Tüm alıştırma soruları

Soru 1241Soru →

A florist is making bouquets. Each small bouquet requires $2$ roses, and each large bouquet requires $5$ roses. The florist has at most $40$ roses available for these bouquets. Additionally, the florist wants to make more than $10$ bouquets in total. If $x$ represents the number of small bouquets and $y$ represents the number of large bouquets, which of the following systems of inequalities represents this situation?

2x + 5y \geq 40

and

x + y < 10

5x + 2y \leq 40

and

x + y > 10

2x + 5y \leq 40

and

x + y > 10

2x + 5y \leq 10

and

x + y > 40

Cevabı ve açıklamayı göster

Cevap: $2x + 5y \leq 40$ and $x + y > 10$

Cevap

The system of inequalities

2x + 5y \leq 40

and

x + y > 10

.

The correct system of inequalities matches the constraints given in the problem. The total number of roses used by

x

small bouquets and

y

large bouquets is

2x + 5y

. Since the florist has at most

40

roses, this is represented by

2x + 5y \leq 40

. The total number of bouquets is

x + y

. Since the florist wants to make more than

10

bouquets, this is represented by

x + y > 10

.

Adım Adım Çözüm

1

Identify the inequality representing the constraint on the total number of roses.

2x + 5y \leq 40

Each small bouquet uses

2

roses (

2x

) and each large bouquet uses

5

roses (

5y

). The total roses must be 'at most'

40

, which corresponds to the inequality symbol

\leq

.

2

Identify the inequality representing the constraint on the total number of bouquets.

x + y > 10

The total number of bouquets is the sum of small bouquets (

x

) and large bouquets (

y

). The florist wants 'more than'

10

bouquets, which corresponds to the inequality symbol

>

.

3

Combine the two inequalities into a single system.

The system is

2x + 5y \leq 40

and

x + y > 10

.

Both conditions must be met simultaneously.

Anahtar Kavram

Translating real-world constraints into a system of linear inequalities in two variables.

Soru 1242Soru →

In the late nineteenth century, a prominent coalition of independent labor unions and social reform groups in Chicago campaigned tirelessly for improved working conditions in manufacturing plants. Despite facing significant opposition from influential industrial leaders and local politicians, ______ eventually succeeded in securing the passage of the landmark eight-hour workday law of 1893.

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

they

it

them

its

Cevabı ve açıklamayı göster

Cevap: it

Cevap

it

The singular pronoun 'it' correctly agrees in number with the singular collective noun antecedent 'coalition'. Although there are plural nouns ('unions', 'groups') in the intervening prepositional phrase, the head of the noun phrase is the singular 'coalition', which requires a singular pronoun.

Adım Adım Çözüm

1

Identify the antecedent of the pronoun needed in the blank.

The antecedent is 'coalition', which is a singular collective noun.

Determining the correct antecedent is necessary to ensure proper pronoun-antecedent agreement in number and gender.

2

Analyze the intervening elements to avoid distraction.

The prepositional phrase 'of independent labor unions and social reform groups in Chicago' contains plural nouns but does not change the singular nature of the subject 'coalition'.

Students often make agreement errors by matching the pronoun to the nearest plural noun in intervening phrases.

3

Determine the correct case and number of the pronoun.

The blank acts as the subject of the verb 'succeeded', so a singular subject pronoun ('it') is required.

The pronoun must match the singular number of the antecedent 'coalition' and be in the subjective case to act as the subject of the clause.

Anahtar Kavram

Pronoun-Antecedent Agreement (Number and Case)

Soru 1243Soru →

In the system of linear equations below,

a

,

b

, and

c

are constants.

\begin{aligned} \frac{3}{2}x - \frac{2}{3}y &= 12 \\ ax + by &= c \end{aligned}

If the system has infinitely many solutions, and the point

(4, b)

lies on the line represented by the second equation, what is the value of

c

?

-162

72

108

162

Cevabı ve açıklamayı göster

Cevap: 162

Cevap

The correct answer is 162.

The correct answer is 162. Since the system of linear equations has infinitely many solutions, the two equations represent the same line. Therefore, any point that lies on the second line must also lie on the first line. Substituting the coordinates of the point (4, b) into the first equation gives 3/2(4) - 2/3(b) = 12, which simplifies to 6 - 2/3(b) = 12. Solving this equation for b yields b = -9. Since the equations represent the same line, the ratio of their corresponding coefficients must be equal: c / 12 = b / (-2/3). Solving this proportion for c gives c = -18b. Substituting the value of b = -9 into this relation yields c = -18(-9) = 162.

Adım Adım Çözüm

1

Determine the relationship between the two equations based on the number of solutions.

Since the system has infinitely many solutions, the two equations represent coincident lines in the coordinate plane. Therefore, any point that lies on the second line must also satisfy the first equation.

Infinitely many solutions in a system of two linear equations indicate that they represent the same line.

2

Substitute the point

(4, b)

into the first equation to solve for

b

.

\frac{3}{2}(4) - \frac{2}{3}b = 12 \implies 6 - \frac{2}{3}b = 12 \implies -\frac{2}{3}b = 6 \implies b = -9

Plugging the coordinates of the point into the first equation allows us to find the value of the unknown coordinate.

3

Set up the ratio of corresponding coefficients for the coincident lines.

\frac{a}{\frac{3}{2}} = \frac{b}{-\frac{2}{3}} = \frac{c}{12}

Equivalent equations must have proportional coefficients and constant terms.

4

Solve for

c

using the ratio containing

b

and

c

.

\frac{c}{12} = \frac{b}{-\frac{2}{3}} \implies c = -18b \implies c = -18(-9) = 162

Substituting the solved value of

b = -9

into the proportion yields the value of

c

.

Anahtar Kavram

Systems of Linear Equations with Infinitely Many Solutions

Soru 1244Soru →

To determine the chemical composition of remote astronomical bodies, astrophysicists analyze the light filtering through their surrounding atmospheres. By studying how a ________ unique chemical signature absorbs specific wavelengths of stellar radiation, researchers can identify the presence of greenhouse gases like methane and carbon dioxide, offering clues about the potential habitability of distant worlds.

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

planet

planets

planet's

planets'

Cevabı ve açıklamayı göster

Cevap: planet's

Cevap

planet's

The correct answer is the singular possessive noun 'planet's'. The singular article 'a' indicates that the noun must be singular. Furthermore, because this noun modifies the noun phrase 'unique chemical signature,' it must be possessive to establish ownership. The spelling 'planet's' correctly denotes singular possession.

Adım Adım Çözüm

1

Determine if the noun in the blank needs to be possessive.

The blank precedes the noun phrase 'unique chemical signature'. Since this signature belongs to the noun in the blank, a possessive form is required.

A possessive noun is needed to modify the noun phrase and show ownership.

2

Determine the grammatical number (singular or plural) of the noun.

The blank is preceded by the singular indefinite article 'a' and the verb 'absorbs' is singular. This requires a singular noun.

The noun must agree in number with its preceding article and context clues.

3

Select the correct singular possessive spelling.

The singular possessive form of 'planet' is 'planet's', formed by adding an apostrophe and 's'.

This spelling correctly identifies singular possession.

Anahtar Kavram

Selecting the correct noun form based on grammatical markers of possession and number (singular vs. plural).

Soru 1245Soru →

Which of the following expressions is equivalent to $(2x^3)^4$ for all values of $x$ ?

16x^7

8x^{12}

16x^{12}

8x^7

Cevabı ve açıklamayı göster

Cevap: $16x^{12}$

Cevap

16x^{12}

The correct answer is the expression

16x^{12}

. When simplifying

(2x^3)^4

, we apply the exponent

4

to both the coefficient

2

and the term

x^3

, resulting in

2^4(x^3)^4

. Evaluating

2^4

gives

16

. Applying the power of a power rule to

(x^3)^4

requires multiplying the exponents (

3 \times 4

), which gives

x^{12}

. Combining these yields

16x^{12}

.

Adım Adım Çözüm

1

Apply the power of a product property to distribute the outer exponent of

4

to both factors inside the parentheses: the coefficient

2

and the variable term

x^3

.

(2x^3)^4 = 2^4 \times (x^3)^4

The power of a product rule states that

(ab)^n = a^n b^n

.

2

Evaluate the constant coefficient

2^4

.

2^4 = 2 \times 2 \times 2 \times 2 = 16

Evaluating the base raised to the exponent.

3

Apply the power of a power property to simplify the variable term

(x^3)^4

.

(x^3)^4 = x^{3 \times 4} = x^{12}

The power of a power rule states that

(x^a)^b = x^{a \times b}

.

4

Combine the simplified coefficient and variable parts to find the final equivalent expression.

16x^{12}

Combining the results of the coefficient simplification and the variable simplification yields the final term.

Anahtar Kavram

Exponent rules, specifically the power of a product property

(ab)^n = a^n b^n

and the power of a power property

(x^a)^b = x^{a \times b}

.

Tahmini Süre:45s

Soru 1246Soru →

In his study of the dwarf planet Ceres, planetary scientist Dr. Hiroshi Sato argues that the bright sodium carbonate deposits in Occator Crater are not merely static geological features. Instead, Sato suggests that these salt deposits serve as critical clues for researchers, functioning as indicators of cryovolcanic activity, remnants of a subsurface ocean, and ________

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

they signal geothermal warming.

to signal geothermal warming.

signals of geothermal warming.

signaling geothermal warming.

Cevabı ve açıklamayı göster

Cevap: signals of geothermal warming.

Cevap

signals of geothermal warming.

The sentence contains a list of functions served by the salt deposits: 'indicators of cryovolcanic activity,' 'remnants of a subsurface ocean,' and the final item. The first two items are plural noun phrases post-modified by prepositional phrases. To maintain parallel structure, the final item must also be a plural noun phrase of the same pattern. The phrase 'signals of geothermal warming' perfectly satisfies this structural requirement.

Adım Adım Çözüm

1

Analyze the sentence to identify the list of items and their grammatical structure.

The text lists three features that the salt deposits function as: 'indicators of cryovolcanic activity,' 'remnants of a subsurface ocean,' and a blank space.

Parallel structure dictates that all items in a list must be in the same grammatical form.

2

Determine the grammatical form of the first two list items.

Both 'indicators of cryovolcanic activity' and 'remnants of a subsurface ocean' are plural noun phrases structured as a plural noun followed by a prepositional phrase starting with 'of'.

Establishing the shared structure of the existing list items determines the form the final item must take.

3

Test the options to find the one that maintains this exact plural noun phrase structure.

The phrase 'signals of geothermal warming' is a plural noun phrase with a matching structure. The other options form clauses, infinitives, or participles.

Selecting the option that matches the plural noun phrase structure preserves the sentence's grammatical parallelism.

Anahtar Kavram

Parallel Structure

Soru 1247Soru →

A school club is planning a fundraiser by selling two types of customized keychains: acrylic keychains and metal keychains. Acrylic keychains cost $3 to make and sell for$ 5, while metal keychains cost $5 to make and sell for$ 8. The club has a budget of at most $150 to produce the keychains, and they want to make at least 40 keychains in total. Let$ x $represent the number of acrylic keychains and$ y$ represent the number of metal keychains. Which of the following systems of inequalities represents this situation?

3x + 5y \leq 150

and

x + y \geq 40

3x + 5y \geq 150

and

x + y \leq 40

5x + 8y \leq 150

and

x + y \geq 40

5x + 3y \leq 150

and

x + y \geq 40

Cevabı ve açıklamayı göster

Cevap: $3x + 5y \leq 150$ and $x + y \geq 40$

Cevap

3x + 5y \leq 150

and

x + y \geq 40

The correct system of inequalities must reflect the constraints on the budget and the total number of keychains. The budget is at most

150, which means the total production cost must be less than or equal to 150. Since each acrylic keychain costs

3 to make and each metal keychain costs

5, the cost constraint is represented by

3x + 5y \leq 150

. The club also wants to make at least 40 keychains, which means the total count must be greater than or equal to 40. This constraint is represented by

x + y \geq 40

. Together, these form the system

3x + 5y \leq 150

and

x + y \geq 40$.

Adım Adım Çözüm

1

Identify the variables and their meanings from the problem statement.

x

represents the number of acrylic keychains, and

y

represents the number of metal keychains.

Understanding the variable definitions is necessary to construct the constraints correctly.

2

Set up the budget inequality using the production costs.

3x + 5y \leq 150

Each acrylic keychain costs

3 to make, each metal keychain costs

5, and the total cost must be at most (less than or equal to) $150.

3

Set up the quantity inequality.

x + y \geq 40

The club wants to make a total of at least (greater than or equal to) 40 keychains.

Anahtar Kavram

Systems of linear inequalities in context

Soru 1248Soru →

A linear relationship exists between the variables $p$ and $q$ . When the value of $p$ increases by $6$ , the value of $q$ decreases by $4$ . If $q = 15$ when $p = 2$ , which of the following equations correctly expresses $p$ in terms of $q$ ?

p = -\frac{3}{2}q + \frac{49}{2}

p = -\frac{2}{3}q + \frac{49}{3}

p = \frac{3}{2}q - \frac{41}{2}

p = -\frac{2}{3}q + 12

Cevabı ve açıklamayı göster

Cevap: $p = -\frac{3}{2}q + \frac{49}{2}$

Cevap

p = -\frac{3}{2}q + \frac{49}{2}

The correct equation is found by first calculating the rate of change of

q

with respect to

p

, which is

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

. Using the point-slope form with

(p, q) = (2, 15)

, we set up the equation

q - 15 = -\frac{2}{3}(p - 2)

. To express

p

in terms of

q

, we isolate

p

by multiplying both sides by

-\frac{3}{2}

to get

-\frac{3}{2}q + \frac{45}{2} = p - 2

, and then adding

2

to both sides, which yields

p = -\frac{3}{2}q + \frac{49}{2}

.

Adım Adım Çözüm

1

Calculate the rate of change of

q

relative to

p

using the given changes.

The rate of change is

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

.

Since

q

decreases by

4

when

p

increases by

6

, the slope

m

in terms of

\Delta q / \Delta p

is

-\frac{2}{3}

.

2

Set up the point-slope form of the linear equation using the point

(p, q) = (2, 15)

.

q - 15 = -\frac{2}{3}(p - 2)

The point-slope form is

q - q_1 = m(p - p_1)

, where

(p_1, q_1) = (2, 15)

and

m = -\frac{2}{3}

.

3

Solve the equation for

p

by first clearing the coefficient of the

p

-term.

-\frac{3}{2}(q - 15) = p - 2 \implies -\frac{3}{2}q + \frac{45}{2} = p - 2

Multiplying both sides by

-\frac{3}{2}

simplifies isolation of the variable

p

.

4

Complete the isolation of

p

by adding

2

to both sides.

p = -\frac{3}{2}q + \frac{45}{2} + 2 \implies p = -\frac{3}{2}q + \frac{49}{2}

Adding

2

(which is

\frac{4}{2}

) to

\frac{45}{2}

isolates

p

on one side of the equation.

Anahtar Kavram

Linear relationships and isolating variables in two-variable linear equations.

Soru 1249Soru →

The passage below discusses desert survival adaptations. Which word completes the passage so that it conforms to the conventions of Standard English?

Aşağıdaki boşlukları doldurun

In a 2025 study of desert survival adaptations in the Sahara, botanist Dr. Elena Vance observed that the resilient *Welwitschia mirabilis* plant successfully conserves water under extreme conditions. According to Vance, the species maintains hydration not only by closing its stomata during peak daytime heat hours but also by moisture from morning fog that rolls in from the Atlantic coast.

Cevabı ve açıklamayı göster

Cevap

A gerund (a verb ending in -ing, such as 'absorbing' or 'harvesting') that completes the parallel construction established by the gerund 'closing'.

The sentence uses the correlative conjunction 'not only... but also' to connect two parallel actions. The first action is introduced as 'by closing its stomata...', where 'closing' is a gerund. To maintain parallel structure, the second action must also be a gerund (a verb ending in -ing). Thus, a word like 'absorbing' or 'harvesting' correctly completes the parallel series.

Adım Adım Çözüm

1

Identify the coordinating elements in the sentence.

The sentence utilizes the correlative conjunction structure 'not only by... but also by...'.

Correlative conjunctions require the phrases they connect to be grammatically parallel.

2

Analyze the grammatical form of the first element in the parallel structure.

The first element is the gerund phrase 'closing its stomata'.

Knowing the form of the first element determines the required form for the second element to achieve parallelism.

3

Select a word for the blank that matches the form of the first element.

The blank must be filled with a gerund (an -ing verb) such as 'absorbing', 'harvesting', or 'collecting'.

Using a gerund maintains parallel structure, conforming to the conventions of Standard English.

Anahtar Kavram

Parallel Structure with Correlative Conjunctions

Soru 1250Soru →

In the $xy$ -plane, the graph of a linear function $f$ passes through the point $(1, 3)$ and has a slope of $m$ , where $m \neq 0$ . The graph of a second linear function, $g$ , is obtained by reflecting the graph of $f$ across the $y$ -axis and then translating it up by $4$ units. If the graphs of $f$ and $g$ intersect at a point on the $x$ -axis, what is the value of $m$ ?

1

-2

5

3

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

The correct value is

5

. By using the point-slope form, the first function is

f(x) = mx - m + 3

. Reflecting it across the

y

-axis gives

f(-x) = -mx - m + 3

, and translating it up by

4

units yields

g(x) = -mx - m + 7

. Since they intersect on the

x

-axis, the

y

-value at their intersection point is

0

. This gives the system of equations

mx_0 = m - 3

and

-mx_0 = m - 7

. Adding these equations eliminates the

mx_0

term, leaving

2m - 10 = 0

, which simplifies to

m = 5

.

Adım Adım Çözüm

1

Write the equation of the linear function

f(x)

using the point-slope form.

f(x) = m(x - 1) + 3 = mx - m + 3

The function passes through the point

(1, 3)

and has a slope of

m

.

2

Determine the equation of the linear function

g(x)

by applying the reflection and translation transformations.

g(x) = f(-x) + 4 = -mx - m + 7

Reflecting across the

y

-axis replaces

x

with

-x

, and translating up by

4

units adds

4

to the function.

3

Set the

y

-coordinates of the intersection point

(x_0, 0)

on the

x

-axis to

0

for both functions.

mx_0 - m + 3 = 0 \implies mx_0 = m - 3

and

-mx_0 - m + 7 = 0 \implies -mx_0 = m - 7

An intersection on the

x

-axis means that the

y

-coordinate of the intersection point is

0

.

4

Solve the system of equations to find the value of

m

.

0 = (m - 3) + (m - 7) \implies 2m = 10 \implies m = 5

Adding the two equations eliminates the term

mx_0

, allowing us to solve directly for

m

.

Anahtar Kavram

Applying transformations (reflections and translations) to linear functions and finding their intercepts in the coordinate plane.

Soru 1251Soru →

A chef is preparing portions of roasted vegetables and mashed potatoes for a catered event. Each portion of roasted vegetables requires $3$ ounces of potatoes, and each portion of mashed potatoes requires $6$ ounces of potatoes. The chef has a total of at most $90$ ounces of potatoes available. The chef must prepare at least $8$ portions of roasted vegetables and at least $5$ portions of mashed potatoes. If $v$ represents the number of portions of roasted vegetables that the chef prepares, what is the maximum possible value of $v$ ?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The maximum possible value of

v

is 20.

The maximum number of portions of roasted vegetables the chef can prepare is 20 because minimizing the number of mashed potato portions to its boundary constraint of 5 maximizes the remaining resources, yielding

3v \leq 90 - 6(5) \implies 3v \leq 60 \implies v \leq 20

.

Adım Adım Çözüm

1

Formulate the inequality representing the potato weight constraint.

3v + 6p \leq 90

, where

v

is the number of portions of roasted vegetables and

p

is the number of portions of mashed potatoes.

Each portion of roasted vegetables uses

3

ounces of potatoes, each portion of mashed potatoes uses

6

ounces of potatoes, and the total amount used cannot exceed

90

ounces.

2

Formulate the inequalities representing the minimum quantity constraints.

v \geq 8

and

p \geq 5

The chef must make at least

8

portions of roasted vegetables and at least

5

portions of mashed potatoes.

3

Isolate the variable

v

in the potato limit inequality.

v \leq 30 - 2p

Subtracting

6p

from both sides of

3v + 6p \leq 90

yields

3v \leq 90 - 6p

, and dividing the entire inequality by

3

gives

v \leq 30 - 2p

.

4

Determine the maximum value of

v

by substituting the minimum possible value of

p

.

v \leq 30 - 2(5) \implies v \leq 20

To maximize

v

, we must minimize the subtracted term

2p

. The minimum allowed value of

p

is

5

.

Anahtar Kavram

Optimization in systems of linear inequalities

Soru 1252Soru →

For decades, researchers in cognitive neuroscience have investigated mirror neurons, which are specialized brain cells that fire both when an action is performed and when it is observed. Many scientists hypothesize that this neural mirroring is essential for empathy ______ argue that its primary function is simply to facilitate motor learning and imitation.

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

; others

, others

others

since others

Cevabı ve açıklamayı göster

Cevap: ; others

Cevap

The option containing '; others' is correct because it properly uses a semicolon to separate two independent clauses.

The correct option properly uses a semicolon to link the two independent clauses: 'Many scientists hypothesize that this neural mirroring is essential for empathy' and 'others argue that its primary function is simply to facilitate motor learning and imitation.' Because no coordinating conjunction is used, a semicolon is the standard way to separate these two complete thoughts.

Adım Adım Çözüm

1

Identify the clauses surrounding the blank.

The first clause is 'Many scientists hypothesize that this neural mirroring is essential for empathy' and the second is 'others argue that its primary function is simply to facilitate motor learning and imitation.' Both are independent clauses because they can stand alone as complete sentences.

Determining clause type is necessary to apply the correct punctuation rule.

2

Evaluate how the clauses should be joined.

Two independent clauses must be joined by a period, a semicolon, a colon, or a comma followed by a coordinating conjunction (FANBOYS).

This rules out options that introduce comma splices or run-on sentences.

3

Test the options against grammar and logic rules.

The option with a semicolon successfully links the two clauses. The option with a comma creates a comma splice. The option with no punctuation creates a run-on. The option with a subordinating conjunction creates an illogical causal relationship between contrasting statements.

Selecting the correct grammatical and logical connector ensures standard English conventions are met.

Anahtar Kavram

Clause Boundaries and Linking

Tahmini Süre:45s

Soru 1253Soru →

Read the passage below. Which punctuation mark should be placed in the blank to correctly set off the nonessential element?

Aşağıdaki boşlukları doldurun

Although pulsars are now recognized as vital tools for studying the cosmos, their discovery in 1967 was initially met with intense skepticism from the scientific establishment. Astrophysicist Jocelyn Bell Burnell first identified these rapidly rotating neutron stars, which emit regular beams of radio waveswhile analyzing massive amounts of data from a newly constructed radio telescope at Cambridge University.

Cevabı ve açıklamayı göster

Cevap

A comma (`,`) must be placed in the blank to correctly set off the nonessential relative clause.

A comma is correct because it matches the opening comma after the word 'stars' that introduces the nonessential relative clause 'which emit regular beams of radio waves.' In Standard English, nonessential elements must be set off with a consistent pair of matching punctuation marks.

Adım Adım Çözüm

1

Identify the nonessential relative clause within the sentence.

The clause 'which emit regular beams of radio waves' is a nonessential relative clause that modifies the noun phrase 'these rapidly rotating neutron stars' by adding descriptive detail that is not grammatically essential to the main sentence.

Recognizing nonessential elements is necessary to apply rules of punctuation symmetry.

2

Examine the punctuation mark that introduces the nonessential clause.

The clause begins with a comma after the word 'stars.'

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

3

Determine the appropriate closing punctuation mark.

Since the nonessential relative clause is introduced with a comma, it must close with a comma to maintain grammatical symmetry before continuing with the rest of the independent clause.

Using mismatched punctuation (such as a comma and a dash) creates an ungrammatical shift.

Anahtar Kavram

Punctuation of Nonessential Elements

Tahmini Süre:1m 0s

Soru 1254Soru →

For the linear function $f$ , the graph of $y = f(x)$ has a slope of $3$ and contains the point $(2, 7)$ . The graph of another linear function, $g$ , is parallel to the graph of $f$ . If $g(0) = -4$ , what is the value of $g(5)$ ?

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

11

Since the graph of

g

is parallel to the graph of

f

, it has the same slope of

3

. The value

g(0) = -4

indicates that the

y

-intercept of the graph of

g

is

-4

. Thus, the equation for

g(x)

is

g(x) = 3x - 4

. Evaluating this function at

x = 5

gives

g(5) = 3(5) - 4 = 11

.

Adım Adım Çözüm

1

Determine the slope of the function

g

.

The slope of

g

is

3

.

Parallel lines in the coordinate plane have equal slopes, and the slope of the graph of

f

is given as

3

.

2

Write the equation for

g(x)

.

g(x) = 3x - 4

Using the slope-intercept form

g(x) = mx + b

with slope

m = 3

and

y

-intercept

b = g(0) = -4

.

3

Evaluate

g(5)

.

11

Substitute

x = 5

into the equation for

g(x)

to get

g(5) = 3(5) - 4 = 11

.

Anahtar Kavram

Parallel lines have the same slope. A linear function can be written in slope-intercept form,

y = mx + b

, where

m

is the slope and

b

is the

y

-intercept.

Soru 1255Soru →

A local library has two types of study rooms: small rooms, which can accommodate up to $3$ people, and large rooms, which can accommodate up to $8$ people. The library has a total of $15$ study rooms. If the maximum capacity of all the study rooms combined is $80$ people, how many large study rooms does the library have?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The correct answer is 7.

By representing the number of small rooms as

s

and large rooms as

l

, we set up the system of linear equations

s + l = 15

and

3s + 8l = 80

. Solving the first equation for

s

gives

s = 15 - l

. Substituting this into the second equation yields

3(15 - l) + 8l = 80

. Distributing and combining like terms results in

45 + 5l = 80

. Subtracting 45 from both sides gives

5l = 35

, which simplifies to

l = 7

. Thus, there are 7 large study rooms.

Adım Adım Çözüm

1

Set up the system of linear equations representing the total number of rooms and their total capacity.

s + l = 15

and

3s + 8l = 80

We define

s

as the number of small study rooms and

l

as the number of large study rooms. The total number of rooms is 15, and the total capacity is 80.

2

Express the number of small rooms,

s

, in terms of the number of large rooms,

l

, using the first equation.

s = 15 - l

This allows for substitution into the capacity equation so that we can solve directly for the number of large study rooms.

3

Substitute the expression for

s

into the capacity equation and solve the resulting single-variable equation for

l

.

3(15 - l) + 8l = 80 \implies 45 - 3l + 8l = 80 \implies 45 + 5l = 80 \implies 5l = 35 \implies l = 7

Substituting

s

eliminates one variable, leaving a linear equation in terms of

l

that can be solved using standard algebraic steps.

Anahtar Kavram

Solving systems of linear equations in a real-world context using substitution.

Soru 1256Soru →

For the constants $p$ and $q$ , the given system of linear equations in $x$ and $y$ has infinitely many solutions:

\begin{aligned} px - 3y &= q \\ 4x + (p - 7)y &= 12 \end{aligned}

Which of the following is a possible value of $q$ ?

3

6

9

16

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

9

For the system of linear equations to have infinitely many solutions, the equations must be equivalent. This means the ratio of the coefficients of

x

, the coefficients of

y

, and the constant terms must all be equal:

\frac{p}{4} = \frac{-3}{p-7} = \frac{q}{12}

. Cross-multiplying the first two ratios gives

p(p-7) = -12

, which simplifies to

p^2 - 7p + 12 = 0

. Factoring this quadratic equation gives

(p-3)(p-4) = 0

, which yields

p = 3

or

p = 4

. Using the relation between the first and third ratios, we have

\frac{q}{12} = \frac{p}{4}

, which simplifies to

q = 3p

. Substituting the possible values of

p

gives

q = 3(3) = 9

or

q = 3(4) = 12

. Since 9 is one of the options, it is a possible value of

q

.

Adım Adım Çözüm

1

Set up the condition for infinitely many solutions by equating the ratios of the coefficients and the constants.

The coefficients and constants must satisfy the proportion:

\frac{p}{4} = \frac{-3}{p - 7} = \frac{q}{12}

For a system of two linear equations to have infinitely many solutions, the two equations must represent the exact same line, meaning all corresponding coefficients and constant terms must be proportional.

2

Solve the equation formed by the first two ratios to find the possible values of the parameter

p

.

\frac{p}{4} = \frac{-3}{p - 7} \Rightarrow p(p - 7) = -12 \Rightarrow p^2 - 7p + 12 = 0

. Factoring the quadratic yields

(p - 3)(p - 4) = 0

, which gives

p = 3

or

p = 4

.

This step determines the values of

p

for which the two lines are parallel (i.e., they have equal slopes).

3

Find the corresponding values of

q

using the relation between the first and third ratios.

\frac{p}{4} = \frac{q}{12} \Rightarrow q = 3p

. Substituting the values of

p

:
- If

p = 3

, then

q = 3(3) = 9

.
- If

p = 4

, then

q = 3(4) = 12

.

This step ensures that the parallel lines are coincident by equating their y-intercepts.

4

Compare the possible values of

q

with the given options.

The value 9 is a possible value of

q

and matches one of the choices.

To identify which of the two mathematically valid solutions for

q

is listed in the multiple-choice options.

Anahtar Kavram

Conditions for infinitely many solutions in a system of linear equations

Alternatif Yöntem

Instead of using ratios directly, we can write both equations in slope-intercept form:

y = mx + b

. For the first equation,

px - 3y = q \Rightarrow y = \frac{p}{3}x - \frac{q}{3}

. For the second equation,

4x + (p-7)y = 12 \Rightarrow y = -\frac{4}{p-7}x + \frac{12}{p-7}

(assuming

p \neq 7

). For the system to have infinitely many solutions, the two lines must have the same slope and the same y-intercept. Equating the slopes gives

\frac{p}{3} = -\frac{4}{p-7}

, which simplifies to

p^2 - 7p + 12 = 0

, yielding

p = 3

or

p = 4

. Equating the y-intercepts gives

-\frac{q}{3} = \frac{12}{p-7} \Rightarrow q = -\frac{36}{p-7}

. Substituting

p=3

gives

q = -\frac{36}{-4} = 9

, and substituting

p=4

gives

q = -\frac{36}{-3} = 12

.

Tahmini Süre:3m 0s

Soru 1257Soru →

A small factory manufactures two types of toys: wood blocks and toy cars. Let $x$ represent the number of wood blocks produced daily, and let $y$ represent the number of toy cars produced daily. The daily production must satisfy the following constraints:

* The total number of toys produced daily cannot exceed 40: $x + y \le 40$
* Each wood block requires 2 minutes of painting, and each toy car requires 1 minute of painting. The total daily painting time is at most 60 minutes: $2x + y \le 60$
* The number of wood blocks produced cannot exceed the number of toy cars produced by more than 15: $y \ge x - 15$

If the factory must produce a non-negative number of both types of toys, what is the maximum possible number of wood blocks the factory can produce daily?

20

25

30

40

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

The maximum possible number of wood blocks the factory can produce daily is 25.

The correct answer is the value that represents the maximum

x

-coordinate within the feasible region. By graphing the inequalities, we find the vertices of the shaded region are

(0, 0)

,

(15, 0)

,

(25, 10)

,

(20, 20)

, and

(0, 40)

. The maximum

x

-value among these vertices is 25, which occurs at the intersection of the painting time constraint and the demand constraint.

Adım Adım Çözüm

1

Identify the system of inequalities representing the constraints.

The system of inequalities is:
1)

x + y \le 40

2)

2x + y \le 60

3)

y \ge x - 15

4)

x \ge 0

5)

y \ge 0

This establishes the boundaries of the feasible region on the coordinate plane.

2

Find the boundary intersection points (vertices of the feasible region) that could maximize

x

.

The relevant intersections are:
- The intersection of (1) and (2):

x + y = 40

and

2x + y = 60 \implies x = 20, y = 20

.
- The intersection of (2) and (3):

2x + y = 60

and

y = x - 15 \implies 2x + (x - 15) = 60 \implies 3x = 75 \implies x = 25, y = 10

.
- The intersection of (3) and the x-axis (

y = 0

):

0 = x - 15 \implies x = 15, y = 0

.

The maximum value of a variable in a bounded linear feasibility region must occur at one of the vertices of the region.

3

Test the vertices to verify they satisfy all inequalities in the system.

- For

(20, 20)

:

20 + 20 \le 40

(True),

2(20) + 20 \le 60

(True),

20 \ge 20 - 15

(True).
- For

(25, 10)

:

25 + 10 = 35 \le 40

(True),

2(25) + 10 = 60 \le 60

(True),

10 \ge 25 - 15

(True).
- For

(15, 0)

:

15 + 0 \le 40

(True),

2(15) + 0 \le 60

(True),

0 \ge 15 - 15

(True).

This ensures that the intersection points actually lie within the feasible region defined by all constraints.

4

Compare the

x

-values of the valid vertices to find the maximum.

The candidate

x

-values are 20, 25, and 15. The maximum value is 25.

This determines the absolute maximum coordinate value within the bounded region.

Anahtar Kavram

Solving systems of linear inequalities by identifying the vertices of the feasible region to optimize a coordinate value.

Soru 1258Soru →

Based on the rules of Standard English conventions, what punctuation mark should be inserted into both blanks to correctly separate the items in the complex list?

Aşağıdaki boşlukları doldurun

In analyzing the migration patterns of monarch butterflies, researchers tracked coordinates across three distinct staging areas: Angangueo, Mexico Sierra Vista, Arizona and Santa Cruz, California.

Cevabı ve açıklamayı göster

Cevap

A semicolon (;) must be placed in both blanks.

The correct punctuation is a semicolon (;). Under the conventions of Standard English, semicolons are used as 'super-commas' to separate items in a list when one or more of those items already contain internal punctuation, such as commas separating cities and states/countries.

Adım Adım Çözüm

1

Identify that the sentence contains a list of geographical locations, each consisting of a city and a state or country (e.g., 'Angangueo, Mexico').

Recognized that these list items already contain internal punctuation (commas separating the city and state/country).

Understanding the structure of the list items is necessary to determine the correct separator.

2

Determine the punctuation required to separate list items that contain internal commas.

Standard English conventions require using semicolons rather than commas to separate the items to prevent confusion.

Using commas as separators would create a confusing string of commas, making it unclear where each item begins and ends.

Anahtar Kavram

Using Semicolons in a Complex List

Soru 1259Soru →

If the expression $5(x^2 - 2x) - 3(x^2 - 4x)$ is rewritten in the form $ax^2 + bx$ , where $a$ and $b$ are constants, what is the value of $a$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The value of

a

is

2

.

Distributing the constants outside the parentheses yields

5x^2 - 10x - 3x^2 + 12x

. Combining the

x^2

terms (

5x^2 - 3x^2 = 2x^2

) and the

x

terms (

-10x + 12x = 2x

) gives the simplified equivalent expression

2x^2 + 2x

. Comparing this directly to the form

ax^2 + bx

, we find that the coefficient

a

is

2

.

Adım Adım Çözüm

1

Distribute the coefficients to the terms inside the parentheses.

5x^2 - 10x - 3x^2 + 12x

To expand the expression so that like terms can be combined.

2

Combine the coefficients of

x^2

and the coefficients of

x

.

2x^2 + 2x

To write the expression in simplified form.

3

Identify the value of

a

by matching the simplified expression with

ax^2 + bx

.

a = 2

The constant

a

represents the coefficient of the

x^2

term.

Anahtar Kavram

Simplifying polynomial expressions by distributing constants and combining like terms.

Tahmini Süre:45s

Soru 1260Soru →

If $\frac{1}{4}(8x - 12) - \frac{2}{3}(3 - 6x) = 13$ , what is the value of $3x - 2$ ?

2

3

7

-29

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

7

Distributing the coefficients yields

\frac{1}{4}(8x - 12) = 2x - 3

and

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

. Combining these terms gives the simplified equation

6x - 5 = 13

. Adding

5

to both sides yields

6x = 18

, and dividing by

6

gives

x = 3

. Substituting

x = 3

into the expression

3x - 2

results in

3(3) - 2 = 7

.

Adım Adım Çözüm

1

Distribute the fraction coefficients to the terms inside the parentheses.

\frac{1}{4}(8x - 12) = 2x - 3

and

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

.

This simplifies the equation by removing the parentheses.

2

Combine like terms on the left side of the equation.

(2x - 3) + (-2 + 4x) = 6x - 5

, transforming the equation to

6x - 5 = 13

.

Grouping the variable terms and constant terms is necessary to isolate the variable.

3

Isolate the variable by adding 5 to both sides and dividing by 6.

6x = 18 \Rightarrow x = 3

.

This determines the value of the variable.

4

Substitute the value of

x

into the expression

3x - 2

.

3(3) - 2 = 7

.

The question asks for the value of the expression

3x - 2

, not the value of

x

itself.

Anahtar Kavram

Linear Equations in One Variable

Alternatif Yöntem

An alternative method is to multiply both sides of the equation by the least common multiple of the denominators

4

and

3

, which is

12

. Multiplying the entire equation by

12

yields

3(8x - 12) - 8(3 - 6x) = 156

. Distributing gives

24x - 36 - 24 + 48x = 156

, which simplifies to

72x - 60 = 156

. Adding

60

gives

72x = 216

, so

x = 3

. Substituting

x = 3

into

3x - 2

gives

7

.

Tahmini Süre:1m 30s

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