Tüm alıştırma soruları

Soru 141Soru →

If the expression $\frac{3x^3 - 8x^2 + kx - 6}{3x - 2}$ is equivalent to $x^2 - 2x + 3$ for all $x \neq \frac{2}{3}$ , where $k$ is a constant, what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 13

Cevap

13

By multiplying both sides of the equation by

3x - 2

, the rational expression simplifies to a polynomial identity:

3x^3 - 8x^2 + kx - 6 = (x^2 - 2x + 3)(3x - 2)

. Expanding the right side yields

3x^3 - 8x^2 + 13x - 6

. Since the two polynomials are equivalent, their corresponding coefficients must be equal, meaning the coefficient of the linear term,

k

, must be equal to

13

.

Adım Adım Çözüm

1

Multiply both sides of the equivalence by the denominator

(3x - 2)

3x^3 - 8x^2 + kx - 6 = (x^2 - 2x + 3)(3x - 2)

To clear the fraction and align the polynomial expressions for coefficient comparison.

2

Expand the right side of the equation using the distributive property

3x^3 - 8x^2 + 13x - 6

To obtain the expanded form of the polynomial so that we can identify the coefficients of each term.

3

Equate the corresponding coefficients of the linear

x

terms on both sides of the equation

k = 13

Since the two expressions are equivalent for all values of

x

, their coefficients for each corresponding power of

x

must be equal.

Anahtar Kavram

Equivalent Algebraic Expressions

Soru 142Soru →

A nutritionist is designing a diet plan using two food types, Food A and Food B. The table below shows the amount of protein and carbohydrates, in grams, per gram of each food type:

Food Type	Protein (g)	Carbohydrates (g)
Food A	$0.12$	$0.08$
Food B	$0.18$	$0.05$

The nutritionist wants to combine the two foods to obtain a mixture that has a total mass of $250$ grams and contains exactly $36$ grams of protein. How many grams of carbohydrates will be in this mixture?

Cevabı ve açıklamayı göster

Cevap: 17

Cevap

The total amount of carbohydrates in the mixture is 17 grams.

To find the total amount of carbohydrates in the mixture, we first define

x

as the mass in grams of Food A and

y

as the mass in grams of Food B. Since the total mass of the mixture is

250

grams, we write the equation

x + y = 250

. Since the total protein in the mixture must be

36

grams, and Food A and Food B contain

0.12

grams and

0.18

grams of protein per gram respectively, we write the equation

0.12x + 0.18y = 36

. Multiplying the first equation by

12

yields

12x + 12y = 3000

, and multiplying the second equation by

100

yields

12x + 18y = 3600

. Subtracting the first new equation from the second gives

6y = 600

, which simplifies to

y = 100

. Substituting this back into the first equation gives

x + 100 = 250

, so

x = 150

. Using these values, the total amount of carbohydrates is

0.08(150) + 0.05(100) = 12 + 5 = 17

grams.

Adım Adım Çözüm

1

Define variables for the mass of each food type.

Let

x

be the mass, in grams, of Food A, and let

y

be the mass, in grams, of Food B.

To represent the unknown quantities in the system of equations.

2

Set up the system of linear equations.

x + y = 250

(total mass equation) and

0.12x + 0.18y = 36

(total protein equation).

The mixture has a total mass of

250

grams and must contain exactly

36

grams of protein.

3

Solve the system of equations for

x

and

y

.

Multiply the first equation by

12

to get

12x + 12y = 3000

. Multiply the second equation by

100

to get

12x + 18y = 3600

. Subtract the first modified equation from the second to get

6y = 600

, which gives

y = 100

. Substitute

y = 100

back into

x + y = 250

to get

x = 150

.

To find the specific masses of Food A and Food B required for the mixture.

4

Calculate the total mass of carbohydrates in the mixture.

Total carbohydrates

= 0.08x + 0.05y = 0.08(150) + 0.05(100) = 12 + 5 = 17

grams.

The question asks for the total grams of carbohydrates in the final mixture.

Anahtar Kavram

Systems of Linear Equations

Soru 143Soru →

An empty water tank is being filled by two hoses. Hose A fills the tank at a constant rate of 12 gallons per minute. Hose B fills the tank at a constant rate of 15 gallons per minute. Hose A is turned on first, and 5 minutes later, Hose B is also turned on. If both hoses run until the tank contains a total of 330 gallons of water, for how many minutes was Hose A running?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

15

Let

m

be the number of minutes Hose A was running. Since Hose B was turned on 5 minutes after Hose A, Hose B was running for

m - 5

minutes. The volume of water contributed by Hose A is

12m

gallons, and the volume of water contributed by Hose B is

15(m - 5)

gallons. The sum of these volumes is 330 gallons, which gives the equation

12m + 15(m - 5) = 330

. Distributing 15 gives

12m + 15m - 75 = 330

. Combining like terms yields

27m - 75 = 330

. Adding 75 to both sides gives

27m = 405

. Dividing both sides by 27 yields

m = 15

. Thus, Hose A was running for 15 minutes.

Adım Adım Çözüm

1

Define variables for the running time of each hose.

Let Hose A run for

m

minutes. Then Hose B runs for

m - 5

minutes.

Hose B started 5 minutes after Hose A, so it ran for 5 fewer minutes.

2

Set up a linear equation for the total gallons of water.

12m + 15(m - 5) = 330

The sum of the water from both hoses must equal 330 gallons.

3

Simplify and solve the linear equation.

27m = 405

, which gives

m = 15

.

Distribute, combine like terms, and isolate the variable

m

.

Anahtar Kavram

Setting up and solving a multi-step linear equation in one variable from a real-world context.

Soru 144Soru →

A public library is purchasing new books for its collection. The library plans to buy paperback books for $6 each and hardcover books for$ 20 each. The library has a budget of at most $1,200 for this purchase, wants to buy at most 120 total books, and must purchase at least 45 hardcover books. What is the maximum number of paperback books the library can purchase?

Cevabı ve açıklamayı göster

Cevap: 50

Cevap

The maximum number of paperback books the library can purchase is 50.

The correct maximum number of paperback books is 50. By expressing the budget constraint as

p \le 200 - \frac{10}{3}h

, we see that the number of paperback books is maximized when the number of hardcover books,

h

, is minimized. Since the library must buy at least 45 hardcover books, we substitute

h = 45

into the inequality to get

p \le 50

. This combination also satisfies the total book constraint because

50 + 45 = 95

, which is less than or equal to 120.

Adım Adım Çözüm

1

Set up the system of inequalities representing the library's constraints.

Let

p

be the number of paperbacks and

h

be the number of hardcovers. The constraints are

6p + 20h \le 1200

,

p + h \le 120

, and

h \ge 45

.

This translates the word problem into mathematical inequalities.

2

Isolate the variable

p

in the budget inequality to express it in terms of

h

.

p \le 200 - \frac{10}{3}h

This shows how the number of paperbacks depends on the number of hardcovers purchased.

3

Maximize

p

by using the minimum possible value of

h

.

p \le 200 - \frac{10}{3}(45) = 50

To maximize

p

, we must minimize

h

because buying more hardcovers decreases the remaining budget for paperbacks. The minimum value for

h

is 45.

4

Verify if the solution satisfies the remaining total books constraint.

50 + 45 = 95 \le 120

, which is true.

This ensures the solution is feasible and does not violate any other constraints.

Anahtar Kavram

Systems of Linear Inequalities in Two Variables

Soru 145Soru →

For a constant $k$ , the graph of the linear equation $kx - 4y = 15$ in the $xy$ -plane contains the point $(3, 3)$ . What is the slope of the line?

Cevabı ve açıklamayı göster

Cevap: 2.25

Cevap

2.25

Substituting the coordinates of the point

(3, 3)

into the given equation yields

k(3) - 4(3) = 15

, which simplifies to

3k - 12 = 15

. Adding

12

to both sides gives

3k = 27

, and dividing by

3

gives

k = 9

. Substituting

k = 9

back into the original equation gives

9x - 4y = 15

. To find the slope of the line, the equation can be written in slope-intercept form,

y = mx + b

, by isolating

y

. Subtracting

9x

from both sides gives

-4y = -9x + 15

. Dividing all terms by

-4

yields

y = \frac{9}{4}x - \frac{15}{4}

, or

y = 2.25x - 3.75

. Therefore, the slope of the line is

\frac{9}{4}

, or

2.25

.

Adım Adım Çözüm

1

Substitute the point

(3, 3)

into the equation

kx - 4y = 15

3k - 12 = 15

Since the graph of the equation contains the point

(3, 3)

, substituting

x = 3

and

y = 3

must satisfy the equation.

2

Solve for the constant

k

k = 9

Adding

12

to both sides of the equation gives

3k = 27

. Dividing both sides by

3

yields

k = 9

.

3

Substitute

k = 9

into the equation and rewrite it in slope-intercept form (

y = mx + b

)

y = \frac{9}{4}x - \frac{15}{4}

Substituting

k = 9

gives

9x - 4y = 15

. Subtracting

9x

from both sides gives

-4y = -9x + 15

. Dividing both sides by

-4

isolates

y

.

4

Identify the slope from the slope-intercept form

Slope =

2.25

(or

\frac{9}{4}

)

In the slope-intercept form

y = mx + b

, the coefficient

m

represents the slope of the line.

Anahtar Kavram

Determining the slope of a line from a linear equation by first finding an unknown constant using a given point on the line.

Soru 146Soru →

A beverage company produces a fruit juice blend using apple juice and cranberry juice. Let $x$ represent the volume, in liters, of apple juice, and let $y$ represent the volume, in liters, of cranberry juice used in one batch of the blend. The production constraints for each batch are modeled by the system of inequalities below:

x + y \geq 100

y \geq \frac{1}{3}x

y \leq -\frac{1}{2}x + 75

What is the maximum possible volume, in liters, of cranberry juice that can be used in a single batch?

Cevabı ve açıklamayı göster

Cevap: 50

Cevap

50

To find the maximum possible value of

y

, we examine the bounds of the feasible region. Rewriting the inequalities in terms of

x

gives

x \geq 100 - y

and

x \leq 150 - 2y

. Since

x

must satisfy both inequalities simultaneously, the lower bound must be less than or equal to the upper bound, meaning

100 - y \leq 150 - 2y

. Solving this inequality for

y

gives

y \leq 50

. The point

(50, 50)

also satisfies the remaining constraint

y \geq \frac{1}{3}x

, confirming that 50 is the maximum possible value.

Adım Adım Çözüm

1

Isolate

x

in the first inequality:

x + y \geq 100

becomes

x \geq 100 - y

.

x \geq 100 - y

This establishes a lower bound for the volume of apple juice in terms of the volume of cranberry juice.

2

Isolate

x

in the third inequality:

y \leq -\frac{1}{2}x + 75

can be rewritten by subtracting 75 from both sides to get

y - 75 \leq -\frac{1}{2}x

, then multiplying by

-2

and reversing the inequality sign to get

x \leq 150 - 2y

.

x \leq 150 - 2y

This establishes an upper bound for the volume of apple juice in terms of the volume of cranberry juice.

3

Combine the lower and upper bounds of

x

:

100 - y \leq x \leq 150 - 2y

. For a valid solution to exist, the lower bound must be less than or equal to the upper bound.

100 - y \leq 150 - 2y

This allows us to solve directly for the constraint on the cranberry juice volume

y

.

4

Solve the inequality

100 - y \leq 150 - 2y

by adding

2y

to both sides and subtracting 100 from both sides.

y \leq 50

This determines that the maximum possible value for

y

is 50.

5

Substitute

y = 50

back into the compound inequality to find

x

:

100 - 50 \leq x \leq 150 - 2(50) \implies 50 \leq x \leq 50

, which means

x = 50

. Check if the point

(50, 50)

satisfies the remaining inequality

y \geq \frac{1}{3}x

:

50 \geq \frac{1}{3}(50) \approx 16.67

, which is true.

The point

(50, 50)

is a valid solution to the system.

This confirms that 50 is a achievable value within the system of inequalities.

Anahtar Kavram

Finding the boundary limits and optimization points within a system of linear inequalities in two variables.

Soru 147Soru →

A hot air balloon is at an altitude of $h$ meters. The balloon begins to descend at a constant rate. After $3$ minutes, the altitude of the balloon is $540$ meters. After $8$ minutes, the altitude of the balloon is $390$ meters. If the altitude of the balloon is modeled by a linear function of time, what was the initial altitude of the balloon, in meters?

Cevabı ve açıklamayı göster

Cevap: 630

Cevap

The initial altitude of the balloon was 630 meters.

To find the initial altitude, we model the balloon's descent as a linear equation of the form

h(t) = mt + b

, where

h(t)

is the altitude at time

t

,

m

is the constant rate of change, and

b

is the initial altitude. Using the given points

(3, 540)

and

(8, 390)

, we find the slope

m = \frac{390 - 540}{8 - 3} = -30

meters per minute. Substituting

m = -30

and the point

(3, 540)

into

h(t) = mt + b

gives

540 = -30(3) + b

. Solving for

b

yields

b = 540 + 90 = 630

meters.

Adım Adım Çözüm

1

Identify two data points from the problem context

The coordinate points are

(3, 540)

and

(8, 390)

To define the linear relationship, we need at least two coordinate points representing (time, altitude)

2

Calculate the slope (

m

) of the linear function

m = \frac{390 - 540}{8 - 3} = \frac{-150}{5} = -30

The rate of change represents the speed at which the balloon descends each minute

3

Solve for the vertical intercept (

b

) using the slope-intercept form

y = mx + b

540 = -30(3) + b \implies b = 630

The initial altitude corresponds to the altitude at time

t = 0

, which is the vertical intercept of the linear function

Anahtar Kavram

Linear Functions and Graphs

Soru 148Soru →

If the expression $\frac{6x^3 + 7x^2 - 14x + 19}{2x + 5}$ is equivalent to $ax^2 + bx + c + \frac{k}{2x + 5}$ for all $x \neq -2.5$ , where $a$ , $b$ , $c$ , and $k$ are constants, what is the value of $a - b + c + k$ ?

Cevabı ve açıklamayı göster

Cevap: 14

Cevap

14

Dividing the numerator

6x^3 + 7x^2 - 14x + 19

by the denominator

2x + 5

using polynomial division yields a quotient of

3x^2 - 4x + 3

and a remainder of

4

. The equivalent expression is

3x^2 - 4x + 3 + \frac{4}{2x + 5}

. Comparing this with the form

ax^2 + bx + c + \frac{k}{2x + 5}

gives the values

a = 3

,

b = -4

,

c = 3

, and

k = 4

. Substituting these values into

a - b + c + k

gives

3 - (-4) + 3 + 4 = 14

.

Adım Adım Çözüm

1

Perform the first step of polynomial long division by dividing

6x^3

by

2x

.

Quotient term:

3x^2

; Remainder:

-8x^2 - 14x + 19

To eliminate the highest-degree term of the numerator.

2

Perform the second step of division by dividing

-8x^2

by

2x

.

Quotient term:

-4x

; Remainder:

6x + 19

To find the next term of the quotient.

3

Perform the third step of division by dividing

6x

by

2x

.

Quotient term:

3

; Remainder:

4

To find the constant term of the quotient and the final remainder.

4

Compare the resulting expression

3x^2 - 4x + 3 + \frac{4}{2x + 5}

with the given form to identify the constants

a

,

b

,

c

, and

k

.

a = 3

,

b = -4

,

c = 3

,

k = 4

To map the coefficients of equivalent algebraic expressions.

5

Evaluate the expression

a - b + c + k

using the identified values.

3 - (-4) + 3 + 4 = 14

To obtain the final numeric value requested.

Anahtar Kavram

Equivalent Algebraic Expressions via Polynomial Long Division

Soru 149Soru →

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 150Soru →

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 151Soru →

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 152Soru →

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 153Soru →

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 154Soru →

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

Soru 155Soru →

For all positive real numbers $x$ and $y$ , the expression $\frac{(x^{3} y^{2})^{2/3} \cdot (x^{-1} y^{4})^{1/6}}{(x^2 y)^{1/2}}$ can be written in the equivalent form $x^a y^b$ , where $a$ and $b$ are constants. What is the value of $6a + 2b$ ?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The correct answer is 8.

Applying the rules of exponents systematically yields the simplified expression

x^{5/6} y^{3/2}

. By setting

a = \frac{5}{6}

and

b = \frac{3}{2}

, the linear combination

6a + 2b

evaluates to

6\left(\frac{5}{6}\right) + 2\left(\frac{3}{2}\right) = 5 + 3 = 8

.

Adım Adım Çözüm

1

Apply the power of a power rule to the first term in the numerator.

(x^3 y^2)^{2/3} = x^{3 \cdot \frac{2}{3}} y^{2 \cdot \frac{2}{3}} = x^2 y^{4/3}

To raise a product to a power, raise each factor to that power by multiplying their exponents.

2

Apply the power of a power rule to the second term in the numerator.

(x^{-1} y^4)^{1/6} = x^{-1 \cdot \frac{1}{6}} y^{4 \cdot \frac{1}{6}} = x^{-1/6} y^{2/3}

To raise a product to a power, raise each factor to that power by multiplying their exponents.

3

Multiply the two simplified terms in the numerator.

(x^2 y^{4/3})(x^{-1/6} y^{2/3}) = x^{2 - \frac{1}{6}} y^{\frac{4}{3} + \frac{2}{3}} = x^{11/6} y^2

When multiplying expressions with the same base, add their exponents.

4

Simplify the denominator.

(x^2 y)^{1/2} = x^{2 \cdot \frac{1}{2}} y^{1 \cdot \frac{1}{2}} = x y^{1/2}

To raise a product to a power, raise each factor to that power by multiplying their exponents.

5

Divide the numerator by the denominator.

\frac{x^{11/6} y^2}{x y^{1/2}} = x^{\frac{11}{6} - 1} y^{2 - \frac{1}{2}} = x^{5/6} y^{3/2}

When dividing expressions with the same base, subtract the exponent of the denominator from the exponent of the numerator.

6

Identify the values of

a

and

b

and calculate

6a + 2b

.

a = \frac{5}{6}

,

b = \frac{3}{2}

, so

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

Matching the simplified expression

x^{5/6} y^{3/2}

to

x^a y^b

yields the values of the constants

a

and

b

, which are then used to calculate the required expression.

Anahtar Kavram

Simplifying rational expressions with fractional exponents using exponent rules.

Soru 156Soru →

If the expression $(x + 5)^2 - (x - 3)^2$ 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: 16

Cevap

16

Expanding

(x + 5)^2

gives

x^2 + 10x + 25

, and expanding

(x - 3)^2

gives

x^2 - 6x + 9

. Subtracting the second expression from the first requires distributing the negative sign across all terms:

x^2 + 10x + 25 - (x^2 - 6x + 9) = x^2 + 10x + 25 - x^2 + 6x - 9

. Combining like terms yields

16x + 16

. Comparing this to

ax + b

shows that the coefficient of

x

,

a

, is 16.

Adım Adım Çözüm

1

Expand the first squared binomial term

(x + 5)^2 = x^2 + 10x + 25

To express the binomial square as a trinomial using the perfect square identity

(u + v)^2 = u^2 + 2uv + v^2

.

2

Expand the second squared binomial term

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

To express the binomial square as a trinomial using the perfect square identity

(u - v)^2 = u^2 - 2uv + v^2

.

3

Subtract the expanded expressions and distribute the negative sign

x^2 + 10x + 25 - x^2 + 6x - 9

To combine the terms while correctly applying the distributive property to the subtracted expression.

4

Combine like terms to simplify the polynomial

16x + 16

To find the final simplified polynomial of the form

ax + b

.

5

Compare the simplified expression to the standard form to find the value of

a

a = 16

The constant

a

represents the coefficient of the linear term

x

, which is 16.

Anahtar Kavram

Simplifying algebraic expressions by expanding binomial products and combining like terms.

Soru 157Soru →

A software company offers a Basic subscription for $\$15$ per month and a Premium subscription for $\$30$ per month. In April, the company had a total of $1,200$ active subscriptions. In May, the number of Basic subscriptions increased by $20\%$ , and the number of Premium subscriptions increased by $5\%$ compared to April. If the company's total monthly revenue increased by $\$2,700$ from April to May, how many Premium subscriptions did the company have in April?

Cevabı ve açıklamayı göster

Cevap: 600

Cevap

600

To find the number of Premium subscriptions in April, we can set up a system of linear equations. Let

B

represent the number of Basic subscriptions in April and

P

represent the number of Premium subscriptions in April. Since the total number of subscriptions is

1,200

, we have

B + P = 1,200

. The revenue increase from April to May is

\$2,700

. The increase in Basic subscriptions is

20\%

, which contributes

15 \times 0.20B = 3B

dollars to the revenue increase. The increase in Premium subscriptions is

5\%

, which contributes

30 \times 0.05P = 1.5P

dollars to the revenue increase. Thus,

3B + 1.5P = 2,700

. Substituting

B = 1,200 - P

into this equation yields

3(1,200 - P) + 1.5P = 2,700

. Solving for

P

gives

3,600 - 1.5P = 2,700

, which simplifies to

1.5P = 900

, or

P = 600

.

Adım Adım Çözüm

1

Define variables and write the first equation based on the total number of subscriptions in April.

B + P = 1,200

, where

B

is the number of Basic subscriptions and

P

is the number of Premium subscriptions.

To represent the relationship between the two types of subscriptions in April.

2

Write the second equation representing the change in monthly revenue from April to May.

3B + 1.5P = 2,700

The change in revenue is the sum of the increase in revenue from each subscription type:

15(0.20B) + 30(0.05P) = 2,700

.

3

Solve the system of equations by substituting

B = 1,200 - P

into the revenue equation.

3(1,200 - P) + 1.5P = 2,700 \implies 3,600 - 1.5P = 2,700

Substitution eliminates the variable

B

, allowing us to solve for

P

directly.

4

Isolate the variable

P

to find the number of Premium subscriptions in April.

-1.5P = -900 \implies P = 600

Dividing the revenue difference by the coefficient solves for the value of

P

.

Anahtar Kavram

Solving systems of linear equations in real-life contexts involving percentage changes.

Soru 158Soru →

A shipping company offers two types of delivery services: standard and express. The total shipping cost, in dollars, for a package sent via standard service is modeled by the function $C(w) = 1.25w + b_s$ , where $w$ is the weight of the package, in pounds, and $b_s$ is a constant representing the flat handling fee. The total shipping cost, in dollars, for a package sent via express service is modeled by the function $E(k) = m_e k + b_e$ , where $k$ is the weight of the package, in kilograms, and $b_e$ is a constant representing the flat handling fee.

The rate of change of the express shipping cost with respect to the package's weight, in dollars per kilogram, is $2.4$ times the rate of change of the standard shipping cost with respect to the package's weight, in dollars per pound. The flat handling fee for the express service is $4.50$ dollars more than the flat handling fee for the standard service. If it costs $58.50$ dollars to ship a package weighing $26.4$ pounds using the express service, what is the cost, in dollars, to ship a package weighing $20$ pounds using the standard service? (Assume $1\text{ kilogram} = 2.2\text{ pounds}$ .)

Cevabı ve açıklamayı göster

Cevap: 43

Cevap

The cost to ship a package weighing 20 pounds using the standard service is 43 dollars.

The correct answer is obtained by first calculating the rate of change of the express shipping cost (

2.4 \times 1.25 = 3.00

dollars per kilogram). Next, the weight of the package is converted to kilograms (

26.4 / 2.2 = 12

kilograms) to match the express cost function's variable. Using the given cost of

58.50

dollars for the express shipment, the express handling fee is determined to be

58.50 - 3(12) = 22.50

dollars. The standard handling fee is then found by subtracting

4.50

dollars from the express fee (

22.50 - 4.50 = 18.00

dollars). Finally, the cost of a

20

-pound standard package is computed as

1.25(20) + 18.00 = 43.00

dollars.

Adım Adım Çözüm

1

Determine the rate of change for the express service (

m_e

).

m_e = 3.00

dollars per kilogram

The rate of change of the standard shipping cost is

1.25

dollars per pound. Since the rate of change for the express shipping cost is

2.4

times this rate,

m_e = 2.4 \times 1.25 = 3.00

dollars per kilogram.

2

Convert the weight of the express package from pounds to kilograms.

k = 12

kilograms

The weight of the package is given as

26.4

pounds. Using the conversion

1\text{ kilogram} = 2.2\text{ pounds}

, the weight in kilograms is

\frac{26.4}{2.2} = 12

kilograms.

3

Find the express flat handling fee (

b_e

) using the given cost of the express shipment.

b_e = 22.50

dollars

We are given that the cost of shipping a

12

-kilogram package using the express service is

58.50

dollars. Substituting these values into the express cost function:

3(12) + b_e = 58.50 \Rightarrow 36 + b_e = 58.50 \Rightarrow b_e = 22.50

dollars.

4

Find the standard flat handling fee (

b_s

).

b_s = 18.00

dollars

The express handling fee is

4.50

dollars more than the standard handling fee:

b_e = b_s + 4.50 \Rightarrow 22.50 = b_s + 4.50 \Rightarrow b_s = 18.00

dollars.

5

Calculate the cost to ship a

20

-pound package using the standard service.

43.00

dollars

Using the standard cost function

C(w) = 1.25w + b_s

with

w = 20

and

b_s = 18.00

:

C(20) = 1.25(20) + 18.00 = 25.00 + 18.00 = 43.00

dollars.

Anahtar Kavram

Interpreting slope, y-intercept, and rates of change of linear functions in a real-world context with unit conversions.

Soru 159Soru →

Consider the system of equations below.

\begin{aligned} 3(x + y) - 2(x - y) &= 20 \\ 2(x + y) + 3(x - y) &= 22 \end{aligned}

If

(x, y)

is the solution to the system of equations, what is the value of

x

?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The value of

x

is

5

.

Expanding and simplifying the given system yields

x + 5y = 20

and

5x - y = 22

. Multiplying the second equation by 5 results in

25x - 5y = 110

. Adding this to the first equation eliminates

y

, leaving

26x = 130

. Dividing both sides by 26 gives

x = 5

.

Adım Adım Çözüm

1

Expand the algebraic expressions in both equations to rewrite the system in standard form.

x + 5y = 20

and

5x - y = 22

Expanding the first equation gives

3x + 3y - 2x + 2y = 20

, which simplifies to

x + 5y = 20

. Expanding the second equation gives

2x + 2y + 3x - 3y = 22

, which simplifies to

5x - y = 22

.

2

Multiply the second equation by 5 to align the coefficients of the y-terms.

25x - 5y = 110

Multiplying

5x - y = 22

by 5 allows the subtraction of

y

to cancel with the addition of

5y

in the first equation.

3

Add the first equation to the modified second equation to eliminate the y-variable and solve for x.

26x = 130

, which simplifies to

x = 5

.

Adding

(x + 5y) + (25x - 5y) = 20 + 110

yields

26x = 130

. Dividing by 26 gives the final value of

x

.

Anahtar Kavram

Solving systems of linear equations using expansion and elimination.

Alternatif Yöntem

Define substitution variables

u = x + y

and

v = x - y

. The system simplifies to

3u - 2v = 20

and

2u + 3v = 22

. Multiplying the first equation by 3 and the second by 2 gives

9u - 6v = 60

and

4u + 6v = 44

. Adding these equations yields

13u = 104 \implies u = 8

. Substituting

u = 8

back in gives

16 + 3v = 22 \implies 3v = 6 \implies v = 2

. Now, solve the system

x + y = 8

and

x - y = 2

. Adding these two equations gives

2x = 10 \implies x = 5

.

Tahmini Süre:1m 30s

Soru 160Soru →

A worker at a distribution center packages boxes at a constant rate. The total number of boxes, $B$ , the worker has packaged $h$ hours after starting their shift can be modeled by the equation $B = 12h + 15$ . According to the model, how many boxes were already packaged at the start of the worker's shift?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

At the start of the worker's shift,

15

boxes were already packaged.

In the linear model

B = 12h + 15

, the term

15

is the constant term (y-intercept), which represents the value of

B

when

h = 0

. In this context,

h = 0

represents the start of the worker's shift. Therefore,

15

boxes were already packaged at the start of the shift.

Adım Adım Çözüm

1

Identify the value of

h

that represents the start of the shift.

h = 0

The variable

h

represents the number of hours since the shift started, so the start of the shift corresponds to

0

hours.

2

Substitute

h = 0

into the given equation to find the value of

B

.

B = 15

Evaluating the equation at

h = 0

gives the initial number of packaged boxes, which is represented by the constant term of the linear equation.

Anahtar Kavram

Interpreting the y-intercept of a linear relationship in context

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