear Equations in One Variable

Question 1226Question →

The linear function $f$ is defined such that its graph in the $xy$ -plane is perpendicular to the line $3x - 4y = 20$ . The graph of another linear function, $g$ , is the result of shifting the graph of $f$ right by $5$ units and down by $4$ units. If $f(0) = 8$ and the graph of $g$ intersects the $x$ -axis at the point $(k, 0)$ , what is the value of $k$ ?

Show answer & explanation

Answer: 8

Answer

8

The slope of the line

3x - 4y = 20

is

\frac{3}{4}

. The slope of a line perpendicular to it is the negative reciprocal,

-\frac{4}{3}

. With a

y

-intercept of

(0, 8)

, the function is

f(x) = -\frac{4}{3}x + 8

. Translating this function

5

units right and

4

units down yields

g(x) = f(x - 5) - 4 = -\frac{4}{3}(x - 5) + 4 = -\frac{4}{3}x + \frac{32}{3}

. Setting

g(k) = 0

to find the

x

-intercept gives

-\frac{4}{3}k + \frac{32}{3} = 0

, which solves to

k = 8

.

Step-by-Step Solution

1

Find the slope of the line

3x - 4y = 20

.

The slope is

\frac{3}{4}

.

Converting the line equation to slope-intercept form

y = mx + b

gives

y = \frac{3}{4}x - 5

, where the coefficient of

x

represents the slope.

2

Find the slope of

f

.

The slope of

f

is

-\frac{4}{3}

.

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

3

Write the function

f(x)

.

f(x) = -\frac{4}{3}x + 8

.

Since the

y

-intercept is given by

f(0) = 8

and the slope is

-\frac{4}{3}

, the slope-intercept form is

f(x) = -\frac{4}{3}x + 8

.

4

Determine the function

g(x)

by translating

f(x)

5

units right and

4

units down.

g(x) = -\frac{4}{3}x + \frac{32}{3}

.

A translation of

f(x)

right by

5

units and down by

4

units corresponds to

g(x) = f(x - 5) - 4

. Substituting

x-5

into

f(x)

gives

g(x) = -\frac{4}{3}(x - 5) + 8 - 4 = -\frac{4}{3}x + \frac{32}{3}

.

5

Set

g(k) = 0

and solve for

k

.

k = 8

.

The graph of

g

intersects the

x

-axis at

(k, 0)

, which means

g(k) = 0

. Solving

-\frac{4}{3}k + \frac{32}{3} = 0

yields

k = 8

.

Key Concept

Writing linear functions using perpendicular slopes and performing vertical and horizontal transformations.

Question 1227Question →

In the

xy

-plane, a system of inequalities is defined as follows:

\begin{aligned} y &\geq 2x - 4 \\ y &\leq -x + 8 \\ y &\geq \frac{1}{2}x \end{aligned}

How many points

(x, y)

with integer coordinates satisfy this system of inequalities?

Show answer & explanation

Answer: 5

Answer

The total number of points with integer coordinates that satisfy the system is 5.

The system of inequalities defines a closed triangular region in the coordinate plane. Finding the vertices of this triangle gives the horizontal boundaries for

x

, which are

\frac{8}{3} \approx 2.67

and

\frac{16}{3} \approx 5.33

. The only integers in this range are

x = 3

,

x = 4

, and

x = 5

. Testing each of these integers in the inequalities shows that when

x = 3

,

y

can be

2, 3, 4,

or

5

(4 points); when

x = 4

,

y

can only be

4

(1 point); and when

x = 5

, there are no valid solutions. Adding these together gives a total of 5 points.

Step-by-Step Solution

1

Find the intersection points of the three boundary lines to define the vertices of the solution region.

The intersection of

y = 2x - 4

and

y = \frac{1}{2}x

is at

x = \frac{8}{3} \approx 2.67, y = \frac{4}{3} \approx 1.33

.
The intersection of

y = -x + 8

and

y = \frac{1}{2}x

is at

x = \frac{16}{3} \approx 5.33, y = \frac{8}{3} \approx 2.67

.
The intersection of

y = 2x - 4

and

y = -x + 8

is at

x = 4, y = 4

.

Finding the vertices determines the exact boundaries of the solution region in the coordinate plane.

2

Identify the integer

x

-coordinates that lie within the horizontal span of the region.

The

x

-coordinates of the region range from

\frac{8}{3} \approx 2.67

to

\frac{16}{3} \approx 5.33

. The integers in this interval are

x = 3

,

x = 4

, and

x = 5

.

Since both coordinates must be integers, we only need to test integer values of

x

that fall within the boundaries of the region.

3

Find the integer

y

-values for each candidate

x

-value.

For

x = 3

: The inequalities require

y \geq 2(3) - 4 = 2

,

y \leq -3 + 8 = 5

, and

y \geq \frac{1}{2}(3) = 1.5

. Thus,

2 \leq y \leq 5

. The integer solutions are

y = 2, 3, 4, 5

(4 points).
For

x = 4

: The inequalities require

y \geq 2(4) - 4 = 4

,

y \leq -4 + 8 = 4

, and

y \geq \frac{1}{2}(4) = 2

. Thus,

4 \leq y \leq 4

, which means

y = 4

(1 point).
For

x = 5

: The inequalities require

y \geq 2(5) - 4 = 6

and

y \leq -5 + 8 = 3

. No real number

y

can satisfy both

y \geq 6

and

y \leq 3

(0 points).

Evaluating the system at each candidate

x

-value reveals the set of matching integer

y

-values.

4

Sum the number of points found for each integer

x

-value.

4 points (when

x = 3

) + 1 point (when

x = 4

) + 0 points (when

x = 5

) = 5 points.

This yields the total number of integer coordinate pairs

(x, y)

that satisfy the system of inequalities.

Key Concept

Analyzing a bounded region defined by a system of linear inequalities to find discrete integer solutions (lattice points).

Alternative Method

Graph the three boundary lines on a grid:

y = 2x - 4

(solid line, shaded above),

y = -x + 8

(solid line, shaded below), and

y = \frac{1}{2}x

(solid line, shaded above). Identify the triangular intersection region on the grid and count the grid intersections (lattice points) that lie within or on the boundaries of this shaded triangle.

Estimated Time:2m 30s

Question 1228Question →

In the high-altitude regions of the White Mountains of California, dendrochronologists study ancient bristlecone pines to reconstruct historical precipitation patterns spanning several millennia. Because these long-lived trees grow extremely slowly in arid environments, they are highly sensitive to climate changes ______ their narrow annual rings provide a precise, year-by-year record of local drought history.

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

changes; consequently,

changes,

changes

changes, but

Show answer & explanation

Answer: changes; consequently,

Answer

The correct choice is the punctuation and transition that uses a semicolon followed by the conjunctive adverb 'consequently' and a comma to link the two independent clauses.

The text contains two independent clauses: the first ends with the word 'changes' and the second begins with the word 'their'. Standard English grammar requires that two independent clauses be separated by a period, a semicolon, or a coordinating conjunction preceded by a comma. The punctuation in the correct option ('changes; consequently,') appropriately uses a semicolon and a transition word showing cause-and-effect to connect the two clauses.

Step-by-Step Solution

1

Identify the structure of the clauses surrounding the blank.

The clause before the blank ('Because these long-lived trees grow extremely slowly in arid environments, they are highly sensitive to climate changes') and the clause after the blank ('their narrow annual rings provide a precise, year-by-year record of local drought history') are both independent clauses.

Determining the clause types helps establish the punctuation rules that must be applied to avoid structural errors.

2

Evaluate the grammatical rules for joining independent clauses.

Two independent clauses must be joined using a period, a semicolon, or a comma combined with a coordinating conjunction (such as 'and').

This step eliminates grammatically incorrect options that lead to run-on sentences or comma splices.

3

Analyze the logical relationship between the two clauses to select the correct transition.

The first clause explains the trees' sensitivity to climate, and the second clause states the result (their rings providing a record). This is a cause-and-effect relationship, so 'consequently' is correct, whereas the contrast coordinator 'but' is logically incorrect.

This ensures the chosen option satisfies both grammatical and semantic constraints.

Key Concept

Clause Boundaries and Linking

Estimated Time:1m 30s

Question 1229Question →

For decades, paleoanthropologists debated whether Neanderthals possessed the cognitive capacity for symbolic thought, often portraying them as lacking artistic expression. Recent discoveries of decorated raptor talons and pigment-stained shells at multiple archaeological sites across Europe have challenged these older assumptions ______ these symbolic artifacts strongly suggest that Neanderthals engaged in complex cultural practices long before the arrival of modern humans.

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

, indeed,

indeed

; indeed,

; although

Show answer & explanation

Answer: ; indeed,

Answer

The choice with '; indeed,' is correct because it uses a semicolon to properly separate two independent clauses and the conjunctive adverb 'indeed' to emphasize the connection between them.

The correct answer contains a semicolon followed by the transition word 'indeed' and a comma. This structure is grammatically correct because a semicolon is used to connect two independent clauses, and 'indeed' logically serves to emphasize the evidence presented in the second clause.

Step-by-Step Solution

1

Analyze the sentence structure before and after the blank to identify the types of clauses.

The clause before the blank ('Recent discoveries... assumptions') is independent. The clause after the blank ('these symbolic artifacts... modern humans') is also independent.

Knowing that we have two independent clauses dictates that we must use appropriate punctuation or a coordinating conjunction to separate them.

2

Evaluate punctuation and transition options for linking two independent clauses.

A semicolon is a grammatically correct way to link two independent clauses. Combining a semicolon with the conjunctive adverb 'indeed' and a comma correctly establishes the logical flow.

Standard English conventions require a period, semicolon, colon, or comma plus a coordinating conjunction to link independent clauses.

3

Eliminate grammatically incorrect alternatives.

The option with ', indeed,' creates a comma splice. The option with 'indeed' creates a run-on sentence. The option with '; although' is incorrect because it uses a semicolon before a dependent clause.

A comma alone cannot link two independent clauses, nor can they be joined without punctuation. A semicolon cannot precede a dependent clause started by a subordinating conjunction.

Key Concept

Clause Boundaries and Linking

Estimated Time:1m 0s

Question 1230Question →

In the system of equations below, $a$ and $b$ are constants.

\begin{aligned} ax + by &= 24 \\ 2x - 5y &= -7 \end{aligned}

If the system has the same unique solution $(x, y)$ for all values of $a$ and $b$ that satisfy the equation $4a + 3b = 24$ , what is the value of $x + y$ ?

Show answer & explanation

Answer: 7

Answer

The value of

x + y

is

7

.

The correct answer is

7

. The solution to the system must satisfy

ax + by = 24

for any constants

a

and

b

that satisfy the constraint

4a + 3b = 24

. By matching the coefficients of

a

and

b

in both equations, we find

x = 4

and

y = 3

. To verify, we substitute these coordinates into the second equation:

2(4) - 5(3) = 8 - 15 = -7

, which is correct. The sum of the coordinates is

x + y = 4 + 3 = 7

.

Step-by-Step Solution

1

Analyze the constraint on the constants

a

and

b

given by

4a + 3b = 24

and relate it to the first equation

ax + by = 24

.

Since the first equation

ax + by = 24

must hold for all pairs of

(a, b)

satisfying

4a + 3b = 24

, the coefficients must correspond directly, meaning

x = 4

and

y = 3

.

This shows that

(4, 3)

is a point that lies on the line

ax + by = 24

regardless of the specific values of

a

and

b

, as long as they satisfy the given constraint.

2

Prove the uniqueness of the point

(4, 3)

by testing specific values for

(a, b)

that satisfy the constraint

4a + 3b = 24

.

If

b = 0

, then

4a = 24 \implies a = 6

. The equation becomes

6x = 24 \implies x = 4

. If

a = 0

, then

3b = 24 \implies b = 8

. The equation becomes

8y = 24 \implies y = 3

. This confirms

(x, y) = (4, 3)

is the unique point.

Choosing convenient test values for

a

and

b

helps isolate the variables

x

and

y

to verify the coordinates of the solution.

3

Substitute the point

(4, 3)

into the second equation of the system,

2x - 5y = -7

, to verify that it is consistent.

2(4) - 5(3) = 8 - 15 = -7

. Since this statement is true,

(4, 3)

is the consistent unique solution to the system of equations.

A solution to a system of equations must satisfy all equations in the system.

4

Find the requested value of

x + y

using the coordinates of the solution.

x + y = 4 + 3 = 7

.

The question asks for the sum of the coordinates of the solution.

Key Concept

Systems of linear equations with parameter constraints

Question 1231Question →

In the $xy$ -plane, the graph of the linear equation $y = mx + b$ , where $m$ and $b$ are constants, passes through the point $(2, 7)$ . If this graph is translated $3$ units to the right and $4$ units down, the resulting graph passes through the point $(4, 2)$ . What is the value of $b$ ?

Show answer & explanation

Answer: 5

Answer

5

The correct answer is

5

. Since the original graph passes through

(2, 7)

and undergoes a translation of

3

units to the right and

4

units down, the point

(2, 7)

is translated to

(2 + 3, 7 - 4) = (5, 3)

on the new graph. The translated graph also passes through

(4, 2)

. Using the slope formula on the two points

(5, 3)

and

(4, 2)

of the translated line gives a slope of

1

. Since a translation preserves the slope, the original line also has a slope of

1

. Substituting

m = 1

and

(2, 7)

into the original equation

y = mx + b

yields

7 = 1(2) + b

, which simplifies to

b = 5

.

Step-by-Step Solution

1

Determine a point on the translated graph by translating the given point

(2, 7)

on the original graph.

The point

(5, 3)

is on the translated graph.

Since the entire graph is translated

3

units to the right and

4

units down, every point on the original graph is translated by the same vector

(+3, -4)

.

2

Calculate the slope of the translated graph using the points

(5, 3)

and

(4, 2)

.

The slope

m = 1

.

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by the change in y divided by the change in x.

3

Find the slope of the original graph.

The slope of the original graph is also

1

.

Translating a line horizontally and vertically shifts its position but does not alter its slope.

4

Substitute the slope

m = 1

and the point

(2, 7)

into the slope-intercept form of the original line,

y = mx + b

, to solve for

b

.

b = 5

.

This determines the value of the y-intercept of the original line.

Key Concept

Linear Equations in Two Variables

Question 1232Question →

An online store sells small gift boxes for $10$ dollars each and large gift boxes for $15$ dollars each. A customer wants to buy a total of at most $10$ boxes. If the customer must spend at least $120$ dollars on the boxes, what is the minimum number of large gift boxes the customer must buy?

Show answer & explanation

Answer: 4

Answer

The minimum number of large gift boxes the customer must buy is 4.

Using the constraints from the problem, we set up the system of inequalities:

x + y \leq 10

and

10x + 15y \geq 120

. Expressing the first inequality as

x \leq 10 - y

and substituting it into the second gives

10(10 - y) + 15y \geq 120

. Simplifying this results in

100 + 5y \geq 120

, which simplifies to

5y \geq 20

, or

y \geq 4

. Therefore, the minimum number of large gift boxes the customer must buy is

4

.

Step-by-Step Solution

1

Define variables for the quantities of each box.

Let

x

be the number of small boxes and

y

be the number of large boxes.

Establishing variables is necessary to translate the word problem into algebraic inequalities.

2

Write the system of inequalities representing the constraints.

x + y \leq 10

and

10x + 15y \geq 120

The total number of boxes is at most

10

, and the total cost must be at least

120

dollars.

3

Express

x

in terms of

y

using the first inequality.

x \leq 10 - y

This allows substitution into the second inequality to solve for the target variable

y

.

4

Substitute

x \leq 10 - y

into the second inequality and simplify.

100 + 5y \geq 120

Solving the resulting single-variable inequality will determine the possible values of

y

.

5

Solve the inequality for

y

.

y \geq 4

Subtracting

100

and dividing by

5

isolates

y

to show its minimum possible value is

4

.

Key Concept

Systems of Linear Inequalities in Two Variables

Question 1233Question →

Although the newly established consortium of independent software developers has faced numerous technical challenges and scheduling delays during the initial phases of creating the open-source operating system, the group has reassured its financial investors that ______ will deliver the completed software by the revised deadline. By consolidating resources and streamlining the software testing process, the organization hopes to establish itself as a leader in the industry.

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

they

it

their

themselves

Show answer & explanation

Answer: it

Answer

The singular pronoun 'it'

The singular pronoun 'it' correctly agrees in number with the singular collective noun 'group,' which is the antecedent. Furthermore, 'it' is in the subject case, which is required to function as the subject of the verb phrase 'will deliver.'

Step-by-Step Solution

1

Identify the antecedent of the pronoun needed for the blank.

The antecedent is the collective noun 'group.'

The pronoun in the blank must refer back to this noun to indicate who will deliver the completed software.

2

Determine the grammatical number of the antecedent.

The collective noun 'group' is singular.

Collective nouns representing a single unified entity are treated as singular in Standard English.

3

Select the correct pronoun case and number to fill the blank.

The singular subject pronoun 'it' is the correct choice.

It agrees in number with the singular antecedent 'group' and correctly functions as the subject of the verb 'will deliver.'

Key Concept

Pronoun-Antecedent Agreement in Number

Estimated Time:1m 0s

Question 1234Question →

A theater sells adult tickets for $x$ dollars each and child tickets for $y$ dollars each. For a morning show, the theater sold 2 adult tickets and 1 child ticket, collecting a total of $14$ dollars. For an afternoon show, the theater sold 3 adult tickets and 2 child tickets, collecting a total of $23$ dollars. What is the cost, in dollars, of 1 adult ticket and 1 child ticket combined?

5

6

9

19

Show answer & explanation

Answer: 9

Answer

The combined cost of 1 adult ticket and 1 child ticket is 9 dollars.

The correct answer is 9. Subtracting the equation representing the morning show (

2x + y = 14

) from the equation representing the afternoon show (

3x + 2y = 23

) directly yields

x + y = 9

. Alternatively, solving the system of equations by substitution or elimination gives

x = 5

(cost of an adult ticket) and

y = 4

(cost of a child ticket), and adding these two values results in

9

.

Step-by-Step Solution

1

Write the system of linear equations based on the problem description.

The system of equations is:

\begin{aligned} 2x + y &= 14 \\ 3x + 2y &= 23 \end{aligned}

To mathematically represent the ticket sales and total revenue for the morning and afternoon shows.

2

Subtract the first equation from the second equation to find the value of

x + y

directly.

(3x + 2y) - (2x + y) = 23 - 14 \implies x + y = 9

Subtracting the equations isolates the expression

x + y

immediately, which represents the combined cost of 1 adult ticket and 1 child ticket.

Key Concept

Solving systems of linear equations by elimination or subtraction to find a linear combination of variables.

Alternative Method

Solve the system using elimination: multiply the first equation by 2 to get

4x + 2y = 28

. Subtract the second equation (

3x + 2y = 23

) from this new equation to find

x = 5

. Substitute

x = 5

into the first equation to find

y = 4

. Finally, add the values to get

x + y = 9

.

Estimated Time:1m 30s

Question 1235Question →

Ancient Roman architects created an exceptionally durable form of concrete by mixing volcanic ash with slaked lime, enabling their harbors and piers to withstand centuries of wave action. When exposed to seawater, this unique mixture undergoes a chemical reaction that continually strengthens the material ______ modern concrete lacks this self-healing property and deteriorates rapidly in marine environments.

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

material

material, so

material;

material,

Show answer & explanation

Answer: material;

Answer

material;

The option containing the semicolon correctly links two independent clauses. The first clause ('When exposed to seawater, this unique mixture undergoes a chemical reaction that continually strengthens the material') and the second clause ('modern concrete lacks this self-healing property and deteriorates rapidly in marine environments') are both independent, meaning they can stand alone as complete sentences. A semicolon is a grammatically correct way to join these two clauses.

Step-by-Step Solution

1

Identify the clause boundaries in the sentence.

The text contains two independent clauses: the first ends with 'material' and the second begins with 'modern concrete'.

Understanding where one independent clause ends and the next begins is necessary to determine the correct punctuation.

2

Evaluate the grammatical relationship between the two independent clauses.

The two clauses express closely related, contrasting ideas that can be joined without a coordinating conjunction.

Determining how the clauses relate helps select the appropriate punctuation or conjunction.

3

Choose the punctuation mark that correctly links the two independent clauses.

A semicolon is a standard way to link two independent clauses without a coordinating conjunction.

Using a semicolon avoids run-on sentences and comma splices.

Key Concept

Semicolons to link independent clauses

Question 1236Question →

A sports store orders standard helmets for $20$ dollars each and premium helmets for $50$ dollars each. The store can spend at most $2{}200$ dollars on this order. The distributor requires the store to order at least $60$ helmets in total. If the store decides to order at least $15$ premium helmets, what is the maximum number of standard helmets that the store can order?

Show answer & explanation

Answer: 72

Answer

The maximum number of standard helmets the store can order is 72.

By setting the number of premium helmets to its minimum allowed value of

15

to maximize the budget remaining for standard helmets, we find

20x \leq 1450

, which yields

x \leq 72.5

. Since standard helmets must be ordered in whole numbers, the maximum number is

72

, which also satisfies the minimum total order of

60

helmets (

72 + 15 = 87 \geq 60

).

Step-by-Step Solution

1

Define variables and translate constraints into inequalities

Let

x

be the number of standard helmets and

y

be the number of premium helmets. The constraints are

20x + 50y \leq 2200

,

x + y \geq 60

, and

y \geq 15

.

To represent the problem's mathematical relationships using a system of linear inequalities.

2

Solve for the upper limit of

x

using the budget constraint and the minimum value of

y

Since

20x \leq 2200 - 50y

,

x

is maximized when

y

is at its minimum value,

y = 15

. Substituting

y = 15

yields

20x + 750 \leq 2200 \implies 20x \leq 1450 \implies x \leq 72.5

.

To find the maximum possible value of standard helmets under the budget constraint.

3

Verify with the minimum total order constraint and determine the maximum integer value

Checking

x + y \geq 60

with

y = 15

gives

x + 15 \geq 60 \implies x \geq 45

. Since

x

must satisfy

45 \leq x \leq 72.5

and must be an integer, the maximum integer value is

72

.

To ensure the solution is physically possible as a whole number of items and satisfies all constraints.

Key Concept

Systems of Linear Inequalities in Two Variables

Question 1237Question →

A drone's battery charge is at $98\%$ when it begins a mission. During the mission, the battery charge decreases by $1.5\%$ per minute when the drone is hovering, and by $2.5\%$ per minute when it is flying horizontally. The drone flies horizontally for exactly twice as many minutes as it hovers. If the battery charge is at $33\%$ at the end of the mission, for how many minutes did the drone hover?

Show answer & explanation

Answer: 10

Answer

The drone hovered for 10 minutes.

Let

t

represent the number of minutes the drone hovers. Since the drone flies horizontally for twice as long as it hovers, it flies horizontally for

2t

minutes. The total percentage drop in battery is

98 - 33 = 65

. The equation representing the total decrease in battery charge is

1.5t + 2.5(2t) = 65

. Simplifying the equation yields

1.5t + 5t = 65

, which becomes

6.5t = 65

. Dividing both sides of the equation by

6.5

gives

t = 10

.

Step-by-Step Solution

1

Define the variable for the unknown quantity.

Let

t

be the number of minutes the drone spent hovering. The time spent flying horizontally is then

2t

minutes.

The problem states that the horizontal flight time is exactly twice the hovering time.

2

Set up an equation representing the total decrease in battery charge.

The total percentage decrease is

1.5t + 2.5(2t) = 98 - 33

.

The battery decreases by

1.5\%

per minute of hovering,

2.5\%

per minute of horizontal flight, and the total change is from

98\%

to

33\%

.

3

Simplify the equation and solve for

t

.

6.5t = 65

, which gives

t = 10

.

Combine like terms and divide both sides by

6.5

to isolate the variable.

Key Concept

Setting up and solving a linear equation in one variable from a real-world scenario.

Question 1238Question →

During his excavations in Egypt's Valley of the Kings, British archaeologist Howard Carter made a historic discovery that captured global attention. The tomb of Tutankhamun—a pharaoh of the Eighteenth Dynasty who ruled during the New Kingdom period__________contained a spectacular array of gold artifacts, ornate jewelry, and everyday items that provided researchers with unprecedented insights into ancient Egyptian culture.

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

period

period—

period,

period;

Show answer & explanation

Answer: period—

Answer

period—

The sentence includes a nonessential appositive phrase ('a pharaoh of the Eighteenth Dynasty who ruled during the New Kingdom period') that provides supplementary information about Tutankhamun. Since this phrase is introduced with an em dash, it must also be concluded with a matching em dash to maintain punctuation symmetry.

Step-by-Step Solution

1

Identify the nonessential element in the sentence.

The phrase 'a pharaoh of the Eighteenth Dynasty who ruled during the New Kingdom period' is an appositive that provides extra information about Tutankhamun and is not essential to the main clause.

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

2

Check the punctuation mark that opens the nonessential element.

The nonessential element begins with an em dash ('Tutankhamun—').

To maintain punctuation symmetry, the nonessential element must be closed with the same type of punctuation mark used to open it.

3

Select the option that closes the nonessential phrase with a matching em dash.

The option containing the word followed by an em dash is correct.

This maintains grammatical consistency and adheres to Standard English conventions.

Key Concept

Punctuation of Nonessential Elements

Estimated Time:1m 0s

Question 1239Question →

Under the extreme pressure of the deep ocean, certain species of glass sponges construct highly stable skeletal lattices _______ structures made of interlocking silica spicules that can withstand immense physical forces without collapsing.

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

lattices—structures

lattices; structures

lattices, they are structures

lattices: and structures

Show answer & explanation

Answer: lattices—structures

Answer

lattices—structures

The correct answer correctly uses a single dash to connect the preceding independent clause ('Under the extreme pressure of the deep ocean, certain species of glass sponges construct highly stable skeletal lattices') to the explanatory appositive phrase at the end of the sentence ('structures made of interlocking silica spicules...').

Step-by-Step Solution

1

Analyze the clause structure before and after the blank.

The clause before the blank ('Under the extreme pressure... lattices') is an independent clause. The phrase after the blank ('structures made of... collapsing') is a noun phrase acting as an explanatory appositive, not an independent clause.

Identifying clause boundaries determines the appropriate punctuation marks that can link them.

2

Evaluate the punctuation choices based on grammatical conventions.

A single dash is appropriate to connect an independent clause to a trailing explanatory appositive. Semicolons and colons followed by coordinating conjunctions are incorrect because the second part is not an independent clause and the conjunction disrupts the grammatical flow, respectively. A comma followed by a subject-verb construction creates a comma splice.

Selecting the option that correctly applies the conventions of punctuation to connect the clauses.

Key Concept

Using a single dash to connect an independent clause to explanatory details or a summary appositive at the end of a sentence.

Question 1240Question →

If the expression $(2x + 3)(3x + 1) - 6x^2$ is rewritten in the form $bx + c$ , where $b$ and $c$ are constants, what is the value of $b$ ?

Show answer & explanation

Answer: 11

Answer

The value of

b

is

11

.

Expanding the expression

(2x + 3)(3x + 1) - 6x^2

yields

(6x^2 + 2x + 9x + 3) - 6x^2

. Combining the linear terms simplifies the expression to

(6x^2 + 11x + 3) - 6x^2

, which further simplifies to

11x + 3

. Comparing this to the form

bx + c

, we see that the coefficient

b

of the

x

term is

11

.

Step-by-Step Solution

1

Expand the product of the binomials

(2x + 3)(3x + 1)

using the distributive property.

6x^2 + 2x + 9x + 3

To rewrite the factored part of the expression in polynomial form.

2

Combine the linear terms

2x

and

9x

.

6x^2 + 11x + 3

To simplify the expanded polynomial expression.

3

Subtract

6x^2

from the simplified polynomial expression.

11x + 3

To complete the subtraction indicated in the original expression.

4

Compare the resulting expression

11x + 3

to the form

bx + c

to identify the coefficient

b

.

b = 11

To determine the constant coefficient of the

x

term.

Key Concept

Expanding products of binomials and combining like terms to find equivalent expressions.

Answer

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Key Concept

Common Mistakes

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Common Mistakes

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