Linear Equations in Two Variables Alıştırma Soruları — SAT

Soru 1Soru →

A florist sells carnations for $1.50$ each and roses for $3.00$ each. The equation $1.50c + 3.00r = 45.00$ represents the possible number of carnations, $c$ , and roses, $r$ , that a customer can buy for exactly $45.00$ . If the customer buys 8 carnations, how many roses can they buy?

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

11

To find the number of roses the customer can buy, substitute the number of carnations bought, which is 8, for

c

in the given equation. This yields

1.50(8) + 3.00r = 45.00

. Simplifying the product gives

12.00 + 3.00r = 45.00

. Subtracting 12.00 from both sides of the equation gives

3.00r = 33.00

. Finally, dividing both sides by 3.00 yields

r = 11

. Thus, the customer can buy 11 roses.

Adım Adım Çözüm

1

Substitute the given value for carnations,

c = 8

, into the linear equation.

1.50(8) + 3.00r = 45.00

We are given that the customer buys 8 carnations, so

c

is replaced by 8 to solve for the number of roses,

r

.

2

Multiply

1.50

by

8

to simplify the term.

12.00 + 3.00r = 45.00

Simplifying the constant term on the left side of the equation.

3

Subtract

12.00

from both sides of the equation to isolate the term with

r

.

3.00r = 33.00

To solve for

r

, we must get the term

3.00r

by itself on one side.

4

Divide both sides of the equation by

3.00

to solve for

r

.

r = 11

Dividing isolates the variable

r

.

Anahtar Kavram

Solving a linear equation in two variables by substitution when one variable's value is known.

Soru 2Soru →

A landscaping company uses the equation $12x + 18y = 240$ to model the total cost, in dollars, of renting a wood chipper for $x$ hours and a soil aerator for $y$ hours. If the wood chipper was rented for 5 more hours than the soil aerator, and the company spent a total of $240$ on these rentals, for how many hours was the soil aerator rented?

5

6

10

11

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The soil aerator was rented for 6 hours.

The correct answer is 6. By writing the relationship between the rental hours as

x = y + 5

and substituting this expression into the cost equation, we obtain

12(y + 5) + 18y = 240

. Distributing the 12 results in

12y + 60 + 18y = 240

. Combining the like terms gives

30y + 60 = 240

. Subtracting 60 from both sides of the equation yields

30y = 180

. Dividing both sides by 30 results in

y = 6

. Therefore, the soil aerator was rented for 6 hours.

Adım Adım Çözüm

1

Express the relationship between the rental hours of the wood chipper (

x

) and the soil aerator (

y

) as an equation.

x = y + 5

The wood chipper was rented for 5 more hours than the soil aerator.

2

Substitute the expression for

x

into the cost equation

12x + 18y = 240

.

12(y + 5) + 18y = 240

Substitution reduces the equation to a single variable, allowing us to solve for

y

.

3

Distribute the 12 and combine like terms.

12y + 60 + 18y = 240 \implies 30y + 60 = 240

Simplification isolates the variable terms on one side.

4

Isolate

y

by subtracting 60 from both sides and then dividing by 30.

30y = 180 \implies y = 6

This calculation yields the rental hours for the soil aerator.

Anahtar Kavram

Solving a system of linear equations in two variables where one equation represents a budget constraint and the other relates the quantities of the two variables.

Soru 3Soru →

The graph of the linear equation $y = 3x - 7$ contains the point $(k, 5)$ . What is the value of $k$ ?

-\frac{2}{3}

4

8

-4

Cevabı ve açıklamayı göster

Cevap: $4$

Cevap

The value of

k

is

4

.

Substituting the coordinates of the point

(k, 5)

into the linear equation

y = 3x - 7

gives

5 = 3k - 7

. Solving for

k

involves adding

7

to both sides of the equation to get

12 = 3k

, and then dividing both sides by

3

to find

k = 4

.

Adım Adım Çözüm

1

Substitute the point

(k, 5)

into the equation of the line

y = 3x - 7

by setting

x = k

and

y = 5

.

5 = 3k - 7

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

2

Isolate the term with

k

by adding

7

to both sides of the equation.

12 = 3k

Adding

7

to both sides cancels the

-7

on the right side.

3

Solve for

k

by dividing both sides of the equation by

3

.

k = 4

Dividing by

3

isolates

k

.

Anahtar Kavram

Determining an unknown coordinate of a point on a line by substituting the point's coordinates into the linear equation in two variables.

Soru 4Soru →

A community garden charges a one-time registration fee plus a monthly fee to plot and maintain a garden bed. The total cost, $y$ , in dollars, of maintaining a garden bed for $x$ months is given by the equation $y = 15x + 45$ . If a gardener spent a total of $165$ dollars, for how many months did they maintain the garden bed?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The correct answer is 8. Setting the total cost y equal to 165 in the linear equation yields 165 = 15x + 45. Subtracting 45 from both sides gives 120 = 15x, and dividing by 15 results in x = 8 months.

To find the number of months the gardener maintained the garden bed, substitute the total spent,

165

, for

y

in the equation

y = 15x + 45

. This results in

165 = 15x + 45

. Subtracting

45

from both sides of the equation yields

120 = 15x

. Dividing both sides by

15

gives

x = 8

. Thus, the garden bed was maintained for 8 months.

Adım Adım Çözüm

1

Substitute 165 for y in the equation.

165 = 15x + 45

The variable y represents the total cost in dollars, which is given as 165.

2

Subtract 45 from both sides of the equation.

120 = 15x

This isolates the variable term by subtracting the constant registration fee.

3

Divide both sides of the equation by 15.

x = 8

This solves for x, the number of months.

Anahtar Kavram

Solving for a variable in a linear equation representing a real-world scenario.

Soru 5Soru →

An online bookstore charges a flat shipping fee plus a fixed price per book purchased. The total cost, $y$ , in dollars, for purchasing $x$ books is given by the equation $y = 8.5x + 4.5$ . If a customer's total cost was $47$ dollars, how many books did the customer purchase?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The customer purchased 5 books.

To find the number of books purchased, substitute the total cost of 47 for

y

in the equation

y = 8.5x + 4.5

, yielding

47 = 8.5x + 4.5

. Subtracting

4.5

from both sides gives

42.5 = 8.5x

. Dividing both sides by

8.5

results in

x = 5

. Thus, the customer purchased 5 books.

Adım Adım Çözüm

1

Substitute the total cost of 47 into the equation for y

47 = 8.5x + 4.5

Since the total cost is represented by y and is given as 47 dollars, we substitute 47 for y in the equation.

2

Subtract 4.5 from both sides of the equation

42.5 = 8.5x

Subtracting the flat fee from both sides isolates the variable term representing the total cost of the books.

3

Divide both sides by 8.5

x = 5

Dividing the remaining cost by the price per book yields the total number of books purchased.

Anahtar Kavram

Linear Equations in Two Variables

Soru 6Soru →

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 7Soru →

A project manager uses the linear equation $ax + by = c$ to model the allocation of resources, where $x$ represents the number of hours spent on research, $y$ represents the number of hours spent on development, and $a, b,$ and $c$ are positive constants. Initially, the project manager allocates $40$ hours for research and $30$ hours for development. To optimize the project, the manager decides to increase the research hours by $25\%$ and decrease the development hours by $20\%$ such that the total resource value, $c$ , remains unchanged. Which of the following equations correctly expresses $a$ in terms of $b$ ?

a = 1.67b

a = 0.6b

a = 0.64b

a = 1.6b

Cevabı ve açıklamayı göster

Cevap: $a = 0.6b$

Cevap

The equation

a = 0.6b

correctly expresses

a

in terms of

b

.

The correct equation is

a = 0.6b

. The initial resource allocation is represented by

40a + 30b = c

. Increasing

40

by

25\%

gives

50

, and decreasing

30

by

20\%

gives

24

, which yields the new equation

50a + 24b = c

. Equating these two expressions since

c

is constant gives

40a + 30b = 50a + 24b

. Simplifying this equation results in

6b = 10a

, which gives

a = 0.6b

.

Adım Adım Çözüm

1

Write the linear equation representing the initial allocation of resources.

40a + 30b = c

The initial values are

40

hours for research (

x

) and

30

hours for development (

y

).

2

Calculate the new values of

x

and

y

after the percentage changes are applied.

x = 40 \times 1.25 = 50

and

y = 30 \times 0.80 = 24

Research hours increase by

25\%

and development hours decrease by

20\%

.

3

Write the new linear equation with the updated resource allocations.

50a + 24b = c

The total resource value

c

remains unchanged.

4

Equate the two expressions for

c

and solve for

a

in terms of

b

.

40a + 30b = 50a + 24b \implies 6b = 10a \implies a = 0.6b

Since both equations equal the same constant

c

, we set them equal to each other and isolate the variable

a

.

Anahtar Kavram

Expressing and manipulating linear relationships in two variables with constraints.

Alternatif Yöntem

Instead of keeping

c

general, we can choose a convenient value for

c

, such as

c = 120

. This gives two linear equations in terms of

a

and

b

:

40a + 30b = 120

and

50a + 24b = 120

. Solving this system for the ratio of

a

to

b

yields

a = 1.33

and

b = 2.22

, which confirms that

a/b = 0.6

, or

a = 0.6b

.

Tahmini Süre:3m 0s

Soru 8Soru →

A student tracks the remaining battery percentage, $y$ , of a tablet after $x$ hours of use. The table below shows the battery percentage at three different times:

Time ( $x$ , hours)	Battery percentage ( $y$ )
0	90
3	75
6	60

Which of the following equations represents the relationship between $x$ and $y$ ?

y = -15x + 90

y = -\frac{1}{5}x + 90

y = -5x + 90

y = 90x - 5

Cevabı ve açıklamayı göster

Cevap: $y = -5x + 90$

Cevap

y = -5x + 90

The correct equation is

y = -5x + 90

. According to the table, when time

x = 0

, the battery percentage

y = 90

, which indicates that the y-intercept of the line is

90

. The slope can be determined using any two points from the table, such as

(0, 90)

and

(3, 75)

:

\text{slope} = \frac{75 - 90}{3 - 0} = \frac{-15}{3} = -5

. Writing this in slope-intercept form gives

y = -5x + 90

.

Adım Adım Çözüm

1

Identify the y-intercept from the table.

The y-intercept

b

is

90

.

When the time

x = 0

hours, the battery percentage

y = 90

. This corresponds to the y-intercept of the linear equation.

2

Calculate the slope (rate of change) using two points from the table.

The slope

m

is

-5

.

Using the points

(0, 90)

and

(3, 75)

, the slope is calculated as the change in

y

divided by the change in

x

:

m = \frac{75 - 90}{3 - 0} = \frac{-15}{3} = -5

.

3

Substitute the slope and y-intercept into the slope-intercept form equation.

y = -5x + 90

Plugging

m = -5

and

b = 90

into

y = mx + b

yields the final equation.

Anahtar Kavram

Determining a linear equation in two variables from a table of values.

Tahmini Süre:1m 0s

Soru 9Soru →

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 10Soru →

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 11Soru →

A linear relationship between $x$ and $y$ is defined by the values in the table below.

$x$	$y$
2	7
4	13
6	19

Which of the following equations represents this relationship?

y = 3x - 5

y = \frac{1}{3}x + \frac{19}{3}

y = 3x + 1

y = 6x - 5

Cevabı ve açıklamayı göster

Cevap: $y = 3x + 1$

Cevap

The equation representing the relationship is

y = 3x + 1

.

The correct equation is

y = 3x + 1

. The slope

m

of the linear function can be determined from the table values

(2, 7)

and

(4, 13)

as

\frac{13 - 7}{4 - 2} = \frac{6}{2} = 3

. Substituting the slope

m = 3

and the point

(2, 7)

into the slope-intercept form

y = mx + b

gives

7 = 3(2) + b

, which simplifies to

7 = 6 + b

, so

b = 1

. Therefore, the equation is

y = 3x + 1

.

Adım Adım Çözüm

1

Find the slope (

m

) of the linear relationship using the points

(2, 7)

and

(4, 13)

from the table.

m = 3

The slope is the ratio of the change in

y

to the change in

x

between any two points on the line:

m = \frac{13 - 7}{4 - 2} = \frac{6}{2} = 3

.

2

Use the slope

m = 3

and the point

(2, 7)

to find the

y

-intercept (

b

) of the line.

b = 1

Substituting the coordinates and slope into the slope-intercept equation

y = mx + b

gives

7 = 3(2) + b

, which simplifies to

7 = 6 + b

, leading to

b = 1

.

3

Substitute the slope and

y

-intercept into the slope-intercept form.

y = 3x + 1

Combining the slope of

3

and

y

-intercept of

1

yields the equation

y = 3x + 1

.

Anahtar Kavram

Determining a linear equation from a table of values

Tahmini Süre:45s

Soru 12Soru →

A line in the $xy$ -plane is represented by the equation $3x - 4y = 24$ . What is the $y$ -coordinate of the $y$ -intercept of this line?

-6

6

8

\frac{3}{4}

Cevabı ve açıklamayı göster

Cevap: -6

Cevap

The

y

-coordinate of the

y

-intercept is

-6

.

The correct answer is

-6

. The

y

-intercept of a line is the point where it intersects the

y

-axis, which occurs when the

x

-coordinate is

0

. Substituting

x = 0

into the given equation

3x - 4y = 24

yields

-4y = 24

. Dividing both sides by

-4

gives

y = -6

.

Adım Adım Çözüm

1

Substitute

x = 0

into the equation to find the

y

-intercept.

3(0) - 4y = 24

By definition, the

y

-intercept of a line in the coordinate plane is the point where the line crosses the

y

-axis, which always has an

x

-coordinate of

0

.

2

Simplify the equation.

-4y = 24

Multiplying

3

by

0

yields

0

, leaving only the term containing

y

on the left side of the equation.

3

Solve for

y

by isolating the variable.

y = -6

Dividing both sides of the simplified equation by

-4

isolates

y

and gives the final coordinate value.

Anahtar Kavram

Finding the y-intercept of a linear equation in two variables by setting x = 0.

Tahmini Süre:45s

Soru 13Soru →

In the $xy$ -plane, line $L_1$ is defined by the equation $ax + by = c$ , where $a$ , $b$ , and $c$ are non-zero constants. Line $L_2$ is perpendicular to line $L_1$ and has a $y$ -intercept that is twice the $y$ -intercept of line $L_1$ . If the $x$ -intercept of line $L_2$ is $d$ , which of the following expressions represents $d$ in terms of $a$ , $b$ , and $c$ ?

-\frac{2ac}{b^2}

\frac{2ac}{b^2}

-\frac{2bc}{a^2}

-\frac{2c}{a}

Cevabı ve açıklamayı göster

Cevap: $-\frac{2ac}{b^2}$

Cevap

The expression that represents

d

in terms of

a

,

b

, and

c

is

-\frac{2ac}{b^2}

.

The correct answer is found by first rewriting the equation of the first line in slope-intercept form to determine its slope,

-\frac{a}{b}

, and its

y

-intercept,

\frac{c}{b}

. The second line is perpendicular to the first, so its slope is the negative reciprocal,

\frac{b}{a}

. Its

y

-intercept is twice that of the first line, which is

\frac{2c}{b}

. Setting

y = 0

in the equation of the second line,

y = \frac{b}{a}x + \frac{2c}{b}

, and solving for

x

yields the

x

-intercept

d = -\frac{2ac}{b^2}

.

Adım Adım Çözüm

1

Rewrite the equation of line

L_1

in slope-intercept form.

y = -\frac{a}{b}x + \frac{c}{b}

To identify the slope and

y

-intercept of the first line.

2

Determine the slope and

y

-intercept of line

L_2

.

Slope of

L_2

is

\frac{b}{a}

;

y

-intercept of

L_2

is

\frac{2c}{b}

.

Line

L_2

is perpendicular to

L_1

(so its slope is the negative reciprocal of

-\frac{a}{b}

) and its

y

-intercept is twice that of

L_1

.

3

Write the equation of line

L_2

.

y = \frac{b}{a}x + \frac{2c}{b}

Using the slope-intercept form

y = mx + k

for the second line.

4

Find the

x

-intercept

d

of line

L_2

by setting

y = 0

.

d = -\frac{2ac}{b^2}

The

x

-intercept is the value of

x

when

y = 0

.

Anahtar Kavram

Linear Equations in Two Variables

Alternatif Yöntem

Instead of converting to slope-intercept form, one can use the general form relationships. For

ax + by = c

, the slope is

-a/b

and the

y

-intercept is

c/b

. A perpendicular line has slope

b/a

. With double the

y

-intercept, its equation can be written as

y - \frac{2c}{b} = \frac{b}{a}x

. Setting

y=0

directly gives

-\frac{2c}{b} = \frac{b}{a}d \implies d = -\frac{2ac}{b^2}

.

Tahmini Süre:2m 0s

Soru 14Soru →

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 15Soru →

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.

Soru 16Soru →

In the $xy$ -plane, the graph of the linear equation $ax + by = c$ , where $a$ , $b$ , and $c$ are non-zero constants, has a positive slope and a negative $y$ -intercept. If the line passes through the point $(b, a)$ , which of the following inequalities must be true?

ab > 0

ac > 0

bc > 0

ac < 0

Cevabı ve açıklamayı göster

Cevap: $ac > 0$

Cevap

The inequality

ac > 0

must be true.

The correct inequality is

ac > 0

. Rearranging

ax + by = c

into slope-intercept form gives

y = -\frac{a}{b}x + \frac{c}{b}

. A positive slope means

-\frac{a}{b} > 0

, so

\frac{a}{b} < 0

, indicating that

a

and

b

have opposite signs (

ab < 0

). A negative

y

-intercept means

\frac{c}{b} < 0

, indicating that

b

and

c

have opposite signs (

bc < 0

). Substituting the point

(b, a)

into the line's equation gives

2ab = c

. Since

ab < 0

, it follows that

c < 0

. Using the relation

bc < 0

with

c < 0

, we find

b > 0

. Using

ab < 0

with

b > 0

, we find

a

is negative. Since both

a

and

c

are negative, their product

ac

must be positive.

Adım Adım Çözüm

1

Rearrange the standard form equation

ax + by = c

into slope-intercept form.

y = -\frac{a}{b}x + \frac{c}{b}

To identify the expressions for the slope and the

y

-intercept in terms of the constants

a

,

b

, and

c

.

2

Determine the sign relationships between the coefficients based on the given signs of the slope and

y

-intercept.

Since the slope is positive,

-\frac{a}{b} > 0 \implies ab < 0

. Since the

y

-intercept is negative,

\frac{c}{b} < 0 \implies bc < 0

.

This establishes that

a

and

b

have opposite signs, and

b

and

c

have opposite signs.

3

Substitute the coordinates of the point

(b, a)

into the original equation

ax + by = c

.

a(b) + b(a) = c \implies 2ab = c

Since the point lies on the line, substituting its coordinates must satisfy the equation.

4

Analyze the sign of

c

using the relation

c = 2ab

.

Since

ab < 0

, it follows that

2ab < 0

, which means

c < 0

.

To determine whether the constant

c

is positive or negative.

5

Determine the individual signs of

b

and

a

.

Since

bc < 0

and

c < 0

,

b

must be positive (

b > 0

). Since

ab < 0

and

b > 0

,

a

must be negative (

a < 0

).

To find the sign of each coefficient individually.

6

Determine the sign of the product

ac

.

Since both

a < 0

and

c < 0

, their product

ac

must be positive (

ac > 0

).

To evaluate which inequality must be true.

Anahtar Kavram

Analyzing slope, intercepts, and points on a linear equation in two variables using sign analysis of its constants.

Alternatif Yöntem

Instead of solving for the individual signs of all variables, we can choose test values that satisfy the conditions. For example, let

a = -1

and

b = 2

, which gives a positive slope of

-\frac{a}{b} = \frac{1}{2}

. Since the line passes through

(b, a) = (2, -1)

, we can find

c

:

ax + by = c \implies (-1)(2) + (2)(-1) = -4

, so

c = -4

. The

y

-intercept is

\frac{c}{b} = \frac{-4}{2} = -2

, which is negative, satisfying all conditions. Evaluating the product of

a

and

c

gives

(-1)(-4) = 4 > 0

, confirming that the product of these constants must be positive.

Tahmini Süre:3m 0s

Soru 17Soru →

In the $xy$ -plane, a line with a positive slope $m$ passes through the point $(4, -3)$ and intersects the $x$ -axis at $(p, 0)$ and the $y$ -axis at $(0, q)$ , where $p$ and $q$ are non-zero constants. If $p + q = 5$ , what is the value of $m$ ?

Cevabı ve açıklamayı göster

Cevap: 0.5

Cevap

The value of

m

is

0.5

(or

1/2

).

To find the slope

m

, write the equation of the line in point-slope form:

y - (-3) = m(x - 4)

, which simplifies to

y = mx - 4m - 3

. The

y

-intercept is found by setting

x = 0

, giving

q = -4m - 3

. The

x

-intercept is found by setting

y = 0

, giving

p = 4 + \frac{3}{m}

. Substituting these expressions into

p + q = 5

gives

4 + \frac{3}{m} - 4m - 3 = 5

, which simplifies to

-4m - 4 + \frac{3}{m} = 0

. Multiplying this equation by

-m

yields the quadratic equation

4m^2 + 4m - 3 = 0

. Factoring the quadratic gives

(2m - 1)(2m + 3) = 0

. Since the line is defined to have a positive slope, the value of

m

must be positive, which is

0.5

(or

1/2

).

Adım Adım Çözüm

1

Write the general equation of a line passing through

(4, -3)

with slope

m

.

y = mx - 4m - 3

Using the point-slope form

y - y_1 = m(x - x_1)

with

(x_1, y_1) = (4, -3)

establishes the relationship between

x

,

y

, and

m

.

2

Determine the value of

q

by finding the

y

-intercept of the line.

q = -4m - 3

The

y

-intercept

(0, q)

occurs when

x = 0

.

3

Determine the value of

p

by finding the

x

-intercept of the line.

p = 4 + \frac{3}{m}

The

x

-intercept

(p, 0)

occurs when

y = 0

.

4

Substitute the expressions for

p

and

q

into the given equation

p + q = 5

.

4 + \frac{3}{m} - 4m - 3 = 5

This sets up a single equation with the variable

m

using the given relationship.

5

Simplify the equation and rewrite it in standard quadratic form.

4m^2 + 4m - 3 = 0

Multiplying the simplified expression

-4m - 4 + \frac{3}{m} = 0

by

-m

eliminates the fraction and aligns it to standard quadratic form.

6

Factor the quadratic equation to find the possible values of

m

.

m = 0.5

or

m = -1.5

Factoring

(2m - 1)(2m + 3) = 0

gives the roots of the equation.

7

Filter the results based on the positive slope constraint.

m = 0.5

The problem states that the line has a positive slope, so we must reject the negative root.

Anahtar Kavram

Using linear intercepts to solve system constraints

Alternatif Yöntem

Instead of using point-slope form, use the intercept form of a linear equation:

\frac{x}{p} + \frac{y}{q} = 1

. Since the line passes through the point

(4, -3)

, substitute

x = 4

and

y = -3

to get

\frac{4}{p} - \frac{3}{q} = 1

. Given

p + q = 5

, substitute

p = 5 - q

into the equation to get

\frac{4}{5 - q} - \frac{3}{q} = 1

. Multiply both sides by the common denominator

q(5 - q)

to obtain

4q - 3(5 - q) = q(5 - q) \implies 7q - 15 = 5q - q^2 \implies q^2 + 2q - 15 = 0

. Factoring gives

(q + 5)(q - 3) = 0

. This yields

q = -5

or

q = 3

. If

q = 3

, then

p = 2

, and the slope is

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

(negative). If

q = -5

, then

p = 10

, and the slope is

m = -\frac{q}{p} = -\frac{-5}{10} = 0.5

(positive). Since the slope must be positive, the slope of the line is

0.5

.

Tahmini Süre:2m 30s

Soru 18Soru →

In the $xy$ -plane, a line has an $x$ -intercept of $(k + 2, 0)$ and a $y$ -intercept of $(0, 2k + 2)$ , where $k$ is a positive constant. If the line passes through the point $(3, 5)$ , what is the value of $k$ ?

1.5

2.5

4

6.5

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

4

The correct answer is the value 4. By substituting the point

(3, 5)

into the intercept form of the line equation

\frac{x}{k+2} + \frac{y}{2k+2} = 1

, we obtain

\frac{3}{k+2} + \frac{5}{2k+2} = 1

. Multiplying by the common denominator

(k+2)(2k+2)

and simplifying results in the quadratic equation

2k^2 - 5k - 12 = 0

. Factoring this equation yields

(2k+3)(k-4) = 0

. Since

k

must be a positive constant, we select

k = 4

.

Adım Adım Çözüm

1

Write the equation of the line using intercept form.

The equation of a line with

x

-intercept

(k+2, 0)

and

y

-intercept

(0, 2k+2)

is

\frac{x}{k+2} + \frac{y}{2k+2} = 1

.

Intercept form

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

is the most direct way to represent a line when both intercepts are given.

2

Substitute the given point

(3, 5)

into the line's equation.

Substituting

x = 3

and

y = 5

yields the equation

\frac{3}{k+2} + \frac{5}{2k+2} = 1

.

Since the line passes through

(3, 5)

, these coordinates must satisfy the equation of the line.

3

Eliminate the denominators by multiplying both sides of the equation by the least common denominator.

Multiplying by

(k+2)(2k+2)

gives

3(2k+2) + 5(k+2) = (k+2)(2k+2)

.

This clears the fractions and allows us to rewrite the equation in polynomial form.

4

Expand both sides of the equation and combine like terms.

Expanding both sides gives

6k + 6 + 5k + 10 = 2k^2 + 2k + 4k + 4 \implies 11k + 16 = 2k^2 + 6k + 4

.

To solve a polynomial equation, we need to simplify the algebraic expressions on both sides.

5

Rearrange the equation into standard quadratic form.

Moving all terms to one side gives

2k^2 - 5k - 12 = 0

.

Standard quadratic form

Ax^2 + Bx + C = 0

is required to apply factoring or the quadratic formula.

6

Factor and solve the quadratic equation.

(2k+3)(k-4) = 0

, which gives

k = -1.5

or

k = 4

.

Factoring determines the values of

k

that satisfy the quadratic relation.

7

Apply the constraint that

k

is a positive constant.

Since

k > 0

, we discard

k = -1.5

, leaving

k = 4

.

The problem specifies that

k

must be a positive constant.

Anahtar Kavram

Using the intercepts of a line to formulate its equation and solving the resulting rational/quadratic equation given a point on the line.

Alternatif Yöntem

Instead of using the intercept form of the line, you can equate the slope calculated between the y-intercept

(0, 2k+2)

and the point

(3,5)

to the slope calculated between the point

(3,5)

and the x-intercept

(k+2,0)

. This yields the equation

\frac{5 - (2k+2)}{3 - 0} = \frac{0 - 5}{(k+2) - 3}

, which simplifies to

\frac{3 - 2k}{3} = \frac{-5}{k - 1}

. Cross-multiplying and simplifying leads to the same quadratic equation,

2k^2 - 5k - 12 = 0

.

Tahmini Süre:2m 30s

Soru 19Soru →

If $3x - 2(y - 5) = 12$ and $x = 8$ , what is the value of $y$ ?

1

6

11

10

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

11

The correct answer is 11. Substituting

x = 8

into the equation gives

3(8) - 2(y - 5) = 12

, which simplifies to

24 - 2y + 10 = 12

. Combining the constants on the left side yields

34 - 2y = 12

. Subtracting 34 from both sides gives

-2y = -22

. Finally, dividing both sides by

-2

yields

y = 11

.

Adım Adım Çözüm

1

Substitute

x = 8

into the given equation.

3(8) - 2(y - 5) = 12

, which simplifies to

24 - 2(y - 5) = 12

.

This isolates the variable

y

by replacing the variable

x

with its given value.

2

Distribute

-2

to the terms inside the parentheses.

24 - 2y + 10 = 12

.

Multiplying

-2

by

y

gives

-2y

, and multiplying

-2

by

-5

gives

+10

because the product of two negative numbers is positive.

3

Combine the constant terms on the left side of the equation and subtract the result from both sides.

34 - 2y = 12 \Rightarrow -2y = -22

.

Combining

24

and

10

gives

34

. Subtracting

34

from both sides isolates the variable term

-2y

.

4

Divide both sides by

-2

to solve for

y

.

y = 11

.

Dividing both sides of the equation by the coefficient of the variable term isolates the variable.

Anahtar Kavram

Solving linear equations in two variables by substitution and isolation.

Tahmini Süre:45s

Soru 20Soru →

In the $xy$ -plane, a line passes through the points $(2, 9)$ and $(5, 21)$ . What is the slope of this line?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The slope of the line is 4.

The slope

m

of a line passing through the points

(x_1, y_1)

and

(x_2, y_2)

is determined by the formula

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

. Substituting the given points

(2, 9)

and

(5, 21)

into the formula gives

m = \frac{21 - 9}{5 - 2} = \frac{12}{3} = 4

.

Adım Adım Çözüm

1

Identify the coordinates of the two points on the line.

(x_1, y_1) = (2, 9)

and

(x_2, y_2) = (5, 21)

These points are used in the slope formula.

2

Apply the slope formula

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

to set up the calculation.

m = \frac{21 - 9}{5 - 2}

The slope is defined as the change in

y

divided by the change in

x

.

3

Simplify the fraction to calculate the final slope.

m = \frac{12}{3} = 4

Dividing the vertical change by the horizontal change yields the slope.

Anahtar Kavram

Calculating the slope of a line given two points in the coordinate plane.

Linear Equations in Two Variables

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap