Tüm alıştırma soruları

Soru 1321Soru →

The rain shadow effect occurs when a mountain range forces moist air upward, causing it to cool and release precipitation on the windward slope. By the time the air mass crosses the summit and descends the leeward side, it is depleted of ______ this dry air creates a semi-arid region on the far side of the mountains.

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

moisture

moisture, although

moisture;

moisture,

Cevabı ve açıklamayı göster

Cevap: moisture;

Cevap

moisture;

The correct option ends with a semicolon, which is standard punctuation used to connect two closely related independent clauses without a coordinating conjunction.

Adım Adım Çözüm

1

Identify the boundary between the two independent clauses in the sentence.

The two clauses are 'By the time the air mass crosses the summit and descends the leeward side, it is depleted of moisture' and 'this dry air creates a semi-arid region on the far side of the mountains.'

Both parts can stand alone as complete sentences, meaning they are independent clauses that must be linked properly.

2

Evaluate the grammar rules for linking independent clauses.

Two independent clauses cannot be joined with just a comma (comma splice) or with no punctuation (run-on). They must be linked with a semicolon, a period, or a comma plus a coordinating conjunction.

This determines the allowable punctuation choices.

3

Choose the option that correctly separates the clauses without introducing logical errors.

The option containing a semicolon is the only choice that correctly separates the two independent clauses without error.

It resolves the clause boundary correctly and maintains standard English conventions.

Anahtar Kavram

Clause boundaries and linking using semicolons to connect independent clauses.

Soru 1322Soru →

Oven A preheats at a constant rate of $15$ degrees Celsius per minute, starting from an initial temperature of $25$ degrees Celsius. Oven B starts preheating $5$ minutes after Oven A begins, starting from an initial temperature of $20$ degrees Celsius and preheating at a constant rate of $20$ degrees Celsius per minute. If both ovens continue to preheat, how many minutes after Oven A begins preheating will both ovens reach the same temperature?

Cevabı ve açıklamayı göster

Cevap: 21

Cevap

The ovens will reach the same temperature

21

minutes after Oven A begins preheating.

To find the number of minutes after Oven A begins preheating when both ovens reach the same temperature, we can write an equation in terms of

t

, the time in minutes since Oven A started. Oven A starts at

25

degrees Celsius and increases by

15

degrees per minute, so its temperature is

25 + 15t

. Oven B starts

5

minutes later, meaning it preheats for

t - 5

minutes. Starting from

20

degrees Celsius and preheating at

20

degrees per minute, Oven B's temperature is

20 + 20(t - 5)

. Setting these two expressions equal gives the equation

25 + 15t = 20 + 20(t - 5)

. Distributing

20

yields

25 + 15t = 20t - 80

. Isolating

t

gives

5t = 105

, which results in

t = 21

.

Adım Adım Çözüm

1

Set up expressions representing the temperature of each oven

t

minutes after Oven A begins preheating.

Oven A:

25 + 15t

; Oven B:

20 + 20(t - 5)

Since Oven B starts

5

minutes after Oven A, it has been preheating for

t - 5

minutes.

2

Set the temperature expressions equal to find the time at which they reach the same temperature.

25 + 15t = 20 + 20(t - 5)

We want to find the value of

t

where the temperatures of the two ovens are equal.

3

Distribute and simplify the equation.

25 + 15t = 20t - 80

Distributing the

20

across

(t - 5)

yields

20t - 100

. Combining the constant terms gives

20 - 100 = -80

.

4

Solve for

t

by isolating the variable term.

5t = 105

, which gives

t = 21

Subtract

15t

and add

80

to both sides, then divide by

5

.

Anahtar Kavram

Solving linear equations in one variable that model real-world situations with a time delay.

Soru 1323Soru →

Consider the system of linear equations below, where $k$ is a constant:

\begin{aligned} ky - 3x &= 4 \\ 4y - (k-1)x &= 3 \end{aligned}

If the system of equations has no solution, what is the sum of all possible values of $k$ ?

1

-1

4

7

Cevabı ve açıklamayı göster

Cevap: 1

Cevap

1

For the system of equations to have no solution, the lines must be parallel. In slope-intercept form, the equations are

y = \frac{3}{k}x + \frac{4}{k}

and

y = \frac{k-1}{4}x + \frac{3}{4}

. Setting the slopes equal gives

\frac{3}{k} = \frac{k-1}{4}

, which simplifies to the quadratic equation

k^2 - k - 12 = 0

. Factoring this equation yields

(k-4)(k+3) = 0

, giving

k = 4

and

k = -3

. Since both values yield different y-intercepts for the two lines, they both result in parallel lines with no intersection. The sum of these values is

4 + (-3) = 1

.

Adım Adım Çözüm

1

Express both equations in slope-intercept form (

y = mx + b

) to find their slopes.

The first equation becomes

y = \frac{3}{k}x + \frac{4}{k}

(for

k \neq 0

). The second equation becomes

y = \frac{k-1}{4}x + \frac{3}{4}

.

For a system of linear equations to have no solution, the lines must be parallel, meaning they have the same slope but different y-intercepts.

2

Set the slopes of the two lines equal to each other to solve for

k

.

\frac{3}{k} = \frac{k-1}{4} \implies k(k-1) = 12 \implies k^2 - k - 12 = 0

.

Equating the slopes allows us to find the values of

k

that make the lines parallel.

3

Solve the quadratic equation

k^2 - k - 12 = 0

by factoring.

(k-4)(k+3) = 0 \implies k = 4

or

k = -3

.

This determines the specific values of

k

that make the slopes equal.

4

Verify that both values of

k

produce different y-intercepts (so the lines do not coincide) and calculate their sum.

For

k = 4

, the y-intercepts are

\frac{4}{4} = 1

and

\frac{3}{4}

, which are different. For

k = -3

, the y-intercepts are

-\frac{4}{3}

and

\frac{3}{4}

, which are also different. The sum of the values of

k

is

4 + (-3) = 1

.

We must confirm the lines are parallel and not identical, then compute the required sum.

Anahtar Kavram

Systems of linear equations with no solution represent parallel lines with equal slopes and unequal y-intercepts.

Soru 1324Soru →

For all real numbers $x$ and $y$ , the expression $(4x^3y^2)^2(2x^2y)$ is equivalent to which of the following?

8x^8y^5

32x^{12}y^4

32x^8y^5

32x^7y^5

Cevabı ve açıklamayı göster

Cevap: $32x^8y^5$

Cevap

The correct equivalent expression is

32x^8y^5

.

To find the equivalent expression, we first apply the power of a product rule to the first term:

(4x^3y^2)^2 = 4^2(x^3)^2(y^2)^2 = 16x^6y^4

. Next, we multiply this result by the second term:

(16x^6y^4)(2x^2y)

. Multiplying the coefficients gives

16 \times 2 = 32

. Adding the exponents of the same base variables gives

x^{6+2} = x^8

and

y^{4+1} = y^5

. This results in

32x^8y^5

.

Adım Adım Çözüm

1

Apply the power of a product rule to square the first expression:

(4x^3y^2)^2

.

16x^6y^4

When raising a product to a power, raise each factor to that power:

4^2 = 16

,

(x^3)^2 = x^{3 \times 2} = x^6

, and

(y^2)^2 = y^{2 \times 2} = y^4

.

2

Multiply the simplified first expression by the second expression:

(16x^6y^4)(2x^2y)

.

32x^8y^5

Multiply the numerical coefficients (

16 \times 2 = 32

) and add the exponents of variables with matching bases (

x^{6+2} = x^8

and

y^{4+1} = y^5

).

Anahtar Kavram

Simplifying equivalent algebraic expressions using rules of exponents

Tahmini Süre:45s

Soru 1325Soru →

Although the local association of independent organic farmers received numerous complaints from consumers about the delayed distribution of agricultural goods during the winter harvest, the organization refused to alter _______ established policy of direct-to-consumer sales. The board of directors argued that maintaining this business model was essential for preserving the financial independence of the small-scale farms involved.

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

their

its

it's

its'

Cevabı ve açıklamayı göster

Cevap: its

Cevap

The singular possessive pronoun 'its' correctly agrees in number with the singular antecedent 'organization'.

The singular possessive pronoun 'its' correctly refers to the singular collective noun antecedent 'organization'. Since the noun is singular and acts as a possessive modifier for 'established policy', the singular possessive form without an apostrophe is correct.

Adım Adım Çözüm

1

Identify the pronoun's antecedent in the sentence.

The antecedent is the singular noun 'organization' (which refers back to the 'association').

A pronoun must agree in number with the noun it replaces or refers to.

2

Determine the grammatical role and case of the required pronoun.

The blank precedes the noun phrase 'established policy', indicating a need for a possessive pronoun.

A possessive pronoun is required to show ownership of the policy.

3

Select the grammatically correct singular possessive pronoun.

The word 'its' is the correct singular possessive pronoun.

It agrees in number with 'organization' and is spelled correctly without an apostrophe.

Anahtar Kavram

Pronoun-Antecedent Agreement and Case

Soru 1326Soru →

A community garden has a total area of $1,200$ square feet available for planting tomatoes and squash. Each tomato plant requires $8$ square feet of space, and each squash plant requires $15$ square feet of space. The garden coordinators want to plant at least $100$ plants in total, but they want the number of squash plants to be no more than twice the number of tomato plants. If $t$ represents the number of tomato plants and $s$ represents the number of squash plants, which of the following systems of inequalities represents this situation?

\begin{cases} 8t + 15s \geq 1,200 \\ t + s \leq 100 \\ s \leq 2t \end{cases}

\begin{cases} 8t + 15s \leq 1,200 \\ t + s \geq 100 \\ t \leq 2s \end{cases}

\begin{cases} 8t + 15s \leq 1,200 \\ t + s \geq 100 \\ s \leq 2t \end{cases}

\begin{cases} 8t + 15s \leq 1,200 \\ t + s \geq 100 \\ s \geq 2t \end{cases}

Cevabı ve açıklamayı göster

Cevap: $\begin{cases} 8t + 15s \leq 1,200 \\ t + s \geq 100 \\ s \leq 2t \end{cases}$

Cevap

The system of inequalities containing the constraints

8t + 15s \leq 1,200

,

t + s \geq 100

, and

s \leq 2t

.

The correct system of inequalities maps the garden's constraints accurately:

8t + 15s \leq 1,200

limits the space usage to a maximum of

1,200

square feet;

t + s \geq 100

ensures there are at least

100

plants in total; and

s \leq 2t

restricts the number of squash plants to at most twice the number of tomato plants.

Adım Adım Çözüm

1

Translate the area constraint.

8t + 15s \leq 1,200

Since the total area available is

1,200

square feet and each tomato plant requires

8

square feet while each squash plant requires

15

square feet, the combined space used by

t

tomato plants and

s

squash plants cannot exceed

1,200

.

2

Translate the total plant count constraint.

t + s \geq 100

The coordinators want at least

100

plants, meaning the sum of the tomato and squash plants must be greater than or equal to

100

.

3

Translate the plant ratio constraint.

s \leq 2t

The number of squash plants (

s

) must be no more than (less than or equal to) twice the number of tomato plants (

2t

).

4

Combine the inequalities into a single system.

\begin{cases} 8t + 15s \leq 1,200 \\ t + s \geq 100 \\ s \leq 2t \end{cases}

Grouping these three constraints forms the mathematical model that represents the given scenario.

Anahtar Kavram

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

Soru 1327Soru →

If the expression $\frac{2x + 1}{x - 3} - \frac{x - 2}{x + 1}$ is rewritten in the equivalent form $a + \frac{bx + c}{x^2 - 2x - 3}$ for all $x > 3$ , where $a$ , $b$ , and $c$ are constants, what is the value of $a + b + c$ ?

-1

11

9

19

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

9

The correct answer is 9. Finding a common denominator yields the combined numerator

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

. Expanding these terms gives

(2x^2 + 3x + 1) - (x^2 - 5x + 6) = x^2 + 8x - 5

. Dividing

x^2 + 8x - 5

by

x^2 - 2x - 3

results in a quotient of

1

and a remainder of

10x - 2

. Thus, the expression is equivalent to

1 + \frac{10x - 2}{x^2 - 2x - 3}

, which gives

a = 1

,

b = 10

, and

c = -2

. The sum

a + b + c = 1 + 10 - 2 = 9

.

Adım Adım Çözüm

1

Find a common denominator for the two rational expressions.

The common denominator is

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

. The combined expression is

\frac{(2x + 1)(x + 1) - (x - 2)(x - 3)}{x^2 - 2x - 3}

.

To combine the fractions, we need to express them with a common denominator.

2

Expand and simplify the numerator.

(2x + 1)(x + 1) = 2x^2 + 3x + 1

and

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

. Subtracting them gives

(2x^2 + 3x + 1) - (x^2 - 5x + 6) = x^2 + 8x - 5

.

Simplifying the numerator allows us to express the combined fraction as a single polynomial over the denominator.

3

Perform polynomial division or rewrite the numerator to match the form

a + \frac{bx + c}{x^2 - 2x - 3}

.

Rewriting the numerator:

x^2 + 8x - 5 = 1(x^2 - 2x - 3) + 10x - 2

. Thus, the expression becomes

1 + \frac{10x - 2}{x^2 - 2x - 3}

.

This separates the rational expression into a constant integer and a proper rational fraction.

4

Identify the values of

a

,

b

, and

c

, and find their sum.

a = 1

,

b = 10

, and

c = -2

. The sum is

a + b + c = 1 + 10 + (-2) = 9

.

We compare the coefficients from our result to the given form and calculate the requested sum.

Anahtar Kavram

Combining rational expressions and rewriting them using polynomial division or algebraic manipulation.

Tahmini Süre:2m 30s

Soru 1328Soru →

The passage below discusses a laboratory setup used to study marine organisms. Which verb completes the sentence so that it conforms to the conventions of parallel structure?

Aşağıdaki boşlukları doldurun

To investigate the ecological consequences of rising ocean acidity, marine biologist Dr. Leo Vance developed a specialized aquarium system. In experiments simulating future atmospheric conditions, this advanced apparatus dynamically adjusts carbon dioxide levels in the water, monitors subtle changes in shell calcification, and the swimming patterns of larval oysters over several weeks.

Cevabı ve açıklamayı göster

Cevap

Any third-person singular present-tense verb that logically fits the context (such as 'tracks', 'records', 'measures', 'observes', 'analyzes', or 'evaluates').

The sentence contains a list of three actions performed by 'this advanced apparatus': 'adjusts,' 'monitors,' and the blank verb. To maintain parallel structure, the blank must contain a verb in the third-person singular present tense (e.g., 'tracks' or 'records') to match the preceding verbs.

Adım Adım Çözüm

1

Identify the parallel elements in the sentence.

The sentence lists three parallel actions performed by the subject 'this advanced apparatus': 'adjusts,' 'monitors,' and the verb in the blank.

To maintain parallel structure, items in a list must share the same grammatical form.

2

Determine the grammatical form of the existing verbs.

Both 'adjusts' and 'monitors' are third-person singular verbs in the present tense.

The final verb in the list must match this grammatical form to be parallel.

3

Select a present-tense, third-person singular verb that fits the context.

Verbs like 'tracks', 'records', or 'observes' fit the context of monitoring larval behavior and match the grammatical pattern.

This satisfies the requirement for parallel structure.

Anahtar Kavram

Parallel Structure

Tahmini Süre:1m 0s

Soru 1329Soru →

During a 2014 excavation in Egypt, a team of archaeologists discovered a previously unknown tomb containing several well-preserved limestone statues and pottery vessels. The researchers carefully documented the size, placement, and condition of each artifact ______ they did not remove any of the delicate items from the site until the structural integrity of the chamber was fully secured.

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

, but

, however,

but

because

Cevabı ve açıklamayı göster

Cevap: , but

Cevap

The option containing a comma followed by the coordinating conjunction 'but'

The correct option uses a comma and the coordinating conjunction 'but' to connect the two independent clauses, establishing a logical contrast between the act of documenting the artifacts and the decision to leave them on-site.

Adım Adım Çözüm

1

Identify the boundary between the clauses.

The text contains two independent clauses: 'The researchers carefully documented the size, placement, and condition of each artifact' and 'they did not remove any of the delicate items...'

Knowing where the clauses meet helps determine what kind of punctuation or conjunction is needed to link them.

2

Analyze the relationship between the two independent clauses.

The relationship is contrastive, as the researchers performed careful documentation but delayed the physical removal of the items.

This helps select a linking option that is both grammatically correct and logically appropriate.

3

Evaluate the choices for grammatical correctness and meaning.

Only the option with a comma and the coordinating conjunction 'but' correctly and logically links the two independent clauses without creating a comma splice or run-on sentence.

This identifies the correct choice according to Standard English conventions.

Anahtar Kavram

Linking independent clauses using coordinating conjunctions and appropriate punctuation.

Tahmini Süre:45s

Soru 1330Soru →

For each real number

k

except

2

, the system of equations below has a unique solution

(x, y)

.

\begin{aligned} kx + (k + 2)y &= 3k + 1 \\ (k - 1)x + ky &= 2k - 1 \end{aligned}

If the solution

(x, y)

to the system also satisfies the equation

x + 2y = 6

, what is the value of

k

?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

k

is 4.

Subtracting the second equation from the first equation yields

x + 2y = k + 2

. Since the solution must also satisfy

x + 2y = 6

, we set

k + 2 = 6

, which gives

k = 4

.

Adım Adım Çözüm

1

Subtract the second equation from the first equation.

x + 2y = k + 2

To find a direct linear combination of

x

and

y

that can be compared directly to the target equation.

2

Equate the resulting expression to the target equation

x + 2y = 6

.

k + 2 = 6

Since the solution

(x, y)

must satisfy

x + 2y = 6

, the value of the linear combination

x + 2y

from the system must equal

6

.

3

Solve for

k

by subtracting 2 from both sides.

k = 4

Isolating the variable

k

yields the final solution.

Anahtar Kavram

Solving systems of linear equations with parameter coefficients by identifying algebraic structure and linear combinations.

Alternatif Yöntem

Solve the system for

x

and

y

in terms of

k

using elimination. Multiplying the first equation by

(k-1)

and the second by

k

, and then subtracting them yields

y = \frac{k^2 - k - 1}{k - 2}

. Substituting this back gives

x = \frac{-k^2 + 2k - 2}{k - 2}

. Substituting these expressions into the equation

x + 2y = 6

results in

\frac{k^2 - 4}{k - 2} = 6

. For

k \neq 2

, factoring

k^2 - 4

as

(k-2)(k+2)

allows simplification to

k + 2 = 6

, which yields

k = 4

.

Tahmini Süre:2m 0s

Soru 1331Soru →

A solution to a system of inequalities in the $xy$ -plane is represented by the point $(2, y)$ . If the system is defined by:

y \geq 4x - 2

y \leq -2x + 10

what is the value of $y$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The value of

y

must be exactly 6 because it is bounded both below and above by 6.

Substituting

x = 2

into the system of inequalities yields two constraints:

y \geq 4(2) - 2

, which simplifies to

y \geq 6

, and

y \leq -2(2) + 10

, which simplifies to

y \leq 6

. The only value of

y

that satisfies both statements is 6.

Adım Adım Çözüm

1

Substitute

x = 2

into the first inequality

y \geq 4x - 2

y \geq 6

To determine the constraint on the

y

-value from the first inequality at the coordinate

x = 2

.

2

Substitute

x = 2

into the second inequality

y \leq -2x + 10

y \leq 6

To determine the constraint on the

y

-value from the second inequality at the coordinate

x = 2

.

3

Find the value of

y

that satisfies both

y \geq 6

and

y \leq 6

y = 6

For a value to be simultaneously greater than or equal to 6 and less than or equal to 6, it must be equal to 6.

Anahtar Kavram

Evaluating a system of linear inequalities at a specific coordinate to determine a unique boundary value

Tahmini Süre:45s

Soru 1332Soru →

To mitigate the potentially destructive structural oscillations caused by high-velocity wind gusts and seismic activity, modern skyscrapers frequently incorporate a mechanism known as a tuned mass damper. This apparatus consists of a massive steel or concrete block suspended near the top of the ______ when the building sways, the block moves in the opposite direction to dissipate the kinetic energy.

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

structure; when

structure, when

structure when

structure, but when

Cevabı ve açıklamayı göster

Cevap: structure; when

Cevap

The correct option is the one that reads 'structure; when'.

The correct option correctly uses a semicolon to separate two independent clauses. Semicolons are appropriate punctuation marks for linking closely related independent clauses without a coordinating conjunction.

Adım Adım Çözüm

1

Identify the clause boundaries in the passage.

The passage contains two independent clauses: the first ends with 'structure' and the second begins with 'when the building sways'.

Locating the boundary is necessary to determine how the two clauses should be linked.

2

Analyze the logical relationship between the clauses.

The second clause describes the function of the object described in the first clause. There is no contrast between them.

This determines whether a contrastive coordinator like 'but' is appropriate.

3

Evaluate the grammar of each option.

A semicolon correctly links the clauses. A comma creates a comma splice, no punctuation creates a run-on, and 'but' introduces an illogical contrast.

This identifies the option that conforms to standard English conventions.

Anahtar Kavram

Clause Boundaries and Linking

Tahmini Süre:1m 15s

Soru 1333Soru →

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

a(2x - 9) - 3(x - b) = 7x + 12

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

-19

10

11

19

Cevabı ve açıklamayı göster

Cevap: 19

Cevap

19

To find the value of

b

for which the equation has infinitely many solutions, we expand the left side to get

2ax - 9a - 3x + 3b = 7x + 12

. Grouping the terms by variable yields

(2a - 3)x + (3b - 9a) = 7x + 12

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

x

and the constant term on both sides must be equal. Setting the

x

-coefficients equal gives

2a - 3 = 7

, which simplifies to

2a = 10

and

a = 5

. Setting the constant terms equal gives

3b - 9a = 12

. Substituting

a = 5

into this equation yields

3b - 9(5) = 12

, which simplifies to

3b - 45 = 12

. Adding

45

to both sides gives

3b = 57

, which results in

b = 19

.

Adım Adım Çözüm

1

Expand and simplify the left side of the equation.

(2a - 3)x + (3b - 9a) = 7x + 12

To compare coefficients with the right side of the equation, we need to group the terms on the left side.

2

Set the coefficients of

x

on both sides equal to each other to solve for

a

.

2a - 3 = 7 \implies 2a = 10 \implies a = 5

For an equation to have infinitely many solutions, the coefficients of the variable on both sides must be equal.

3

Set the constant terms on both sides equal to each other, substitute the value of

a

, and solve for

b

.

3b - 9a = 12 \implies 3b - 9(5) = 12 \implies 3b - 45 = 12 \implies 3b = 57 \implies b = 19

For an equation to have infinitely many solutions, the constant terms on both sides must also be equal.

Anahtar Kavram

Linear equations with infinitely many solutions require the coefficients of the variable to be equal and the constant terms to be equal on both sides of the equation.

Soru 1334Soru →

A delivery truck carries a total of $x$ bags of organic fertilizer and $y$ bags of compost. Each bag of fertilizer weighs $45$ pounds, and each bag of compost weighs $30$ pounds. When the truck is loaded to exactly $80\%$ of its maximum payload capacity, the relationship between $x$ and $y$ can be represented by the equation $3x + 2y = 240$ . What is the truck's maximum payload capacity, in pounds?

2,880

3,000

3,600

4,500

Cevabı ve açıklamayı göster

Cevap: $4,500$

Cevap

The maximum payload capacity of the truck is

4,500

pounds.

The total weight of the bags loaded on the truck is represented by the expression

45x + 30y

pounds. Factoring out

15

from this expression gives

15(3x + 2y)

. We are given that when the truck is loaded to exactly

80\%

of its capacity, the relationship between

x

and

y

is

3x + 2y = 240

. Substituting

240

for

3x + 2y

in the weight expression shows that the weight of the load at

80\%

capacity is

15 \times 240 = 3,600

pounds. To find the maximum capacity, we set up the equation

0.80 \times C = 3,600

, where

C

is the maximum payload capacity. Solving for

C

yields

C = \frac{3,600}{0.80} = 4,500

pounds.

Adım Adım Çözüm

1

Write the expression for the total weight of the bags in terms of

x

and

y

.

Total weight

W = 45x + 30y

pounds.

Each of the

x

bags of fertilizer weighs

45

pounds and each of the

y

bags of compost weighs

30

pounds.

2

Relate the total weight expression to the given equation for

80\%

capacity.

45x + 30y = 15(3x + 2y)

Factoring out the greatest common divisor of

45

and

30

, which is

15

, allows us to write the weight expression in terms of the left-hand side of the given equation

3x + 2y = 240

.

3

Calculate the actual weight of the load when the truck is at

80\%

capacity.

W = 15 \times 240 = 3,600

pounds.

Since the relationship at

80\%

capacity is

3x + 2y = 240

, we can substitute

240

for

3x + 2y

in the factored weight expression.

4

Calculate the maximum payload capacity of the truck.

Maximum capacity

C = \frac{3,600}{0.80} = 4,500

pounds.

The weight of

3,600

pounds represents

80\%

of the maximum payload capacity, so dividing the actual weight by

0.80

gives the total capacity.

Anahtar Kavram

Translating a real-world scenario into a linear equation and interpreting its coefficients and constants to solve for unknown quantities.

Soru 1335Soru →

In 1956, experimental physicist Chien-Shiung Wu performed a landmark experiment to test the law of conservation of parity, which states that physical systems and their mirror images behave identically. Wu ______ cooled radioactive cobalt-60 atoms to near absolute zero, allowing her to observe that beta particles were emitted asymmetrically. This groundbreaking result disproved a widely accepted theory, fundamentally altering the course of modern particle physics.

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

, working in collaboration with the National Bureau of Standards—

—working in collaboration with the National Bureau of Standards—

—working in collaboration with the National Bureau of Standards,

, working in collaboration with the National Bureau of Standards

Cevabı ve açıklamayı göster

Cevap: —working in collaboration with the National Bureau of Standards—

Cevap

The option that uses matching em dashes to set off the nonessential phrase 'working in collaboration with the National Bureau of Standards'.

The phrase 'working in collaboration with the National Bureau of Standards' is a nonessential parenthetical modifier that provides additional context about the physicist Chien-Shiung Wu but is not grammatically necessary for the main clause. Nonessential elements must be set off from the rest of the sentence using a pair of matching punctuation marks, such as two commas, two parentheses, or two em dashes. The correct option uses matching em dashes to set off this parenthetical information, which maintains grammatical consistency and clarity.

Adım Adım Çözüm

1

Identify the role of the phrase 'working in collaboration with the National Bureau of Standards' within the sentence.

The phrase is a nonessential parenthetical modifier that provides extra information about Chien-Shiung Wu but is not essential to the main clause.

Determining whether the phrase is essential or nonessential dictates what punctuation is required to set it off.

2

Determine the punctuation rule for setting off nonessential elements.

Nonessential elements must be enclosed by matching pairs of punctuation: either two commas, two em dashes, or two parentheses.

Asymmetrical punctuation (such as mixing a comma with an em dash) violates Standard English conventions.

3

Evaluate the choices to find the one that uses a matching pair of punctuation marks to enclose the phrase.

The option using matching em dashes at both the beginning and the end of the phrase is correct, while all other options mix different punctuation marks or omit the closing punctuation.

Only matching em dashes correctly set off the parenthetical modifier.

Anahtar Kavram

Punctuation of Nonessential Elements

Soru 1336Soru →

If $(x, y)$ is a solution to the system of equations below and $x > 0$ , what is the value of $x$ ?

y = x^2

y = x + 6

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The correct answer is

3

.

Substituting

y = x^2

into

y = x + 6

gives the quadratic equation

x^2 - x - 6 = 0

. Factoring this expression yields

(x - 3)(x + 2) = 0

, which gives solutions of

x = 3

and

x = -2

. Since the system requires

x > 0

, the only valid solution is

3

.

Adım Adım Çözüm

1

Substitute the expression for

y

from the first equation into the second equation.

x^2 = x + 6

To eliminate the variable

y

and solve for

x

directly.

2

Subtract

x

and

6

from both sides to write the quadratic equation in standard form.

x^2 - x - 6 = 0

Setting the quadratic expression equal to zero allows it to be factored.

3

Factor the quadratic trinomial.

(x - 3)(x + 2) = 0

Finding factors whose product is

-6

and whose sum is

-1

helps find the roots.

4

Solve for

x

and apply the constraint

x > 0

.

x = 3

The equation has solutions

x = 3

and

x = -2

. Because

x

must be greater than

0

, we discard the negative solution.

Anahtar Kavram

Solving a system of nonlinear equations by substitution and factoring the resulting quadratic equation.

Soru 1337Soru →

In the

xy

-plane, a point with coordinates

(x, y)

lies in the solution set of the system of inequalities below:

\begin{aligned} 2(x - 5) + y &\leq 16 \\ x + y &\leq 25 \end{aligned}

If

y \geq 8

, what is the maximum possible integer value of

x

?

6

9

13

17

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

9

To find the maximum possible integer value of

x

in the system's solution set under the constraint

y \geq 8

, we find the boundaries for

x

in both inequalities. Simplifying the first inequality,

2(x - 5) + y \leq 16

, gives

2x - 10 + y \leq 16

, which simplifies to

x \leq 13 - 0.5y

. The second inequality,

x + y \leq 25

, simplifies to

x \leq 25 - y

. Since both boundary lines have negative slopes, the maximum value of

x

for any point in the solution set occurs at the minimum value of

y

, which is

y = 8

. Substituting

y = 8

into both simplified inequalities gives

x \leq 9

and

x \leq 17

. To satisfy both inequalities simultaneously,

x

must be less than or equal to the more restrictive upper bound, which is 9. Thus, the maximum possible integer value of

x

is 9.

Adım Adım Çözüm

1

Simplify the first inequality to isolate x.

2(x - 5) + y \leq 16 \implies 2x - 10 + y \leq 16 \implies 2x \leq 26 - y \implies x \leq 13 - 0.5y

To determine the boundary condition imposed on x by the first inequality.

2

Simplify the second inequality to isolate x.

x + y \leq 25 \implies x \leq 25 - y

To determine the boundary condition imposed on x by the second inequality.

3

Determine the maximum value of x under the condition

y \geq 8

.

For the first inequality, the maximum value of x occurs at the minimum value of y (

y = 8

), which gives

x \leq 13 - 0.5(8) = 9

. For the second inequality, the maximum value of x occurs at

y = 8

, which gives

x \leq 25 - 8 = 17

.

Because both boundary lines have negative slopes, the maximum value of x is achieved when y is at its minimum.

4

Find the intersection of the constraints to find the maximum possible value of x that satisfies both inequalities.

Since x must be less than or equal to 9 and less than or equal to 17, the maximum integer value that satisfies both conditions is 9.

The solution set must satisfy all inequalities in the system simultaneously.

Anahtar Kavram

Systems of Linear Inequalities in Two Variables

Soru 1338Soru →

If the expression $5(2x - 3) - 4(x - 2)$ is equivalent to $ax + b$ for all values of $x$ , where $a$ and $b$ are constants, what is the value of $a$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The value of the constant coefficient is 6.

Expanding the expression

5(2x - 3) - 4(x - 2)

yields

10x - 15 - 4x + 8

. Combining the like terms results in

(10x - 4x) + (-15 + 8) = 6x - 7

. Comparing this to the expression

ax + b

, the constant coefficient

a

is equal to 6.

Adım Adım Çözüm

1

Distribute the multipliers to the terms inside the parentheses.

10x - 15 - 4x + 8

To remove the parentheses and prepare the expression for simplification, multiply each term inside

(2x - 3)

by

5

and each term inside

(x - 2)

by

-4

.

2

Combine the linear terms and the constant terms.

6x - 7

Combine the variable terms (

10x - 4x = 6x

) and the constants (

-15 + 8 = -7

) to rewrite the expression in its simplest form.

3

Compare the simplified expression to the form

ax + b

to identify the value of

a

.

a = 6

The coefficient of the variable

x

in

6x - 7

corresponds directly to

a

in

ax + b

.

Anahtar Kavram

Equivalent Algebraic Expressions

Soru 1339Soru →

A tutor charges a total fee, $C$ , in dollars, for a tutoring session of $h$ hours according to the equation $C = 25h + 40$ . What is the best interpretation of $40$ in this context?

The fee, in dollars, charged for each hour of tutoring

The total fee, in dollars, charged for a 1-hour tutoring session

The flat sign-up fee, in dollars, charged by the tutor

The number of hours of tutoring for which the total fee is $0

Cevabı ve açıklamayı göster

Cevap: The flat sign-up fee, in dollars, charged by the tutor

Cevap

The flat sign-up fee, in dollars, charged by the tutor

The constant term in the linear equation represents the value of the dependent variable when the independent variable is zero. In this context, when the tutoring hours are zero, the total fee is 40 dollars, representing the flat sign-up fee.

Adım Adım Çözüm

1

Identify the structure of the linear equation

The equation

C = 25h + 40

matches the slope-intercept form

y = mx + b

.

The linear model consists of a variable term representing the hourly rate and a constant term representing the flat fee.

2

Determine the value of the constant term when the variable is zero

When

h = 0

, the equation simplifies to

C = 40

.

Setting the hours of tutoring to zero isolates the starting value of the function.

3

Interpret the starting value in the context of the problem

A session of

0

hours corresponds to the cost before any tutoring begins, which represents a flat initial sign-up fee of

40

.

Connecting the mathematical y-intercept to the real-world scenario identifies the interpretation of the constant.

Anahtar Kavram

Interpreting Linear Relationships in Context

Soru 1340Soru →

A student works two part-time jobs: tutoring, which pays $20$ dollars per hour, and working at a bookstore, which pays $12$ dollars per hour. The student can work at most $15$ hours per week and wants to earn at least $220$ dollars per week. If the student works a whole number of hours at each job, what is the minimum number of hours the student must tutor to meet these requirements?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

The system of inequalities modeling the scenario consists of

x + y \leq 15

(representing the limit on total hours) and

20x + 12y \geq 220

(representing the earning target), where

x

is the hours of tutoring and

y

is the hours at the bookstore. To find the minimum hours of tutoring, we assume the maximum bookstore hours

y = 15 - x

and substitute it into the earnings equation:

20x + 12(15 - x) \geq 220

. Simplifying yields

8x + 180 \geq 220

, which reduces to

8x \geq 40

, or

x \geq 5

. The minimum integer value that satisfies this condition is 5.

Adım Adım Çözüm

1

Define variables and set up the constraint for total hours worked.

x + y \leq 15

Let

x

be the number of hours tutoring and

y

be the number of hours working at the bookstore. The total hours cannot exceed 15.

2

Set up the constraint for the minimum weekly earnings.

20x + 12y \geq 220

Tutoring pays

20

dollars per hour and the bookstore pays

12

dollars per hour, and the total earnings must be at least

220

dollars.

3

Substitute the maximum value of

y

in terms of

x

into the earnings inequality.

20x + 12(15 - x) \geq 220

To minimize

x

, we must maximize

y

. From

x + y \leq 15

, the maximum value of

y

is

15 - x

.

4

Solve the inequality for

x

.

x \geq 5

Distribute and simplify:

20x + 180 - 12x \geq 220 \Rightarrow 8x + 180 \geq 220 \Rightarrow 8x \geq 40 \Rightarrow x \geq 5

.

Anahtar Kavram

Solving systems of linear inequalities to find optimal boundary values in context.

Tahmini Süre:1m 30s

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