Tüm alıştırma soruları

Soru 81Soru →

A linear equation models the total cost, in dollars, of purchasing $x$ pounds of almonds and $y$ pounds of walnuts. Under the original pricing, purchasing $8$ pounds of almonds and $15$ pounds of walnuts costs $c$ dollars. If the price per pound of almonds is increased by $25\%$ and the price per pound of walnuts is decreased by $10\%$ , the cost of purchasing $8$ pounds of almonds and $15$ pounds of walnuts is still $c$ dollars. Under the original pricing, a customer can purchase exactly $22$ pounds of almonds and no walnuts for $c$ dollars. How many pounds of walnuts and no almonds can the customer purchase for $c$ dollars under the original pricing?

Cevabı ve açıklamayı göster

Cevap: 16.5

Cevap

16.5

Under the original pricing, the cost of

8

pounds of almonds at

a

dollars per pound and

15

pounds of walnuts at

b

dollars per pound is represented by the linear equation

8a + 15b = c

. Under promotional pricing, the cost of almonds increases by

25\%

to

1.25a

, and the cost of walnuts decreases by

10\%

to

0.90b

. The new cost for the same amounts of nuts is

8(1.25a) + 15(0.90b) = 10a + 13.5b = c

. Setting the two cost equations equal to each other gives

8a + 15b = 10a + 13.5b

, which simplifies to

1.5b = 2a

, or

a = 0.75b

. Since

22

pounds of almonds can be purchased for

c

dollars, we have

22a = c

. Substituting

a = 0.75b

yields

22(0.75b) = c

, which simplifies to

16.5b = c

. Thus, exactly

16.5

pounds of walnuts can be purchased for

c

dollars.

Adım Adım Çözüm

1

Set up the equation representing the total cost of the initial purchase under the original pricing.

8a + 15b = c

, where

a

is the original price per pound of almonds and

b

is the original price per pound of walnuts.

To represent the cost relation using the original individual prices of almonds and walnuts.

2

Modify the individual prices for the promotional rates and write the new total cost equation.

8(1.25a) + 15(0.90b) = c \implies 10a + 13.5b = c

.

The price per pound of almonds increases by

25\%

(multiplied by

1.25

) and the price per pound of walnuts decreases by

10\%

(multiplied by

0.90

).

3

Equate the two expressions representing

c

to find the ratio between the prices

a

and

b

.

8a + 15b = 10a + 13.5b \implies 2a = 1.5b \implies a = 0.75b

.

Since both purchasing combinations yield the same total budget

c

, their cost equations are equal.

4

Use the budget equation for purchasing only almonds to determine the equivalent purchase in walnuts.

22a = c \implies 22(0.75b) = c \implies 16.5b = c

. Therefore,

16.5

pounds of walnuts can be bought for

c

dollars.

Substituting the price relationship

a = 0.75b

allows expressing the budget

c

purely in terms of the price of walnuts

b

.

Anahtar Kavram

Modeling linear relationships in two variables and analyzing changes in coefficients.

Soru 82Soru →

If the equations $x + 2y = 12$ and $x + y = 7$ are true for the same values of $x$ and $y$ , what is the value of $y$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The value of

y

is 5.

Subtracting the equation

x + y = 7

from

x + 2y = 12

gives

(x - x) + (2y - y) = 12 - 7

, which simplifies directly to

y = 5

.

Adım Adım Çözüm

1

Subtract the second equation from the first equation.

y = 5

Subtracting

x + y = 7

from

x + 2y = 12

eliminates the

x

terms, leaving a single linear equation in terms of

y

.

Anahtar Kavram

Solving systems of linear equations using elimination.

Soru 83Soru →

In the $xy$ -plane, the graph of the linear function $f$ passes through the point $(6, 4)$ . Line $g$ is perpendicular to the graph of $f$ and has a $y$ -intercept of $(0, 12)$ . The graph of $f$ and line $g$ intersect at the point $P(h, k)$ , where $h$ and $k$ are integers. If the slope of the graph of $f$ is a positive integer greater than $5$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

The value of

k

is

11

.

By equating the linear equations representing

f

and

g

, we find that the intersection point's

x

-coordinate

h

is given by

h = 6 + \frac{8m - 6}{m^2 + 1}

. Since

h

is an integer and

m

is a positive integer greater than

5

, the inequality

m^2 + 1 \le 8m - 6

limits

m

to the range

1 \le m \le 7

. Testing the integers greater than

5

(which are

6

and

7

) reveals that only

m = 7

yields an integer value for

h

, resulting in

h = 7

. Substituting these values back into the equation of either line yields

k = 11

.

Adım Adım Çözüm

1

Write the equation of the linear function

f

in terms of its slope

m

.

y - 4 = m(x - 6) \implies y = mx + 4 - 6m

Since the graph of

f

passes through the point

(6, 4)

with slope

m

, we can use the point-slope form of a linear equation.

2

Write the equation of line

g

in slope-intercept form.

y = -\frac{1}{m}x + 12

Line

g

is perpendicular to the graph of

f

, so its slope is the negative reciprocal of

m

, which is

-\frac{1}{m}

. Its

y

-intercept is

(0, 12)

.

3

Equate the two equations to find the

x

-coordinate

h

of the intersection point

P(h, k)

in terms of

m

.

h = \frac{6m^2 + 8m}{m^2 + 1}

At the intersection point, the

y

-values of both equations are equal, so we solve

mx + 4 - 6m = -\frac{1}{m}x + 12

for

x

.

4

Analyze the expression for

h

to find the integer constraints on

m

.

h = 6 + \frac{8m - 6}{m^2 + 1} \implies m^2 - 8m + 7 \le 0 \implies 1 \le m \le 7

For

h

to be an integer,

\frac{8m - 6}{m^2 + 1}

must be an integer. Since

m

is a positive integer greater than

5

, the numerator is positive, meaning the denominator

m^2 + 1

must be less than or equal to the numerator

8m - 6

.

5

Test the possible integer values of

m

greater than

5

to find the one that yields integer coordinates.

m = 7 \implies h = 7

The only integers greater than

5

in the range

[1, 7]

are

6

and

7

. Testing

m=6

gives a non-integer, while

m=7

gives

h = 7

, which is an integer.

6

Calculate the

y

-coordinate

k

of the intersection point

P

.

k = 11

Substitute

h = 7

into the equation of line

g

:

k = -\frac{1}{7}(7) + 12 = 11

.

Anahtar Kavram

Linear Functions and Graphs

Soru 84Soru →

In the $xy$ -plane, the graph of the linear function $f$ has a slope of $m$ and a $y$ -intercept of $(0, b)$ . The graph of the linear function $g$ has a slope of $-3m$ and a $y$ -intercept of $(0, b + 15)$ . If the graphs of $f$ and $g$ intersect at the point $(3, 5)$ , what is the value of $f(5)$ ?

Cevabı ve açıklamayı göster

Cevap: 7.5

Cevap

7.5 (or 15/2)

By writing the equations of the two lines as

f(x) = mx + b

and

g(x) = -3mx + b + 15

, and substituting the intersection point

(3, 5)

, we get the system of equations

3m + b = 5

and

-9m + b = -10

. Subtracting these equations gives

12m = 15

, which simplifies to

m = 1.25

. Substituting

m

back into

3m + b = 5

yields

b = 1.25

. The function is therefore

f(x) = 1.25x + 1.25

. Evaluating this at

x = 5

gives

f(5) = 1.25(5) + 1.25 = 7.5

.

Adım Adım Çözüm

1

Formulate equations for both linear functions using slope-intercept form.

f(x) = mx + b

and

g(x) = -3mx + b + 15

The equation of a line with slope

M

and

y

-intercept

(0, B)

is given by

y = Mx + B

.

2

Substitute the coordinates of the intersection point

(3, 5)

into both equations.

3m + b = 5

and

-9m + b = -10

Since the intersection point lies on both lines, its coordinates must satisfy both equations.

3

Solve the system of equations for

m

and

b

.

m = 1.25

and

b = 1.25

Subtracting the equations eliminates

b

, giving

12m = 15 \implies m = 1.25

. Substituting this back into the first equation gives

3(1.25) + b = 5 \implies b = 1.25

.

4

Evaluate

f(5)

using the found values.

f(5) = 1.25(5) + 1.25 = 7.5

Substitute

x = 5

into the equation

f(x) = 1.25x + 1.25

.

Anahtar Kavram

Solving systems of linear equations and evaluating linear functions from graph intersections.

Tahmini Süre:2m 0s

Soru 85Soru →

In the $xy$ -plane, a line passes through the origin and has a slope of $\frac{3}{5}$ . If the point $(k, 9)$ lies on the line, what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

15

A line passing through the origin has a

y

-intercept of

0

. Thus, its equation in slope-intercept form is

y = mx

, where

m

is the slope. Given the slope is

\frac{3}{5}

, the equation is

y = \frac{3}{5}x

. Since the point

(k, 9)

lies on the line, substituting these coordinates yields

9 = \frac{3}{5}k

. Multiplying both sides by

\frac{5}{3}

gives

k = 15

.

Adım Adım Çözüm

1

Write the equation of the line in slope-intercept form.

y = \frac{3}{5}x

The line passes through the origin

(0,0)

, so the

y

-intercept is

0

, and the slope is

\frac{3}{5}

.

2

Substitute the coordinates of the point

(k, 9)

into the line's equation.

9 = \frac{3}{5}k

A point lies on a line if its coordinates satisfy the equation of the line.

3

Solve the equation for

k

.

k = 15

Multiply both sides of the equation by

\frac{5}{3}

to isolate

k

.

Anahtar Kavram

Using the slope and a point on a line to determine an unknown coordinate.

Soru 86Soru →

For the linear function $f$ , the value of $f(0)$ is $3$ and the value of $f(5)$ is $18$ . What is the slope of the graph of $y = f(x)$ in the $xy$ -plane?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The slope of the graph of

y = f(x)

is

3

.

The slope of a linear function can be determined using any two points

(x_1, y_1)

and

(x_2, y_2)

on its graph with the formula

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

. The given function values

f(0) = 3

and

f(5) = 18

correspond to the coordinate points

(0, 3)

and

(5, 18)

respectively. Substituting these coordinates into the formula gives

m = \frac{18 - 3}{5 - 0} = \frac{15}{5} = 3

.

Adım Adım Çözüm

1

Identify the coordinates of two points on the line using the given function values.

The function values

f(0) = 3

and

f(5) = 18

correspond to the points

(0, 3)

and

(5, 18)

on the graph of the function.

In function notation,

f(x) = y

represents a point

(x, y)

on the graph of the function.

2

Calculate the slope using the slope formula.

The slope

m

is calculated as

\frac{18 - 3}{5 - 0} = 3

.

The slope of a line is defined as the change in

y

divided by the change in

x

between any two points on the line.

Anahtar Kavram

Calculating the slope of a linear function from given function values.

Soru 87Soru →

If $3(2.5x - 4) - 1.5(x - 2) = 0.8(5x + 10)$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 8.5

Cevap

8.5

By distributing and simplifying terms, the linear equation simplifies to

2x = 17

, which yields

x = 8.5

.

Adım Adım Çözüm

1

Distribute the constants on both sides of the equation.

7.5x - 12 - 1.5x + 3 = 4x + 8

To simplify the expressions.

2

Combine like terms on the left side.

6x - 9 = 4x + 8

To simplify the left side of the equation.

3

Subtract

4x

from both sides of the equation.

2x - 9 = 8

To group variable terms on one side.

4

Add 9 to both sides of the equation.

2x = 17

To isolate the term with the variable.

5

Divide both sides by 2.

x = 8.5

To find the final value of the variable.

Anahtar Kavram

Linear Equations in One Variable

Alternatif Yöntem

Multiply the entire equation by 10 to clear all decimal values:

30(2.5x - 4) - 15(x - 2) = 8(5x + 10)

. Expanding this yields

75x - 120 - 15x + 30 = 40x + 80

, which simplifies to

60x - 90 = 40x + 80

. Subtracting

40x

and adding 90 to both sides gives

20x = 170

, resulting in

x = 8.5

.

Tahmini Süre:1m 30s

Soru 88Soru →

A linear function $f$ is defined by $f(x) = mx + b$ , where $m$ and $b$ are constants. The graph of $f$ in the $xy$ -plane passes through the point $(2, 5)$ . A second linear function $g$ is defined by $g(x) = f(x - 3) + 4$ . The graph of $g$ has an $x$ -intercept of $(k, 0)$ and a $y$ -intercept of $(0, -k)$ , where $k$ is a nonzero constant. What is the value of $f(8)$ ?

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

11

The correct answer is 11. By using the given point on the graph of

f(x)

, we express its y-intercept

b

in terms of its slope

m

as

b = 5 - 2m

. Substituting this expression into the translation equation

g(x) = f(x - 3) + 4

yields

g(x) = mx - 5m + 9

. Setting the y-intercept of

g(x)

, which is

-5m + 9

, equal to

-k

gives

k = 5m - 9

. Evaluating

g(x)

at its x-intercept

x = k

gives

mk - 5m + 9 = 0

. Substituting the relation

-5m + 9 = -k

into this equation yields

mk - k = 0

, or

k(m - 1) = 0

. Since

k

is nonzero, we find

m = 1

, which in turn gives

b = 3

. Thus, the linear function is

f(x) = x + 3

, and

f(8) = 11

.

Adım Adım Çözüm

1

Express the y-intercept constant

b

in terms of the slope

m

.

b = 5 - 2m

Since the graph of

f(x) = mx + b

passes through

(2, 5)

, substituting

x = 2

and

f(x) = 5

gives

5 = 2m + b

.

2

Write the expression for

g(x)

in terms of

m

.

g(x) = mx - 5m + 9

By definition,

g(x) = f(x - 3) + 4 = m(x - 3) + b + 4

. Substituting

b = 5 - 2m

simplifies the expression to

g(x) = mx - 5m + 9

.

3

Relate the y-intercept of

g(x)

to the parameter

k

.

k = 5m - 9

The y-intercept of

g(x)

is

g(0) = -5m + 9

. Since the y-intercept is given as

(0, -k)

, we set

-k = -5m + 9

, which gives

k = 5m - 9

.

4

Set up an equation using the x-intercept of

g(x)

.

g(k) = mk - 5m + 9 = 0

Since the x-intercept of

g(x)

is

(k, 0)

, substituting

x = k

into the expression for

g(x)

must yield

0

.

5

Solve for the slope

m

.

m = 1

Substituting

-5m + 9 = -k

into the equation

mk - 5m + 9 = 0

gives

mk - k = 0

, which factors as

k(m - 1) = 0

. Since

k

is a nonzero constant, we divide by

k

to get

m - 1 = 0

, so

m = 1

.

6

Find the constant

b

and write the final formula for

f(x)

.

f(x) = x + 3

Substituting

m = 1

back into

b = 5 - 2m

yields

b = 3

. Therefore,

f(x) = x + 3

.

7

Evaluate

f(8)

.

11

Substituting

x = 8

into the function

f(x) = x + 3

gives

f(8) = 8 + 3 = 11

.

Anahtar Kavram

Linear function transformations and intercept properties

Soru 89Soru →

The equation $3x + 4y = 36$ represents the relationship between the number of small boxes, $x$ , and large boxes, $y$ , that can fit in a delivery van. If the van is loaded with exactly 6 large boxes, what is the number of small boxes that can also fit in the van?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The number of small boxes that can also fit in the van is 4.

Substituting

y = 6

into the equation

3x + 4y = 36

gives

3x + 24 = 36

. Subtracting 24 from both sides results in

3x = 12

. Dividing both sides by 3 yields

x = 4

. Thus, the number of small boxes that can fit is 4.

Adım Adım Çözüm

1

Substitute 6 for

y

in the equation

3x + 4y = 36

.

3x + 4(6) = 36

The variable

y

represents the number of large boxes, which is given as 6.

2

Simplify the multiplication in the equation.

3x + 24 = 36

To simplify the terms before solving for

x

.

3

Subtract 24 from both sides of the equation.

3x = 12

To isolate the variable term

3x

on one side of the equation.

4

Divide both sides of the equation by 3.

x = 4

To solve for

x

, the number of small boxes.

Anahtar Kavram

Evaluating and solving a linear equation in two variables given the value of one variable.

Tahmini Süre:45s

Soru 90Soru →

The graph of the linear function $f$ in the $xy$ -plane is defined by $f(x) = kx + 4$ , where $k$ is a positive constant. The graph of the linear function $g$ is obtained by translating the graph of $f$ left by $3$ units and down by $2$ units. If the $x$ -intercept of the graph of $g$ is $5$ times the $x$ -intercept of the graph of $f$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

6

The correct answer is 6. The linear function

g(x)

represents the translated graph, which is expressed as

g(x) = f(x+3) - 2

. Substituting

f(x) = kx + 4

yields

g(x) = k(x+3) + 4 - 2 = kx + 3k + 2

. The

x

-intercept of a graph is the point where the function value is

0

. Solving

f(x) = 0

gives the

x

-intercept of the graph of

f

as

x = -\frac{4}{k}

. Solving

g(x) = 0

gives the

x

-intercept of the graph of

g

as

x = -\frac{3k + 2}{k}

. We are given that the

x

-intercept of the graph of

g

is

5

times the

x

-intercept of the graph of

f

, which translates to the equation

-\frac{3k+2}{k} = 5 \left(-\frac{4}{k}\right)

. Multiplying both sides by

-k

(since

k > 0

,

k \neq 0

) yields

3k + 2 = 20

. Subtracting

2

from both sides gives

3k = 18

, and dividing by

3

results in

k = 6

.

Adım Adım Çözüm

1

Determine the equation of the translated function

g(x)

g(x) = kx + 3k + 2

Shifting the graph of

f(x)

left by

3

units and down by

2

units translates to

g(x) = f(x+3) - 2

.

2

Find the

x

-intercept of the graph of

f

x = -\frac{4}{k}

The

x

-intercept is the input value for which the output of the function is zero, so we solve

f(x) = 0

.

3

Find the

x

-intercept of the graph of

g

x = -\frac{3k + 2}{k}

We set the equation of

g(x)

to zero and solve for

x

in terms of

k

.

4

Set up and solve the equation relating the two

x

-intercepts

k = 6

Using the relation

x_g = 5 \cdot x_f

, we obtain

-\frac{3k + 2}{k} = 5 \left(-\frac{4}{k}\right)

, which simplifies to

3k + 2 = 20

, leading to

k = 6

.

Anahtar Kavram

Function translations and finding intercepts of linear equations

Soru 91Soru →

In the $xy$ -plane, a line with a positive slope $m$ and a $y$ -intercept of $(0, b)$ passes through the point $(b, 12)$ , where $b > 0$ . If the area of the triangle bounded by the line, the $x$ -axis, and the $y$ -axis is $4$ , what is the value of $m$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

2

The line equation is

y = mx + b

. Substituting the point

(b, 12)

gives

12 = b(m + 1)

. The

x

-intercept of the line is at

(-\frac{b}{m}, 0)

, which gives a base length of

\frac{b}{m}

for the right triangle, while the height is

b

. The area of the triangle is

\frac{b^2}{2m} = 4 \implies b^2 = 8m

. Substituting

b = \frac{12}{m + 1}

into this equation yields

\frac{144}{(m + 1)^2} = 8m \implies m(m + 1)^2 = 18

. Solving this for positive values of

m

gives the unique solution

m = 2

.

Adım Adım Çözüm

1

Express the line equation and substitute the given point

(b, 12)

.

b(m + 1) = 12

Since the line has a

y

-intercept of

(0, b)

and a slope of

m

, its equation is

y = mx + b

. Substituting the point

(b, 12)

yields

12 = m(b) + b = b(m + 1)

.

2

Determine the lengths of the base and height of the triangle formed by the line and the coordinate axes.

Base

= \frac{b}{m}

, Height

= b

The

y

-intercept is at

(0, b)

and the

x

-intercept is at

(-\frac{b}{m}, 0)

. Since

b > 0

and

m > 0

, the height is

b

and the base is the absolute value of the

x

-intercept, which is

\frac{b}{m}

.

3

Write the area equation of the triangle and simplify it.

b^2 = 8m

The area of the right triangle is

\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \left(\frac{b}{m}\right)b = \frac{b^2}{2m}

. Setting this equal to the given area of

4

gives

\frac{b^2}{2m} = 4 \implies b^2 = 8m

.

4

Solve the system of equations for

m

.

m = 2

Substituting

b = \frac{12}{m + 1}

into

b^2 = 8m

yields

\left(\frac{12}{m + 1}\right)^2 = 8m \implies \frac{144}{(m + 1)^2} = 8m \implies m(m + 1)^2 = 18

. Since

m(m + 1)^2

is strictly increasing for positive

m

, testing small integers shows

2(3)^2 = 18

, so

m = 2

is the unique positive real solution.

Anahtar Kavram

Formulating linear equations in slope-intercept form and solving non-linear systems of equations derived from geometric constraints.

Alternatif Yöntem

Instead of algebraically expanding the cubic equation, one can test small positive integers for

m

in the equation

m(m + 1)^2 = 18

. Testing

m = 1

yields

1(2)^2 = 4

(too small), and testing

m = 2

yields

2(3)^2 = 18

(correct). Because

m(m + 1)^2

is strictly increasing for

m > 0

,

m = 2

is the only positive real root.

Tahmini Süre:3m 0s

Soru 92Soru →

If $-2(x - 5) \geq 4$ , what is the maximum possible value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

3

Distributing the

-2

on the left side of the inequality

-2(x - 5) \geq 4

yields

-2x + 10 \geq 4

. Subtracting

10

from both sides gives

-2x \geq -6

. Dividing both sides by

-2

and reversing the inequality sign results in

x \leq 3

. The maximum possible value is therefore 3.

Adım Adım Çözüm

1

Distribute

-2

to the terms inside the parentheses.

-2x + 10 \geq 4

Applying the distributive property simplifies the inequality.

2

Subtract

10

from both sides of the inequality.

-2x \geq -6

This isolates the variable term on the left side.

3

Divide both sides of the inequality by

-2

and reverse the inequality sign.

x \leq 3

Dividing both sides of an inequality by a negative number reverses the direction of the inequality.

Anahtar Kavram

Solving linear inequalities in one variable using the distributive property and division by negative numbers.

Soru 93Soru →

A company manufactures custom travel mugs. The daily cost $C$ , in dollars, to manufacture $n$ mugs is given by the formula $C = 1.25n + 250$ . The company sells each mug for $4.50. To make a profit of exactly$ 400$ in a day, how many mugs must the company manufacture?

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Cevap: 200

Cevap

The company must manufacture 200 mugs to make a profit of exactly $400 in a day.

Subtracting the cost expression from the revenue expression gives the profit equation:

4.50n - (1.25n + 250) = 400

. Distributing the negative sign results in

4.50n - 1.25n - 250 = 400

, which simplifies to

3.25n - 250 = 400

. Adding 250 to both sides gives

3.25n = 650

. Dividing both sides by 3.25 yields

n = 200

.

Adım Adım Çözüm

1

Set up the profit equation

4.50n - (1.25n + 250) = 400

Profit is calculated as total revenue (

4.50n

) minus total cost (

1.25n + 250

).

2

Distribute the negative sign and simplify the equation

3.25n - 250 = 400

Subtracting the entire cost expression requires distributing the subtraction to both

1.25n

and

250

, yielding

4.50n - 1.25n - 250

.

3

Isolate the variable term by adding 250 to both sides

3.25n = 650

Adding 250 to both sides of the equation isolates the term containing

n

.

4

Solve for

n

by dividing both sides by 3.25

n = 200

Dividing 650 by 3.25 gives the number of mugs needed to achieve the target profit.

Anahtar Kavram

Solving linear equations in one variable derived from a real-world context.

Soru 94Soru →

In the $xy$ -plane, the graph of the linear function $h$ has a slope of $3$ and passes through the point $(5, 8)$ . What is the value of $h(11)$ ?

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Cevap: 26

Cevap

The value of

h(11)

is 26.

The correct answer is 26. Since the function

h

is linear, its rate of change (slope) is constant. A slope of 3 means that for every 1-unit increase in

x

, the value of

h(x)

increases by 3. The change in

x

from 5 to 11 is

11 - 5 = 6

. Therefore, the corresponding increase in

h(x)

is

3 \times 6 = 18

. Adding this increase to the known value of

h(5) = 8

gives

h(11) = 8 + 18 = 26

.

Adım Adım Çözüm

1

Determine the change in the input variable

x

from the known point to the target point.

The change in

x

is

11 - 5 = 6

.

This identifies how many units the input increases from

x = 5

to

x = 11

.

2

Multiply the change in

x

by the slope to find the corresponding change in the function value.

The change in the function value is

3 \times 6 = 18

.

The slope represents the constant rate of change (change in

y

divided by change in

x

). Therefore, multiplying the slope by the change in

x

gives the total change in the output.

3

Add the calculated change in the function value to the initial function value.

h(11) = 8 + 18 = 26

.

Adding the total increase in output to the starting output value at

x = 5

yields the value of the function at

x = 11

.

Anahtar Kavram

Linear functions have a constant rate of change, which is represented by the slope. The change in the output value is equal to the slope multiplied by the change in the input value.

Alternatif Yöntem

Alternatively, find the equation of the line using the point-slope form:

y - y_1 = m(x - x_1)

. Substituting the slope

m = 3

and the point

(5, 8)

gives

y - 8 = 3(x - 5)

, which simplifies to

y = 3x - 7

. Thus, the function is defined by

h(x) = 3x - 7

. Evaluating this function at

x = 11

yields

h(11) = 3(11) - 7 = 33 - 7 = 26

.

Tahmini Süre:1m 0s

Soru 95Soru →

A gym membership costs $\$35$ per month plus an additional $\$5$ per fitness class attended. If a member wants to spend at most $\$65$ in a single month, what is the maximum number of fitness classes the member can attend?

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Cevap: 6

Cevap

The maximum number of fitness classes the member can attend is

6

.

Let

c

represent the number of classes. The inequality representing the budget constraint is

35 + 5c \le 65

. Subtracting

35

from both sides of the inequality yields

5c \le 30

. Dividing both sides by

5

yields

c \le 6

. Since the number of classes must be a whole number, the maximum number of classes the member can attend is

6

.

Adım Adım Çözüm

1

Set up the inequality representing the situation.

35 + 5c \le 65

The cost of the membership is a fixed

\$35

plus

\$5

per class

c

, and this total must not exceed

\$65

.

2

Subtract

35

from both sides of the inequality to isolate the variable term.

5c \le 30

Isolating the term with the variable allows us to solve for

c

.

3

Divide both sides by

5

to find the solution range for the number of classes.

c \le 6

Dividing by the coefficient of the variable gives the upper limit for the number of classes.

Anahtar Kavram

Solving one-variable linear inequalities in a real-world context.

Soru 96Soru →

In the $xy$ -plane, the graph of the linear function $f$ passes through the points $(k, 12)$ and $(8, k)$ , where $k$ is a constant. If the slope of the graph of $f$ is $-\frac{1}{3}$ , what is the value of $k$ ?

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Cevap: 14

Cevap

14

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by the formula

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

. Substituting the given points

(k, 12)

and

(8, k)

and the slope

-\frac{1}{3}

into the formula gives

-\frac{1}{3} = \frac{k - 12}{8 - k}

. Cross-multiplying yields

3(k - 12) = -1(8 - k)

, which simplifies to

3k - 36 = -8 + k

. Isolating the variable

k

gives

2k = 28

, so

k = 14

.

Adım Adım Çözüm

1

Apply the slope formula

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

using the points

(k, 12)

and

(8, k)

with slope

-\frac{1}{3}

.

-\frac{1}{3} = \frac{k - 12}{8 - k}

The slope of a linear function is constant and defined by the ratio of the change in

y

-values to the change in

x

-values.

2

Cross-multiply to solve the equation for

k

.

3(k - 12) = -1(8 - k)

To eliminate the fractions and solve the rational equation.

3

Distribute and simplify both sides of the equation.

3k - 36 = -8 + k

Distributing the constants on both sides prepares the equation for isolating the variable

k

.

4

Isolate the variable

k

by subtracting

k

from both sides and adding

36

to both sides.

2k = 28

Grouping like terms on opposite sides of the equation.

5

Divide by

2

to find the value of

k

.

k = 14

To find the final numerical value of the constant

k

.

Anahtar Kavram

Linear function slope formula

Tahmini Süre:1m 30s

Soru 97Soru →

An artist sells customized prints online. The total price, $y$ , in dollars, for an order of $x$ prints is given by the equation $y = 8.50x + 4.00$ , where $4.00$ represents a flat shipping fee. If a customer's total order price is $38.00$ , how many prints did the customer order?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

4

Substituting

38.00

for

y

in the given linear equation

y = 8.50x + 4.00

yields

38.00 = 8.50x + 4.00

. Subtracting

4.00

from both sides of the equation results in

34.00 = 8.50x

. Dividing both sides of the equation by

8.50

yields

x = 4

. Therefore, the customer ordered 4 prints.

Adım Adım Çözüm

1

Substitute the given total price of

38.00

for

y

in the equation.

38.00 = 8.50x + 4.00

The variable

y

represents the total price of the order in dollars, which is given as

38.00

.

2

Subtract the shipping fee of

4.00

from both sides of the equation.

34.00 = 8.50x

Subtracting

4.00

from both sides isolates the variable term

8.50x

.

3

Divide both sides of the equation by

8.50

.

x = 4

Dividing the remaining cost of

34.00

by the unit price per print of

8.50

yields the total number of prints ordered.

Anahtar Kavram

Solving a linear equation in two variables for one variable when the value of the other variable is given.

Tahmini Süre:45s

Soru 98Soru →

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

\begin{aligned} 3x - 4y &= 7 \\ ax - 12y &= b \end{aligned}

If the system has infinitely many solutions, what is the value of $a + b$ ?

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

30

For a system of two linear equations in two variables to have infinitely many solutions, the two equations must represent the same line. Comparing the

y

-coefficients shows that

-12

is

3

times

-4

. Multiplying the first equation by

3

gives the equivalent equation

9x - 12y = 21

. For this to be identical to the second equation,

ax - 12y = b

, the corresponding coefficients must be equal, meaning

a = 9

and

b = 21

. Therefore, the value of

a + b

is

9 + 21 = 30

.

Adım Adım Çözüm

1

Identify the condition for a system of linear equations to have infinitely many solutions.

The two equations must represent the same line, making them equivalent equations.

Infinitely many solutions occur when the two equations in a system are scalar multiples of each other.

2

Align the coefficients of the

y

-terms in both equations.

Multiplying the first equation by

3

gives

9x - 12y = 21

.

The coefficient of

y

in the second equation is

-12

, which is

3

times the coefficient of

y

in the first equation (

-4

).

3

Equate the coefficients of the remaining terms to find the values of

a

and

b

.

a = 9

and

b = 21

.

For

9x - 12y = 21

and

ax - 12y = b

to be equivalent, their corresponding coefficients and constants must be equal.

4

Calculate the value of

a + b

.

a + b = 9 + 21 = 30

.

The problem asks for the value of the sum

a + b

.

Anahtar Kavram

Determining constants in a system of linear equations with infinitely many solutions

Tahmini Süre:1m 30s

Soru 99Soru →

In the $xy$ -plane, the graphs of two linear functions, $f$ and $g$ , are perpendicular lines that intersect at the point $(12, k)$ , where $k$ is a positive constant. If the $y$ -intercept of the graph of $f$ is $(0, 24)$ and the $y$ -intercept of the graph of $g$ is $(0, -6)$ , what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

18

To find the value of

k

, we determine the slopes of the two lines. The line representing

f

passes through

(0, 24)

and

(12, k)

, so its slope is

m_f = \frac{k - 24}{12}

. The line representing

g

passes through

(0, -6)

and

(12, k)

, so its slope is

m_g = \frac{k + 6}{12}

. Since the two lines are perpendicular, the product of their slopes is

-1

. This gives the equation

\left(\frac{k - 24}{12}\right)\left(\frac{k + 6}{12}\right) = -1

. Multiplying both sides by

144

yields

(k - 24)(k + 6) = -144

. Expanding the left side gives

k^2 - 18k - 144 = -144

. Adding

144

to both sides results in

k^2 - 18k = 0

, which factors as

k(k - 18) = 0

. Since

k

is a positive constant,

k

must be

18

.

Adım Adım Çözüm

1

Determine the slopes of the lines

f

and

g

in terms of

k

.

The slope of

f

is

m_f = \frac{k - 24}{12 - 0} = \frac{k - 24}{12}

. The slope of

g

is

m_g = \frac{k - (-6)}{12 - 0} = \frac{k + 6}{12}

.

We use the slope formula

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

for each line passing through their respective

y

-intercepts and their intersection point

(12, k)

.

2

Set up an equation using the perpendicular condition.

m_f \cdot m_g = -1 \implies \left(\frac{k - 24}{12}\right)\left(\frac{k + 6}{12}\right) = -1 \implies \frac{(k - 24)(k + 6)}{144} = -1

Since the graphs of

f

and

g

are perpendicular, the product of their slopes must equal

-1

.

3

Solve the quadratic equation for

k

.

(k - 24)(k + 6) = -144 \implies k^2 - 18k - 144 = -144 \implies k^2 - 18k = 0 \implies k(k - 18) = 0

. Since

k

must be positive,

k = 18

.

Multiplying by

144

, expanding, and factoring the quadratic equation gives the possible values

k = 0

and

k = 18

. The problem states

k

is positive, so

k = 18

.

Anahtar Kavram

Perpendicular lines have slopes that are negative reciprocals of each other, meaning their product is

-1

.

Soru 100Soru →

In the $xy$ -plane, a line with a negative slope passes through the point $(4, 3)$ and intersects the positive $x$ -axis at $(a, 0)$ and the positive $y$ -axis at $(0, b)$ . If the area of the triangle formed by this line and the coordinate axes is $24$ , what is the value of $b$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The value of

b

is 6.

The correct value is 6. By representing the linear equation in intercept form as

\frac{x}{a} + \frac{y}{b} = 1

and substituting the given point

(4, 3)

, we obtain

\frac{4}{a} + \frac{3}{b} = 1

. Using the area of the triangle,

\frac{1}{2}ab = 24

, we can write

a = \frac{48}{b}

. Substituting this into the intercept equation yields

\frac{b}{12} + \frac{3}{b} = 1

. Multiplying by

12b

leads to the quadratic equation

b^2 - 12b + 36 = 0

, which factors as

(b-6)^2 = 0

, giving the unique solution

b = 6

.

Adım Adım Çözüm

1

Express the line in intercept form

\frac{x}{a} + \frac{y}{b} = 1

Since the intercepts are

(a, 0)

and

(0, b)

with

a > 0

and

b > 0

, the intercept form of a linear equation is the most direct representation.

2

Substitute the given point

(4, 3)

into the intercept form

\frac{4}{a} + \frac{3}{b} = 1

The line passes through

(4, 3)

, so these coordinates must satisfy the equation of the line.

3

Express

a

in terms of

b

using the area of the triangle

a = \frac{48}{b}

The area of the right triangle with base

a

and height

b

is

\frac{1}{2}ab = 24

, which gives

ab = 48

.

4

Substitute

a = \frac{48}{b}

into the equation from Step 2

\frac{b}{12} + \frac{3}{b} = 1

This reduces the equation to a single variable

b

.

5

Clear denominators and write the equation in standard quadratic form

b^2 - 12b + 36 = 0

Multiplying both sides by

12b

allows us to form a quadratic equation.

6

Factor the quadratic equation to solve for

b

b = 6

The quadratic expression is a perfect square trinomial,

(b-6)^2 = 0

, which yields

b = 6

.

Anahtar Kavram

Using the intercept form of a linear equation and geometric properties of linear graphs to solve for unknown parameters.

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