Tüm alıştırma soruları

Soru 201Soru →

A water utility company charges a flat monthly connection fee plus a constant rate per gallon of water consumed. In a certain month, a household that consumed $2,800$ gallons of water was billed a total of $62.00$ dollars, and a household that consumed $4,500$ gallons of water was billed a total of $87.50$ dollars. What is the flat monthly connection fee, in dollars, charged by the water utility company?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The flat monthly connection fee charged by the water utility company is 20 dollars.

The flat monthly connection fee is 20 dollars. A linear relationship can be modeled by the equation

y = mx + b

, where

x

represents the water consumed in gallons,

y

represents the total bill in dollars,

m

represents the rate per gallon, and

b

represents the flat monthly connection fee. Using the data for the two households, we get the points

(2800, 62)

and

(4500, 87.5)

. The rate per gallon is the slope of the line passing through these points:

m = \frac{87.5 - 62}{4500 - 2800} = \frac{25.5}{1700} = 0.015

dollars per gallon. Substituting this slope and the point

(2800, 62)

into the equation gives

62 = 0.015(2800) + b

, which simplifies to

62 = 42 + b

. Solving for

b

yields

20

dollars.

Adım Adım Çözüm

1

Define the linear relationship using the variables

x

for water consumed in gallons and

y

for the total bill in dollars.

The relationship can be written as

y = mx + b

, where

m

represents the cost per gallon of water and

b

represents the flat monthly connection fee.

This establishes the linear framework needed to solve for the unknown constants using the provided data points.

2

Calculate the constant rate of change (slope

m

) using the coordinates of the two households:

(2800, 62)

and

(4500, 87.5)

.

m = \frac{87.5 - 62}{4500 - 2800} = \frac{25.5}{1700} = 0.015

dollars per gallon.

The slope of a linear function represents the constant rate of change between the two variables.

3

Substitute the calculated slope

m = 0.015

and the point

(2800, 62)

into the equation

y = mx + b

to find the flat fee

b

.

62 = 0.015(2800) + b \implies 62 = 42 + b \implies b = 20

.

The flat monthly connection fee corresponds to the vertical intercept (

y

-intercept) of the linear equation.

Anahtar Kavram

Linear Equations in Two Variables

Soru 202Soru →

For all $x > 0$ and $x \neq 1$ , the expression $\frac{x^{5/2} - x^{3/2} - x + 1}{x^{3/2} - x - x^{1/2} + 1}$ is equivalent to $x + x^a + b$ , where $a$ and $b$ are constants and $a < 1$ . What is the value of $2a + b$ ?

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Cevap: 2

Cevap

The correct answer is 2.

Substituting

u = x^{1/2}

transforms the expression into

\frac{u^5 - u^3 - u^2 + 1}{u^3 - u^2 - u + 1}

. Factoring the numerator by grouping yields

(u^3 - 1)(u^2 - 1)

, and factoring the denominator yields

(u^2 - 1)(u - 1)

. Canceling the common factor

(u^2 - 1)

leaves

\frac{u^3 - 1}{u - 1}

. Applying the difference of cubes formula to factor

u^3 - 1

as

(u - 1)(u^2 + u + 1)

allows us to cancel the

(u - 1)

term, leaving

u^2 + u + 1

. Re-substituting

u = x^{1/2}

results in the equivalent expression

x + x^{1/2} + 1

. Comparing this to

x + x^a + b

with

a < 1

gives

a = 0.5

and

b = 1

. Evaluating

2a + b

yields

2(0.5) + 1 = 2

.

Adım Adım Çözüm

1

Introduce a variable substitution to eliminate fractional exponents.

Let

u = x^{1/2}

, which implies

x = u^2

. The expression becomes

\frac{u^5 - u^3 - u^2 + 1}{u^3 - u^2 - u + 1}

.

Substitution simplifies working with fractional exponents and makes factoring patterns easier to identify.

2

Factor the numerator by grouping terms.

The numerator factors as

u^3(u^2 - 1) - (u^2 - 1) = (u^3 - 1)(u^2 - 1)

.

Grouping common terms helps simplify polynomials into products of lower-degree factors.

3

Factor the denominator by grouping terms.

The denominator factors as

u^2(u - 1) - (u - 1) = (u^2 - 1)(u - 1)

.

Factoring the denominator reveals common terms that can be canceled with the numerator.

4

Combine the factored forms and cancel the common factor.

The expression becomes

\frac{(u^3 - 1)(u^2 - 1)}{(u^2 - 1)(u - 1)} = \frac{u^3 - 1}{u - 1}

for

u \neq 1

.

Canceling

(u^2 - 1)

simplifies the rational expression since

x \neq 1

ensures

u^2 - 1 \neq 0

.

5

Simplify the remaining rational expression using the difference of cubes formula.

Factoring

u^3 - 1

as

(u - 1)(u^2 + u + 1)

and canceling

(u - 1)

yields

u^2 + u + 1

.

The difference of cubes formula allows cancellation of the linear term in the denominator.

6

Substitute back the original variable and determine the constants.

Substituting

u = x^{1/2}

back gives

x + x^{1/2} + 1

, meaning

a = 0.5

and

b = 1

.

This puts the expression in the final requested form

x + x^a + b

to find

a

and

b

.

7

Calculate the requested value

2a + b

.

2(0.5) + 1 = 2

.

This computes the final numeric answer requested by the question.

Anahtar Kavram

Simplifying rational expressions with fractional exponents by substitution and grouping

Soru 203Soru →

\begin{aligned} y &= x^2 - 5 \\ y &= 4x \end{aligned}

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

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

20

Substituting

y = 4x

into the equation

y = x^2 - 5

yields

4x = x^2 - 5

. Rearranging this equation into standard form gives

x^2 - 4x - 5 = 0

. Factoring the quadratic expression gives

(x - 5)(x + 1) = 0

, which means

x = 5

or

x = -1

. Substituting these values back into

y = 4x

gives the coordinates of the two solutions:

(5, 20)

and

(-1, -4)

. Since the question specifies that

y > 0

, the correct value is the positive

y

-value, which is 20.

Adım Adım Çözüm

1

Substitute the expression for

y

from the second equation into the first equation.

4x = x^2 - 5

This eliminates

y

and creates a single equation in terms of

x

.

2

Subtract

4x

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

x^2 - 4x - 5 = 0

Standard quadratic form

ax^2 + bx + c = 0

is required to solve by factoring.

3

Factor the quadratic equation.

(x - 5)(x + 1) = 0

Factoring helps find the values of

x

that satisfy the equation.

4

Solve for

x

by setting each factor to zero.

x = 5

or

x = -1

Applying the zero product property yields the possible

x

-coordinates of the solutions.

5

Calculate the corresponding

y

-values using the equation

y = 4x

.

For

x = 5

,

y = 20

. For

x = -1

,

y = -4

.

This determines the coordinates of the intersection points, which are

(5, 20)

and

(-1, -4)

.

6

Apply the constraint

y > 0

to identify the correct value of

y

.

y = 20

Since

-4

is not greater than zero, the only valid solution is

(5, 20)

, giving

y = 20

.

Anahtar Kavram

Solving a system of linear and quadratic equations using substitution.

Soru 204Soru →

The quadratic equation $x^2 - 4x + 2 = 0$ has roots $r_1$ and $r_2$ . A second quadratic equation, $x^2 - px + q = 0$ , has roots $\left(r_1 + \frac{1}{r_2}\right)$ and $\left(r_2 + \frac{1}{r_1}\right)$ , where $p$ and $q$ are constants. What is the value of $p$ ?

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Cevap: 6

Cevap

The value of

p

is 6.

By Vieta's formulas, the sum of the roots of

x^2 - 4x + 2 = 0

is

r_1 + r_2 = 4

and the product of the roots is

r_1 r_2 = 2

. The sum of the roots of the second equation

x^2 - px + q = 0

is

p

. Therefore,

p = \left(r_1 + \frac{1}{r_2}\right) + \left(r_2 + \frac{1}{r_1}\right) = (r_1 + r_2) + \frac{r_1 + r_2}{r_1 r_2}

. Substituting the known values yields

p = 4 + \frac{4}{2} = 6

.

Adım Adım Çözüm

1

Determine the sum and product of the roots of the first equation.

r_1 + r_2 = 4

and

r_1 r_2 = 2

Vieta's formulas state that for

x^2 - Bx + C = 0

, the sum of the roots is

B

and the product is

C

.

2

Express the sum of the roots of the second equation, which is

p

.

p = (r_1 + r_2) + \left(\frac{1}{r_1} + \frac{1}{r_2}\right)

The sum of the roots of

x^2 - px + q = 0

is

p

.

3

Simplify the fractional part of the equation.

\frac{1}{r_1} + \frac{1}{r_2} = \frac{r_1 + r_2}{r_1 r_2}

Finding a common denominator allows us to write the sum of reciprocals in terms of the sum and product of the roots.

4

Substitute the values of the sum and product into the expression for

p

.

p = 4 + \frac{4}{2} = 6

This evaluates the expression to find the final value of

p

.

Anahtar Kavram

Relating the roots and coefficients of quadratic equations using Vieta's formulas.

Soru 205Soru →

If $(2y - 1)^2 = 49$ and $y > 0$ , what is the value of $y$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The correct answer is

4

.

By taking the square root of both sides of the equation

(2y - 1)^2 = 49

, we get

2y - 1 = 7

or

2y - 1 = -7

. Solving

2y - 1 = 7

gives

2y = 8

, or

y = 4

. Solving

2y - 1 = -7

gives

2y = -6

, or

y = -3

. Since it is given that

y > 0

, the value of

y

must be

4

.

Adım Adım Çözüm

1

Take the square root of both sides of the equation.

2y - 1 = 7

or

2y - 1 = -7

Applying the square root property of equality to isolate the linear term.

2

Solve each linear equation for

y

.

y = 4

or

y = -3

Isolating

y

in each equation by adding

1

and then dividing by

2

.

3

Apply the constraint

y > 0

to identify the final value.

y = 4

The value

y = -3

is negative, so it does not satisfy the constraint

y > 0

.

Anahtar Kavram

Solving quadratic equations of the form

(ax + b)^2 = c

using the square root property.

Soru 206Soru →

In the $xy$ -plane, a circle has center $(6, 8)$ and is tangent to the $y$ -axis. A line with a positive slope $m$ passes through the origin and is tangent to the circle. If $m$ is written as a fraction in simplest form, $\frac{p}{q}$ , what is the value of $p + q$ ?

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Cevap: 31

Cevap

The value of

p + q

is 31.

The radius of the circle is 6 since the center is

(6, 8)

and it is tangent to the

y

-axis. Substituting the line

y = mx

into the circle's equation

(x-6)^2 + (y-8)^2 = 36

yields a quadratic equation in

x

:

(1+m^2)x^2 - (12+16m)x + 64 = 0

. Since the line is tangent to the circle, there is exactly one solution, meaning the discriminant of this quadratic must equal zero:

(12+16m)^2 - 4(1+m^2)(64) = 0

. Simplifying this equation yields

24m - 7 = 0

, which gives a slope of

m = \frac{7}{24}

. Since the fraction is in simplest form,

p=7

and

q=24

, and their sum is

31

.

Adım Adım Çözüm

1

Determine the radius of the circle.

r = 6

Since the center is at

(6, 8)

and the circle is tangent to the

y

-axis, the radius is the horizontal distance from the center to the

y

-axis.

2

Write the equation of the circle.

(x-6)^2 + (y-8)^2 = 36

A circle with center

(h, k)

and radius

r

has the equation

(x-h)^2 + (y-k)^2 = r^2

.

3

Substitute

y = mx

into the circle's equation.

(1+m^2)x^2 - (12+16m)x + 64 = 0

Substituting the equation of the line

y = mx

into the circle's equation and expanding gives a quadratic equation in terms of

x

.

4

Set the discriminant of the quadratic equation to zero.

\Delta = (12+16m)^2 - 256(1+m^2) = 0

For the line to be tangent to the circle, the system must have exactly one real solution, meaning the quadratic equation must have a discriminant of zero.

5

Solve for the positive slope

m

.

m = \frac{7}{24}

Expanding and simplifying the discriminant equation yields

24m - 7 = 0

, which gives

m = \frac{7}{24}

.

6

Calculate

p + q

using the simplest form fraction.

31

Since

m = \frac{7}{24}

is in simplest form,

p = 7

and

q = 24

. Therefore,

p + q = 7 + 24 = 31

.

Anahtar Kavram

Nonlinear Systems of Equations

Soru 207Soru →

In the inequality $-2(3x - k) - 5(x + 1) > 17$ , $k$ is an integer constant. If the maximum integer value of $x$ that satisfies the inequality is $2$ , what is the least possible value of $k$ ?

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Cevap: 23

Cevap

The correct answer is 23.

By simplifying the inequality to

x < \frac{2k - 22}{11}

, we establish that the upper bound of the solution interval must be strictly greater than

2

but less than or equal to

3

for

2

to be the maximum integer solution. Solving the resulting compound inequality

2 < \frac{2k - 22}{11} \le 3

yields

22 < k \le 27.5

. The smallest integer within this range is

23

.

Adım Adım Çözüm

1

Distribute the constants on the left side of the inequality.

-6x + 2k - 5x - 5 > 17

Applying the distributive property simplifies the expression and removes the parentheses.

2

Combine like terms on the left side and isolate the

x

term.

-11x + 2k - 5 > 17 \implies -11x > 22 - 2k

Grouping

x

terms together and moving the constant terms to the other side prepares the inequality for division.

3

Divide both sides by

-11

and flip the inequality sign.

x < \frac{22 - 2k}{-11} \implies x < \frac{2k - 22}{11}

Dividing by a negative number reverses the direction of the inequality sign from greater-than (

>

) to less-than (

<

).

4

Set up the inequality for the maximum integer solution to be

2

.

2 < \frac{2k - 22}{11} \le 3

For

2

to be the largest integer satisfying

x < L

(where

L

is the boundary),

2

must be strictly less than

L

, and

L

must be less than or equal to the next integer,

3

.

5

Solve the compound inequality for the parameter

k

.

22 < 2k - 22 \le 33 \implies 44 < 2k \le 55 \implies 22 < k \le 27.5

Multiplying all parts by 11, adding 22, and dividing by 2 isolates

k

.

6

Find the least integer value of

k

in the interval

(22, 27.5]

.

23

The integers that satisfy

22 < k \le 27.5

are

23, 24, 25, 26,

and

27

. The least of these values is

23

.

Anahtar Kavram

Solving linear inequalities in one variable involving parameter bounds, negative coefficients, and integer solution constraints.

Soru 208Soru →

For a wireless sensor node in an environmental monitoring network, the battery life $B$ , in days, when transmitting $n$ data packets per day is modeled by a linear function. Under standard operating conditions, the node's battery lasts for $120$ days when transmitting $80$ packets per day, and it lasts for $90$ days when transmitting $140$ packets per day. Under a new energy-saving firmware, the battery consumption rate per packet transmitted is reduced by $40\%$ , and the static power draw is also reduced, increasing the maximum battery life (when zero packets are transmitted) by $10\%$ . Under this new firmware, what is the battery life, in days, of a sensor node that transmits $150$ data packets per day?

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Cevap: 131

Cevap

131

To solve this problem, we first determine the linear relationship

B = mn + B_0

under standard operating conditions. The rate of change (slope

m

) represents the change in battery life per packet transmitted per day:

m = \frac{90 - 120}{140 - 80} = -0.5

days per packet. Substituting this back into the linear equation gives

120 = -0.5(80) + B_0

, which simplifies to

120 = -40 + B_0

, yielding a y-intercept of

B_0 = 160

days. Under the energy-saving firmware, the battery consumption rate per packet is reduced by

40\%

. This means the rate at which battery life decreases per packet changes from

0.5

to

0.5 \times (1 - 0.40) = 0.3

, giving us a new slope of

-0.3

. The maximum battery life (y-intercept) increases by

10\%

, making the new y-intercept

160 \times 1.10 = 176

. The new linear model is

B_{\text{new}} = -0.3n + 176

. Evaluating this equation for

n = 150

packets per day yields

B_{\text{new}} = -0.3(150) + 176 = -45 + 176 = 131

days.

Adım Adım Çözüm

1

Determine the linear relationship representing standard operating conditions by using the two given points,

(80, 120)

and

(140, 90)

.

Slope

m = -0.5

days per packet and y-intercept

B_0 = 160

days.

Establishing the initial linear equation is necessary to obtain the base battery consumption rate and maximum battery capacity.

2

Calculate the new slope and y-intercept parameters under the energy-saving firmware by applying the specified percentage changes.

New slope

m_{\text{new}} = -0.3

days per packet and new y-intercept

B_{0,\text{new}} = 176

days.

To model the linear relationship under the updated power management firmware.

3

Formulate the new linear function

B_{\text{new}} = -0.3n + 176

and substitute

150

for

n

.

Battery life

B = 131

days.

To find the expected battery life at the target daily packet transmission rate.

Anahtar Kavram

Interpreting how physical rates and initial values map to the slope and y-intercept of a linear model, and applying transformations to these parameters.

Soru 209Soru →

A moving company uses the equation $C = 2.5d + 75$ to determine the total charge $C$ , in dollars, for renting a small truck and driving it $d$ miles. According to the relationship, what is the charge, in dollars, for each mile the truck is driven?

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Cevap: 2.5

Cevap

The charge for each mile the truck is driven is 2.5 dollars.

In the linear model

C = 2.5d + 75

, the total cost

C

is a function of the number of miles driven

d

. The rate of change of this function represents the cost per mile driven. In the equation, this rate of change is the coefficient of

d

, which is 2.5. Therefore, the charge for each mile the truck is driven is 2.5 dollars.

Adım Adım Çözüm

1

Analyze the linear equation

C = 2.5d + 75

to determine the relationship between variables.

The total charge

C

depends on the number of miles

d

driven, with a rate of change of 2.5 dollars per mile and a base fee of 75 dollars.

Understanding the components of a linear equation helps isolate the rate of change.

2

Identify the coefficient of the independent variable

d

.

The coefficient of

d

is 2.5.

In a linear equation of the form

y = mx + b

, the coefficient of the independent variable represents the rate of change.

3

Interpret the meaning of this coefficient in the context of the problem.

The coefficient 2.5 represents the cost, in dollars, incurred per mile driven.

The question asks for the charge per mile, which corresponds to the rate of change.

Anahtar Kavram

Interpreting Linear Relationships in Context

Soru 210Soru →

A system of two equations is given:

y = x^2 + kx + 4

y = 4x - k

In these equations, $k$ represents a positive constant. If the system has a single real solution, what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

12

Setting the two equations equal yields the quadratic equation

x^2 + (k - 4)x + (k + 4) = 0

. For the system to have exactly one real solution, the discriminant of this quadratic equation must be equal to

0

. The discriminant is

(k - 4)^2 - 4(1)(k + 4) = k^2 - 12k

. Solving

k^2 - 12k = 0

yields

k = 0

or

k = 12

. Since

k

must be positive, the value of

k

is

12

.

Adım Adım Çözüm

1

Set the two equations equal to each other to form a single quadratic equation in terms of

x

.

x^2 + (k - 4)x + (k + 4) = 0

Equating the expressions for

y

allows us to find the

x

-coordinates where the graphs of the two equations intersect.

2

Set the discriminant of the quadratic equation to zero.

(k - 4)^2 - 4(1)(k + 4) = 0

A system of equations consisting of a line and a parabola has a single real solution if and only if the line is tangent to the parabola, which corresponds to a quadratic equation with a discriminant of zero.

3

Expand and simplify the equation for

k

.

k^2 - 12k = 0

Expanding

(k - 4)^2

yields

k^2 - 8k + 16

, and distributing

-4

yields

-4k - 16

. Combining like terms simplifies the relation.

4

Solve for

k

and apply the constraint that

k > 0

.

k = 12

Factoring

k(k - 12) = 0

gives

k = 0

or

k = 12

. Since

k

is specified to be positive,

k = 12

is the correct value.

Anahtar Kavram

Solving nonlinear systems of equations by substitution and using the discriminant to find conditions for a single real solution.

Soru 211Soru →

A software programmer is writing test cases for a new application. The programmer must write at least 15 test cases in total, consisting of $x$ unit tests and $y$ integration tests. Each unit test takes 10 minutes to write, and each integration test takes 30 minutes to write. If the programmer has at most 300 minutes to write all the test cases, what is the maximum number of integration tests the programmer can write?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

7

Adım Adım Çözüm

1

Set up the system of inequalities representing the given constraints.

x + y \ge 15

and

10x + 30y \le 300

The total number of tests must be at least 15, and the total time taken by writing

x

unit tests (10 minutes each) and

y

integration tests (30 minutes each) cannot exceed 300 minutes.

2

Simplify the time inequality and combine it with the total test count constraint to isolate

y

.

x + 3y \le 30

. Substituting

x \ge 15 - y

into this inequality yields

(15 - y) + 3y \le 30 \implies 15 + 2y \le 30

.

Simplification and substitution help find the upper bound for the number of integration tests.

3

Solve for

y

and determine the maximum integer value.

2y \le 15 \implies y \le 7.5

. The largest integer satisfying this inequality is 7.

The number of integration tests must be a whole number, so we round down to the nearest integer.

4

Verify that a valid integer number of unit tests (

x

) exists when

y = 7

.

When

y = 7

, we get

x \ge 15 - 7 \implies x \ge 8

and

x + 3(7) \le 30 \implies x \le 9

. The integers

x = 8

and

x = 9

both satisfy the conditions.

We must confirm that the maximum value of

y

is achievable with an integer number of unit tests.

Anahtar Kavram

Systems of Linear Inequalities in Two Variables

Soru 212Soru →

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

\begin{aligned} x^2 + y^2 &= 50 \\ x - y &= 6 \end{aligned}

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

8

Solving the linear equation for

x

gives

x = y + 6

. Substituting this into the circle equation yields

(y + 6)^2 + y^2 = 50

, which simplifies to

2y^2 + 12y - 14 = 0

. Dividing by

2

gives

y^2 + 6y - 7 = 0

, which factors as

(y + 7)(y - 1) = 0

. This yields

y = 1

(with

x = 7

) or

y = -7

(with

x = -1

). The constraint

x > 0

requires selecting the solution

(7, 1)

. The sum of

x

and

y

for this solution is

7 + 1 = 8

.

Adım Adım Çözüm

1

Express

x

in terms of

y

using the linear equation.

x = y + 6

This allows for substitution into the second equation to reduce the system to a single variable.

2

Substitute the expression for

x

into the equation of the circle and expand.

2y^2 + 12y - 14 = 0

Substituting

x = y + 6

into

x^2 + y^2 = 50

yields

(y + 6)^2 + y^2 = 50

. Expanding gives

y^2 + 12y + 36 + y^2 = 50

.

3

Simplify and solve the quadratic equation by factoring.

y = 1

or

y = -7

Dividing the equation by 2 gives

y^2 + 6y - 7 = 0

, which factors as

(y + 7)(y - 1) = 0

.

4

Calculate the corresponding values of

x

for each

y

.

For

y = 1

,

x = 7

; for

y = -7

,

x = -1

Using the relation

x = y + 6

yields the coordinates of the two intersection points.

5

Apply the constraint

x > 0

and calculate

x + y

.

x + y = 8

Since

x > 0

, the only valid solution is

(7, 1)

. The sum of the coordinates is

7 + 1 = 8

.

Anahtar Kavram

Solving a nonlinear system of equations representing a circle and a line using substitution and factoring.

Soru 213Soru →

In the $xy$ -plane, the graph of the linear equation $3x + 5y = c$ , where $c$ is a constant, passes through the point $(4, 3)$ . What is the $x$ -coordinate of the point on this graph where the $y$ -coordinate is $-3$ ?

Cevabı ve açıklamayı göster

Cevap: 14

Cevap

The correct answer is 14.

To find the correct answer, first substitute the given point

(4, 3)

into the equation

3x + 5y = c

to determine the value of the constant

c

:

3(4) + 5(3) = 12 + 15 = 27

. Thus, the equation is

3x + 5y = 27

. Next, substitute

-3

for

y

in this equation to find the corresponding

x

-coordinate:

3x + 5(-3) = 27 \implies 3x - 15 = 27

. Adding

15

to both sides gives

3x = 42

, and dividing by

3

yields

x = 14

.

Adım Adım Çözüm

1

Substitute the point

(4, 3)

into the equation

3x + 5y = c

to solve for

c

.

c = 27

Since the point lies on the graph of the equation, its coordinates must satisfy the equation.

2

Substitute

y = -3

and

c = 27

into the equation

3x + 5y = c

.

3x - 15 = 27

We want to find the

x

-coordinate of the point on the line when the

y

-coordinate is

-3

.

3

Solve the equation

3x - 15 = 27

for

x

.

x = 14

Isolating

x

gives the

x

-coordinate of the point.

Anahtar Kavram

Using a known point on a line to find a constant coefficient or constant term, and using the resulting equation to find other coordinates.

Alternatif Yöntem

Alternatively, you can write the equation in slope-intercept form. Solving

3x + 5y = c

for

y

gives

y = -\frac{3}{5}x + \frac{c}{5}

. The slope of the line is

-\frac{3}{5}

. Since the slope is constant, the change in

y

divided by the change in

x

between

(4, 3)

and

(x, -3)

is equal to the slope:

\frac{-3 - 3}{x - 4} = -\frac{3}{5} \implies \frac{-6}{x - 4} = -\frac{3}{5}

. Cross-multiplying gives

-30 = -3(x - 4) \implies 10 = x - 4 \implies x = 14

.

Tahmini Süre:1m 30s

Soru 214Soru →

For all $x > 3$ , the expression $\frac{x^3 - 2x^2 - 9x + 18}{x^2 - 5x + 6} - \frac{x^3 + 3x^2 - 4x - 12}{x^2 + 5x + 6}$ is equivalent to the constant $k$ . What is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The value of the constant

k

is 5.

Factoring the numerators by grouping yields

x^3 - 2x^2 - 9x + 18 = (x-3)(x+3)(x-2)

and

x^3 + 3x^2 - 4x - 12 = (x-2)(x+2)(x+3)

. Dividing out their respective denominators

(x-2)(x-3)

and

(x+2)(x+3)

leaves the simplified linear expressions

x+3

and

x-2

. Subtracting these yields

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

.

Adım Adım Çözüm

1

Factor the numerator and denominator of the first rational expression and cancel common factors.

x + 3

To simplify the first fraction by reducing it to its lowest terms.

2

Factor the numerator and denominator of the second rational expression and cancel common factors.

x - 2

To simplify the second fraction by reducing it to its lowest terms.

3

Subtract the second simplified expression from the first simplified expression.

5

To find the constant value equivalent to the given difference of rational expressions, ensuring to distribute the negative sign to all parts of the subtracted binomial.

Anahtar Kavram

Simplifying rational expressions by factoring cubic polynomials by grouping and quadratic trinomials

Soru 215Soru →

If the expression $\frac{6x^2 + 7x - 20}{2x - 3}$ is equivalent to $ax + b + \frac{c}{2x-3}$ for all $x \neq 1.5$ , where $a$ , $b$ , and $c$ are constants, what is the value of $a + b + c$ ?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

The value of

a + b + c

is 15.

By dividing the numerator

6x^2 + 7x - 20

by the denominator

2x - 3

, we find that the quotient is

3x + 8

and the remainder is

4

. Thus, the expression can be rewritten as

3x + 8 + \frac{4}{2x-3}

. Comparing this to the given expression

ax + b + \frac{c}{2x-3}

, we obtain

a = 3

,

b = 8

, and

c = 4

. Their sum is

3 + 8 + 4 = 15

.

Adım Adım Çözüm

1

Set up the polynomial division of the numerator

6x^2 + 7x - 20

by the denominator

2x - 3

.

Dividing

6x^2 + 7x - 20

by

2x - 3

.

To express the rational expression in terms of a polynomial quotient and a remainder.

2

Divide the first term of the numerator by the first term of the denominator to determine the first quotient term.

The first term is

3x

. Subtracting

3x(2x - 3)

from the numerator leaves

16x - 20

.

\frac{6x^2}{2x} = 3x

, and subtracting

6x^2 - 9x

from the polynomial leaves the next term to be divided.

3

Divide the leading term of the remaining expression by the leading term of the denominator to determine the constant term of the quotient.

The constant term is

8

. Subtracting

8(2x - 3)

from

16x - 20

leaves a remainder of

4

.

\frac{16x}{2x} = 8

, and subtracting

16x - 24

from

16x - 20

gives the final constant remainder.

4

Compare the quotient and remainder to the given form to identify

a

,

b

, and

c

.

a = 3

,

b = 8

, and

c = 4

.

The quotient is

3x + 8

and the remainder is

4

, which matches the form

ax + b + \frac{c}{2x-3}

.

5

Calculate the sum of

a

,

b

, and

c

.

3 + 8 + 4 = 15

.

The question asks for the value of

a + b + c

.

Anahtar Kavram

Polynomial division and rewrite of rational expressions

Soru 216Soru →

A commercial cargo aircraft's total weight $W$ , in kilograms, is modeled as a linear function of the volume of fuel $F$ , in liters, in its fuel tanks. The equation modeling this relationship is:

W = 0.8F + 74,000

To comply with runway safety regulations at a destination airport, the aircraft's total weight must not exceed $85,000$ kilograms upon landing. The aircraft takes off with $18,000$ liters of fuel and consumes fuel at a constant rate of $1,700$ liters per hour of flight. What is the minimum number of hours the aircraft must fly before it can safely land?

Cevabı ve açıklamayı göster

Cevap: 2.5

Cevap

The minimum number of hours the aircraft must fly is 2.5.

To satisfy runway safety regulations, the aircraft's weight must be at most

85,000

kg. According to the weight model

W = 0.8F + 74,000

, we set

0.8F + 74,000 \leq 85,000

, which simplifies to

0.8F \leq 11,000

. Dividing by the density coefficient

0.8

yields

F \leq 13,750

liters as the maximum amount of fuel the aircraft can contain at landing. Since the aircraft takes off with

18,000

liters of fuel and consumes

1,700

liters per hour, the fuel remaining after

t

hours is

18,000 - 1,700t

. To ensure the remaining fuel is less than or equal to

13,750

liters, we solve

18,000 - 1,700t \leq 13,750

, which simplifies to

4,250 \leq 1,700t

, or

t \geq 2.5

hours. Therefore, the minimum duration of the flight is 2.5 hours.

Adım Adım Çözüm

1

Formulate the weight limit inequality using the linear relationship.

0.8F + 74,000 \leq 85,000

The aircraft's weight

W

is given by

0.8F + 74,000

, where

0.8

represents the density of the fuel in kg/L and

74,000

represents the weight of the empty aircraft and its cargo. This total weight must not exceed

85,000

kg.

2

Calculate the maximum fuel capacity allowed at landing by solving the inequality.

F \leq 13,750

Subtracting

74,000

from both sides yields

0.8F \leq 11,000

. Dividing by

0.8

gives

F \leq 13,750

liters as the fuel ceiling for landing.

3

Use the fuel consumption rate to find the minimum flight time.

t \geq 2.5

With an initial fuel volume of

18,000

liters and a burn rate of

1,700

liters/hour, the fuel remaining after

t

hours is

18,000 - 1,700t

. Setting this expression to be at most

13,750

liters yields

18,000 - 1,700t \leq 13,750

. Subtracting

18,000

gives

-1,700t \leq -4,250

, and dividing by

-1,700

results in

t \geq 2.5

hours.

Anahtar Kavram

Interpreting linear coefficients in context and setting up linear inequalities with rates to model real-world constraints.

Tahmini Süre:2m 30s

Soru 217Soru →

A scientist studying a sample of a radioactive isotope determines that its mass decays exponentially. The mass of the isotope, in grams, is modeled by the function $M(t) = a \cdot b^t$ , where $t$ is the time, in days, since the study began, and $a$ and $b$ are constants. The table below shows the mass of the isotope for selected values of $t$ .

$t$ (days)	$M(t)$ (grams)
0	128
1	96
2	72

What is the value of $b$ ?

Cevabı ve açıklamayı göster

Cevap: 0.75

Cevap

0.75

To find the constant

b

in the exponential decay function

M(t) = a \cdot b^t

, we can use the given table values. For

t = 0

,

M(0) = a \cdot b^0 = a = 128

. For

t = 1

,

M(1) = a \cdot b^1 = a \cdot b = 96

. Substituting

a = 128

into the second equation gives

128b = 96

. Dividing both sides by

128

yields

b = \frac{96}{128} = 0.75

(or

\frac{3}{4}

). We can verify this with

t = 2

:

M(2) = 128 \cdot (0.75)^2 = 128 \cdot 0.5625 = 72

, which matches the table.

Adım Adım Çözüm

1

Determine the initial value

a

using the data point for

t = 0

.

a = 128

Substituting

t = 0

into

M(t) = a \cdot b^t

gives

M(0) = a \cdot b^0 = a

. Since the table shows

M(0) = 128

,

a

must equal

128

.

2

Use the data point for

t = 1

to write an equation for

b

.

128b = 96

Substituting

t = 1

and

a = 128

into the function gives

M(1) = 128 \cdot b^1 = 128b

. The table shows

M(1) = 96

.

3

Solve the equation to find

b

.

b = 0.75

Dividing both sides of

128b = 96

by

128

yields

b = \frac{96}{128}

, which simplifies to

0.75

(or

\frac{3}{4}

).

Anahtar Kavram

Finding the decay factor (base) of an exponential function from a table of values.

Soru 218Soru →

A logistics company uses two types of boxes, small and large, to ship items. A shipment of $8$ small boxes and $5$ large boxes has a total weight of $180$ pounds. A second shipment of $6$ small boxes and $10$ large boxes has a total weight of $260$ pounds. What is the weight, in pounds, of one large box?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The weight of one large box is 20 pounds.

The correct weight of one large box is 20 pounds. By formulating the system of linear equations representing the total weight of each shipment (

8s + 5L = 180

and

6s + 10L = 260

), we can eliminate

L

by multiplying the first equation by 2, resulting in

16s + 10L = 360

. Subtracting the second equation from this gives

10s = 100

, which solves to

s = 10

. Substituting

s = 10

back into

8s + 5L = 180

yields

80 + 5L = 180

, which simplifies to

5L = 100

, meaning

L = 20

.

Adım Adım Çözüm

1

Set up a system of two linear equations based on the shipments.

Let

s

be the weight of a small box and

L

be the weight of a large box. The system of equations is:

\begin{aligned} 8s + 5L &= 180 \\ 6s + 10L &= 260 \end{aligned}

Translating the verbal statements into mathematical symbols represents the problem systemically.

2

Multiply the first equation by 2 to align the coefficients of

L

.

16s + 10L = 360

This sets up the variable

L

to have the same coefficient in both equations, allowing for elimination.

3

Subtract the second equation from the modified first equation to solve for

s

.

(16s + 10L) - (6s + 10L) = 360 - 260

10s = 100

s = 10

Subtracting eliminates the variable

L

, leaving a single-variable linear equation to solve.

4

Substitute

s = 10

back into the first equation to solve for

L

.

8(10) + 5L = 180

80 + 5L = 180

5L = 100

L = 20

Substituting the value of the solved variable determines the value of the remaining variable.

Anahtar Kavram

Solving systems of linear equations in two variables using elimination or substitution.

Soru 219Soru →

If $x^2 - 9x + 18 = 0$ and $x > 4$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The value of

x

is 6.

Factoring the equation

x^2 - 9x + 18 = 0

gives

(x - 3)(x - 6) = 0

, which yields the solutions

x = 3

and

x = 6

. Applying the constraint

x > 4

, the only valid solution is 6.

Adım Adım Çözüm

1

Factor the quadratic equation

x^2 - 9x + 18 = 0

.

(x - 3)(x - 6) = 0

Finding two numbers that multiply to 18 and add to -9 allows us to write the quadratic expression in its factored form.

2

Solve for the roots of the equation.

x = 3

or

x = 6

By the zero product property, setting each factor to zero yields the possible solutions for the equation.

3

Apply the given constraint

x > 4

.

x = 6

Since the question specifies that

x

must be greater than 4, the root

x = 3

is discarded, leaving

x = 6

as the only valid solution.

Anahtar Kavram

Solving quadratic equations by factoring and applying constraints.

Soru 220Soru →

The quadratic function $f$ is defined by $f(x) = (x - 5)(x - 11)$ . In the $xy$ -plane, the graph of $y = f(x)$ is a parabola with vertex $(h, k)$ . What is the value of $h$ ?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The value of

h

is

8

.

The given quadratic function is in factored form,

f(x) = (x - 5)(x - 11)

, which shows that the

x

-intercepts of the graph are at

x = 5

and

x = 11

. Because a parabola is symmetric, the

x

-coordinate of its vertex,

h

, is located exactly halfway between the two

x

-intercepts. Calculating the average of the

x

-intercepts gives

h = \frac{5 + 11}{2} = 8

.

Adım Adım Çözüm

1

Find the

x

-intercepts of the parabola.

The

x

-intercepts are

x = 5

and

x = 11

.

The

x

-intercepts of a quadratic function in the form

f(x) = (x - r_1)(x - r_2)

occur at

x = r_1

and

x = r_2

because setting

f(x) = 0

yields these roots.

2

Calculate the

x

-coordinate of the vertex,

h

.

h = 8

Since a parabola is symmetric, its vertex lies on the axis of symmetry, which is the vertical line halfway between the

x

-intercepts. The average of

5

and

11

is

\frac{5 + 11}{2} = 8

.

Anahtar Kavram

Finding the vertex of a parabola from its factored form

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Alternatif Yöntem

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