Tüm alıştırma soruları

Soru 121Soru →

In the $xy$ -plane, the graph of the linear equation $y = mx + b$ , where $m$ and $b$ are constants, passes through the point $(2, 7)$ . If this graph is translated $3$ units to the right and $4$ units down, the resulting graph passes through the point $(4, 2)$ . What is the value of $b$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

The correct answer is

5

. Since the original graph passes through

(2, 7)

and undergoes a translation of

3

units to the right and

4

units down, the point

(2, 7)

is translated to

(2 + 3, 7 - 4) = (5, 3)

on the new graph. The translated graph also passes through

(4, 2)

. Using the slope formula on the two points

(5, 3)

and

(4, 2)

of the translated line gives a slope of

1

. Since a translation preserves the slope, the original line also has a slope of

1

. Substituting

m = 1

and

(2, 7)

into the original equation

y = mx + b

yields

7 = 1(2) + b

, which simplifies to

b = 5

.

Adım Adım Çözüm

1

Determine a point on the translated graph by translating the given point

(2, 7)

on the original graph.

The point

(5, 3)

is on the translated graph.

Since the entire graph is translated

3

units to the right and

4

units down, every point on the original graph is translated by the same vector

(+3, -4)

.

2

Calculate the slope of the translated graph using the points

(5, 3)

and

(4, 2)

.

The slope

m = 1

.

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by the change in y divided by the change in x.

3

Find the slope of the original graph.

The slope of the original graph is also

1

.

Translating a line horizontally and vertically shifts its position but does not alter its slope.

4

Substitute the slope

m = 1

and the point

(2, 7)

into the slope-intercept form of the original line,

y = mx + b

, to solve for

b

.

b = 5

.

This determines the value of the y-intercept of the original line.

Anahtar Kavram

Linear Equations in Two Variables

Soru 122Soru →

An online store sells small gift boxes for $10$ dollars each and large gift boxes for $15$ dollars each. A customer wants to buy a total of at most $10$ boxes. If the customer must spend at least $120$ dollars on the boxes, what is the minimum number of large gift boxes the customer must buy?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The minimum number of large gift boxes the customer must buy is 4.

Using the constraints from the problem, we set up the system of inequalities:

x + y \leq 10

and

10x + 15y \geq 120

. Expressing the first inequality as

x \leq 10 - y

and substituting it into the second gives

10(10 - y) + 15y \geq 120

. Simplifying this results in

100 + 5y \geq 120

, which simplifies to

5y \geq 20

, or

y \geq 4

. Therefore, the minimum number of large gift boxes the customer must buy is

4

.

Adım Adım Çözüm

1

Define variables for the quantities of each box.

Let

x

be the number of small boxes and

y

be the number of large boxes.

Establishing variables is necessary to translate the word problem into algebraic inequalities.

2

Write the system of inequalities representing the constraints.

x + y \leq 10

and

10x + 15y \geq 120

The total number of boxes is at most

10

, and the total cost must be at least

120

dollars.

3

Express

x

in terms of

y

using the first inequality.

x \leq 10 - y

This allows substitution into the second inequality to solve for the target variable

y

.

4

Substitute

x \leq 10 - y

into the second inequality and simplify.

100 + 5y \geq 120

Solving the resulting single-variable inequality will determine the possible values of

y

.

5

Solve the inequality for

y

.

y \geq 4

Subtracting

100

and dividing by

5

isolates

y

to show its minimum possible value is

4

.

Anahtar Kavram

Systems of Linear Inequalities in Two Variables

Soru 123Soru →

A sports store orders standard helmets for $20$ dollars each and premium helmets for $50$ dollars each. The store can spend at most $2{}200$ dollars on this order. The distributor requires the store to order at least $60$ helmets in total. If the store decides to order at least $15$ premium helmets, what is the maximum number of standard helmets that the store can order?

Cevabı ve açıklamayı göster

Cevap: 72

Cevap

The maximum number of standard helmets the store can order is 72.

By setting the number of premium helmets to its minimum allowed value of

15

to maximize the budget remaining for standard helmets, we find

20x \leq 1450

, which yields

x \leq 72.5

. Since standard helmets must be ordered in whole numbers, the maximum number is

72

, which also satisfies the minimum total order of

60

helmets (

72 + 15 = 87 \geq 60

).

Adım Adım Çözüm

1

Define variables and translate constraints into inequalities

Let

x

be the number of standard helmets and

y

be the number of premium helmets. The constraints are

20x + 50y \leq 2200

,

x + y \geq 60

, and

y \geq 15

.

To represent the problem's mathematical relationships using a system of linear inequalities.

2

Solve for the upper limit of

x

using the budget constraint and the minimum value of

y

Since

20x \leq 2200 - 50y

,

x

is maximized when

y

is at its minimum value,

y = 15

. Substituting

y = 15

yields

20x + 750 \leq 2200 \implies 20x \leq 1450 \implies x \leq 72.5

.

To find the maximum possible value of standard helmets under the budget constraint.

3

Verify with the minimum total order constraint and determine the maximum integer value

Checking

x + y \geq 60

with

y = 15

gives

x + 15 \geq 60 \implies x \geq 45

. Since

x

must satisfy

45 \leq x \leq 72.5

and must be an integer, the maximum integer value is

72

.

To ensure the solution is physically possible as a whole number of items and satisfies all constraints.

Anahtar Kavram

Systems of Linear Inequalities in Two Variables

Soru 124Soru →

A drone's battery charge is at $98\%$ when it begins a mission. During the mission, the battery charge decreases by $1.5\%$ per minute when the drone is hovering, and by $2.5\%$ per minute when it is flying horizontally. The drone flies horizontally for exactly twice as many minutes as it hovers. If the battery charge is at $33\%$ at the end of the mission, for how many minutes did the drone hover?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

The drone hovered for 10 minutes.

Let

t

represent the number of minutes the drone hovers. Since the drone flies horizontally for twice as long as it hovers, it flies horizontally for

2t

minutes. The total percentage drop in battery is

98 - 33 = 65

. The equation representing the total decrease in battery charge is

1.5t + 2.5(2t) = 65

. Simplifying the equation yields

1.5t + 5t = 65

, which becomes

6.5t = 65

. Dividing both sides of the equation by

6.5

gives

t = 10

.

Adım Adım Çözüm

1

Define the variable for the unknown quantity.

Let

t

be the number of minutes the drone spent hovering. The time spent flying horizontally is then

2t

minutes.

The problem states that the horizontal flight time is exactly twice the hovering time.

2

Set up an equation representing the total decrease in battery charge.

The total percentage decrease is

1.5t + 2.5(2t) = 98 - 33

.

The battery decreases by

1.5\%

per minute of hovering,

2.5\%

per minute of horizontal flight, and the total change is from

98\%

to

33\%

.

3

Simplify the equation and solve for

t

.

6.5t = 65

, which gives

t = 10

.

Combine like terms and divide both sides by

6.5

to isolate the variable.

Anahtar Kavram

Setting up and solving a linear equation in one variable from a real-world scenario.

Soru 125Soru →

If the expression $(2x + 3)(3x + 1) - 6x^2$ is rewritten in the form $bx + c$ , where $b$ and $c$ are constants, what is the value of $b$ ?

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

The value of

b

is

11

.

Expanding the expression

(2x + 3)(3x + 1) - 6x^2

yields

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

. Combining the linear terms simplifies the expression to

(6x^2 + 11x + 3) - 6x^2

, which further simplifies to

11x + 3

. Comparing this to the form

bx + c

, we see that the coefficient

b

of the

x

term is

11

.

Adım Adım Çözüm

1

Expand the product of the binomials

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

using the distributive property.

6x^2 + 2x + 9x + 3

To rewrite the factored part of the expression in polynomial form.

2

Combine the linear terms

2x

and

9x

.

6x^2 + 11x + 3

To simplify the expanded polynomial expression.

3

Subtract

6x^2

from the simplified polynomial expression.

11x + 3

To complete the subtraction indicated in the original expression.

4

Compare the resulting expression

11x + 3

to the form

bx + c

to identify the coefficient

b

.

b = 11

To determine the constant coefficient of the

x

term.

Anahtar Kavram

Expanding products of binomials and combining like terms to find equivalent expressions.

Soru 126Soru →

A chef is preparing portions of roasted vegetables and mashed potatoes for a catered event. Each portion of roasted vegetables requires $3$ ounces of potatoes, and each portion of mashed potatoes requires $6$ ounces of potatoes. The chef has a total of at most $90$ ounces of potatoes available. The chef must prepare at least $8$ portions of roasted vegetables and at least $5$ portions of mashed potatoes. If $v$ represents the number of portions of roasted vegetables that the chef prepares, what is the maximum possible value of $v$ ?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The maximum possible value of

v

is 20.

The maximum number of portions of roasted vegetables the chef can prepare is 20 because minimizing the number of mashed potato portions to its boundary constraint of 5 maximizes the remaining resources, yielding

3v \leq 90 - 6(5) \implies 3v \leq 60 \implies v \leq 20

.

Adım Adım Çözüm

1

Formulate the inequality representing the potato weight constraint.

3v + 6p \leq 90

, where

v

is the number of portions of roasted vegetables and

p

is the number of portions of mashed potatoes.

Each portion of roasted vegetables uses

3

ounces of potatoes, each portion of mashed potatoes uses

6

ounces of potatoes, and the total amount used cannot exceed

90

ounces.

2

Formulate the inequalities representing the minimum quantity constraints.

v \geq 8

and

p \geq 5

The chef must make at least

8

portions of roasted vegetables and at least

5

portions of mashed potatoes.

3

Isolate the variable

v

in the potato limit inequality.

v \leq 30 - 2p

Subtracting

6p

from both sides of

3v + 6p \leq 90

yields

3v \leq 90 - 6p

, and dividing the entire inequality by

3

gives

v \leq 30 - 2p

.

4

Determine the maximum value of

v

by substituting the minimum possible value of

p

.

v \leq 30 - 2(5) \implies v \leq 20

To maximize

v

, we must minimize the subtracted term

2p

. The minimum allowed value of

p

is

5

.

Anahtar Kavram

Optimization in systems of linear inequalities

Soru 127Soru →

For the linear function $f$ , the graph of $y = f(x)$ has a slope of $3$ and contains the point $(2, 7)$ . The graph of another linear function, $g$ , is parallel to the graph of $f$ . If $g(0) = -4$ , what is the value of $g(5)$ ?

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

11

Since the graph of

g

is parallel to the graph of

f

, it has the same slope of

3

. The value

g(0) = -4

indicates that the

y

-intercept of the graph of

g

is

-4

. Thus, the equation for

g(x)

is

g(x) = 3x - 4

. Evaluating this function at

x = 5

gives

g(5) = 3(5) - 4 = 11

.

Adım Adım Çözüm

1

Determine the slope of the function

g

.

The slope of

g

is

3

.

Parallel lines in the coordinate plane have equal slopes, and the slope of the graph of

f

is given as

3

.

2

Write the equation for

g(x)

.

g(x) = 3x - 4

Using the slope-intercept form

g(x) = mx + b

with slope

m = 3

and

y

-intercept

b = g(0) = -4

.

3

Evaluate

g(5)

.

11

Substitute

x = 5

into the equation for

g(x)

to get

g(5) = 3(5) - 4 = 11

.

Anahtar Kavram

Parallel lines have the same slope. A linear function can be written in slope-intercept form,

y = mx + b

, where

m

is the slope and

b

is the

y

-intercept.

Soru 128Soru →

A local library has two types of study rooms: small rooms, which can accommodate up to $3$ people, and large rooms, which can accommodate up to $8$ people. The library has a total of $15$ study rooms. If the maximum capacity of all the study rooms combined is $80$ people, how many large study rooms does the library have?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The correct answer is 7.

By representing the number of small rooms as

s

and large rooms as

l

, we set up the system of linear equations

s + l = 15

and

3s + 8l = 80

. Solving the first equation for

s

gives

s = 15 - l

. Substituting this into the second equation yields

3(15 - l) + 8l = 80

. Distributing and combining like terms results in

45 + 5l = 80

. Subtracting 45 from both sides gives

5l = 35

, which simplifies to

l = 7

. Thus, there are 7 large study rooms.

Adım Adım Çözüm

1

Set up the system of linear equations representing the total number of rooms and their total capacity.

s + l = 15

and

3s + 8l = 80

We define

s

as the number of small study rooms and

l

as the number of large study rooms. The total number of rooms is 15, and the total capacity is 80.

2

Express the number of small rooms,

s

, in terms of the number of large rooms,

l

, using the first equation.

s = 15 - l

This allows for substitution into the capacity equation so that we can solve directly for the number of large study rooms.

3

Substitute the expression for

s

into the capacity equation and solve the resulting single-variable equation for

l

.

3(15 - l) + 8l = 80 \implies 45 - 3l + 8l = 80 \implies 45 + 5l = 80 \implies 5l = 35 \implies l = 7

Substituting

s

eliminates one variable, leaving a linear equation in terms of

l

that can be solved using standard algebraic steps.

Anahtar Kavram

Solving systems of linear equations in a real-world context using substitution.

Soru 129Soru →

If the expression $5(x^2 - 2x) - 3(x^2 - 4x)$ is rewritten in the form $ax^2 + bx$ , where $a$ and $b$ are constants, what is the value of $a$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The value of

a

is

2

.

Distributing the constants outside the parentheses yields

5x^2 - 10x - 3x^2 + 12x

. Combining the

x^2

terms (

5x^2 - 3x^2 = 2x^2

) and the

x

terms (

-10x + 12x = 2x

) gives the simplified equivalent expression

2x^2 + 2x

. Comparing this directly to the form

ax^2 + bx

, we find that the coefficient

a

is

2

.

Adım Adım Çözüm

1

Distribute the coefficients to the terms inside the parentheses.

5x^2 - 10x - 3x^2 + 12x

To expand the expression so that like terms can be combined.

2

Combine the coefficients of

x^2

and the coefficients of

x

.

2x^2 + 2x

To write the expression in simplified form.

3

Identify the value of

a

by matching the simplified expression with

ax^2 + bx

.

a = 2

The constant

a

represents the coefficient of the

x^2

term.

Anahtar Kavram

Simplifying polynomial expressions by distributing constants and combining like terms.

Tahmini Süre:45s

Soru 130Soru →

An artist sells small prints for $4$ dollars each and large prints for $9$ dollars each. The artist wants to sell at least $15$ prints in total and earn at least $80$ dollars from these sales. If the artist sells $5$ small prints, what is the minimum number of large prints the artist must sell to meet both conditions?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

The minimum number of large prints the artist must sell is 10.

Substituting the value of small prints (

x = 5

) into the total count inequality

x + y \geq 15

gives

5 + y \geq 15

, which simplifies to

y \geq 10

. Substituting it into the earnings inequality

4x + 9y \geq 80

yields

20 + 9y \geq 80

, which simplifies to

y \geq 6.67

. To satisfy both conditions, the value of

y

must be at least 10.

Adım Adım Çözüm

1

Set up the inequalities for the system.

Let

x

represent the number of small prints and

y

represent the number of large prints. The constraint for the minimum number of prints is

x + y \geq 15

. The constraint for the minimum earnings is

4x + 9y \geq 80

.

To represent the given conditions as a system of linear inequalities in two variables.

2

Substitute the known value of small prints (

x = 5

) into the inequalities.

Substituting

x = 5

into the first inequality gives

5 + y \geq 15

, which simplifies to

y \geq 10

. Substituting

x = 5

into the second inequality gives

4(5) + 9y \geq 80

, which simplifies to

20 + 9y \geq 80

, then

9y \geq 60

, resulting in

y \geq \frac{20}{3} \approx 6.67

.

To determine the range of values for the number of large prints (

y

) under both constraints.

3

Determine the minimum integer value for

y

that satisfies both inequalities.

The first constraint requires

y \geq 10

, and the second constraint requires

y \geq 6.67

. Since the number of prints must be an integer and both conditions must be satisfied, the minimum value is

10

.

To satisfy both inequalities simultaneously with the smallest possible integer value.

Anahtar Kavram

Systems of Linear Inequalities in Two Variables

Tahmini Süre:1m 0s

Soru 131Soru →

The system of linear equations shown below contains constants

a

and

b

.

\begin{aligned} \frac{1}{2}(ax - 4y) &= x - \frac{1}{2}y + 6 \\ 2x - y &= b \end{aligned}

If the system has infinitely many solutions, what is the value of

a + b

?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The correct answer is 12.

To find the value of

a + b

that yields infinitely many solutions, we rewrite both equations in the standard form

Ax + By = C

. Simplifying the first equation gives

(a - 2)x - 3y = 12

. Multiplying the second equation

2x - y = b

by

3

gives

6x - 3y = 3b

. For the system to have infinitely many solutions, the two equations must be equivalent, meaning

a - 2 = 6

(which gives

a = 8

) and

3b = 12

(which gives

b = 4

). The sum of these values is

8 + 4 = 12

.

Adım Adım Çözüm

1

Distribute the fraction

\frac{1}{2}

on the left side of the first equation.

\frac{1}{2}ax - 2y = x - \frac{1}{2}y + 6

To expand the expression and prepare it for simplification.

2

Group the

x

and

y

terms on the left side of the equation and the constants on the right side.

\left(\frac{1}{2}a - 1\right)x - \frac{3}{2}y = 6

To write the equation in standard linear form.

3

Multiply the entire equation by

2

to eliminate the fractional coefficients.

(a - 2)x - 3y = 12

To simplify comparison with the second equation by working with integer coefficients.

4

Multiply the second equation,

2x - y = b

, by

3

to align the

y

-coefficients with the first equation.

6x - 3y = 3b

Two linear equations have infinitely many solutions if they represent the same line, which requires matching coefficients and constants.

5

Equate the corresponding

x

-coefficients and constant terms from

(a - 2)x - 3y = 12

and

6x - 3y = 3b

.

a - 2 = 6 \implies a = 8

and

3b = 12 \implies b = 4

To solve for the values of the constants

a

and

b

.

6

Calculate the sum of

a

and

b

.

8 + 4 = 12

To find the final requested value of

a + b

.

Anahtar Kavram

Determining parameters for infinitely many solutions in a system of linear equations

Soru 132Soru →

A company plans to install standard-charging ports and fast-charging ports at its office building. The company must install at least $15$ charging ports in total. Each fast-charging port requires $12$ kilowatts of power, and each standard-charging port requires $4$ kilowatts of power. The electrical grid can supply a maximum of $120$ kilowatts of power for these ports. Additionally, the number of standard-charging ports must be at least twice the number of fast-charging ports. What is the maximum number of fast-charging ports the company can install?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The maximum number of fast-charging ports the company can install is 6.

Representing the number of fast-charging ports by

x

and standard-charging ports by

y

yields the system of inequalities:

x + y \geq 15

,

12x + 4y \leq 120

, and

y \geq 2x

. The power constraint simplifies to

3x + y \leq 30

, or

y \leq 30 - 3x

. Combining this with

y \geq 2x

gives

2x \leq 30 - 3x

, which simplifies to

5x \leq 30

, or

x \leq 6

. Substituting

x = 6

into the constraints gives

y = 12

, which satisfies the total port constraint because

6 + 12 = 18 \geq 15

. Therefore, the maximum number of fast-charging ports is 6.

Adım Adım Çözüm

1

Define variables and write the system of inequalities representing the constraints.

Let

x

be the number of fast-charging ports and

y

be the number of standard-charging ports. The constraints are:
1)

x + y \geq 15

2)

12x + 4y \leq 120

3)

y \geq 2x

4)

x \geq 0, y \geq 0

This translates the word problem constraints into a system of linear inequalities.

2

Simplify the power capacity inequality.

3x + y \leq 30

, which can be rewritten as

y \leq 30 - 3x

.

Dividing the terms by 4 simplifies the coefficients, making algebraic manipulation easier.

3

Combine the simplified inequality with the charging port ratio constraint to find the upper limit for

x

.

Since

2x \leq y

and

y \leq 30 - 3x

, we have

2x \leq 30 - 3x

.
Adding

3x

to both sides yields

5x \leq 30

.
Dividing by 5 gives

x \leq 6

.

This determines the maximum possible value for the number of fast-charging ports.

4

Verify that the upper limit

x = 6

satisfies all system constraints with integer values.

If

x = 6

, then

y \geq 2(6) = 12

and

y \leq 30 - 3(6) = 12

, which means

y = 12

.
Checking the total port constraint:

x + y = 6 + 12 = 18

. Since

18 \geq 15

, the point

(6, 12)

satisfies all constraints.

Since the number of ports must be integers, we must confirm that

x = 6

yields a valid integer coordinate

(6, 12)

that lies within the feasible region.

Anahtar Kavram

Maximizing a variable under a system of linear inequalities in two variables.

Soru 133Soru →

In the $xy$ -plane, the graph of the equation $ax + by = 24$ , where $a$ and $b$ are constants, is a line. The line has an $x$ -intercept of $(d, 0)$ and a $y$ -intercept of $(0, d - 6)$ , where $d > 6$ is a constant. If the line passes through the point $(2, 5)$ , what is the value of $a + b$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

6

Substituting the intercepts

(d, 0)

and

(0, d-6)

into

ax + by = 24

gives

a = 24/d

and

b = 24/(d-6)

. Using the point

(2, 5)

, we get

2(24/d) + 5(24/(d-6)) = 24

, which simplifies to

2/d + 5/(d-6) = 1

. Solving for

d

yields

d^2 - 13d + 12 = 0

, giving

d = 12

or

d = 1

. Given

d > 6

, we have

d = 12

. Substituting

d = 12

back gives

a = 2

and

b = 4

, so

a + b = 6

.

Adım Adım Çözüm

1

Find expressions for

a

and

b

in terms of

d

by substituting the given intercepts.

a = \frac{24}{d}

and

b = \frac{24}{d-6}

The

x

-intercept

(d, 0)

and

y

-intercept

(0, d-6)

lie on the line

ax + by = 24

.

2

Substitute the point

(2, 5)

into the line's equation.

2a + 5b = 24

The point

(2, 5)

lies on the line.

3

Substitute the expressions for

a

and

b

into

2a + 5b = 24

and simplify.

\frac{2}{d} + \frac{5}{d-6} = 1

To create a single equation in terms of the variable

d

.

4

Solve the rational equation for

d

.

d = 12

Multiplying by the common denominator

d(d-6)

leads to the quadratic equation

d^2 - 13d + 12 = 0

, which factors as

(d-12)(d-1) = 0

. Since the problem states

d > 6

, the only valid solution is

d = 12

.

5

Calculate

a

and

b

using

d = 12

, and find their sum.

a = 2

,

b = 4

, and

a + b = 6

Substituting

d = 12

into the expressions for

a

and

b

yields the constant coefficients, and summing them provides the final requested value.

Anahtar Kavram

Using intercepts and a given point on a line in the coordinate plane to solve for the parameters of its linear equation.

Soru 134Soru →

In the $xy$ -plane, the system of linear equations below has infinitely many solutions, where $a$ , $b$ , and $c$ are constants and $c > 0$ :

\begin{aligned} 3x - 4y &= 10 \\ ax + by &= c \end{aligned}

If the graph of the second equation in the system passes through the point $(a, b)$ , what is the value of $c$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The correct answer is 4.

Since the system has infinitely many solutions, the two equations are equivalent. This means the coefficients are proportional, so we can write

a = 3k

,

b = -4k

, and

c = 10k

for some constant

k

. Because the line passes through

(a, b)

, substituting these coordinates into the second equation gives

a^2 + b^2 = c

. Substituting the expressions in terms of

k

results in

(3k)^2 + (-4k)^2 = 10k

, which simplifies to

25k^2 = 10k

. Since

c > 0

, we have

k > 0

, and dividing by

k

gives

25k = 10

, so

k = 0.4

. Finally,

c = 10(0.4) = 4

.

Adım Adım Çözüm

1

Set up a proportionality constant to relate the coefficients of the two equations.

a = 3k

,

b = -4k

, and

c = 10k

for a constant

k

.

Since the system has infinitely many solutions, the equations represent the same line, meaning their coefficients and constants must be proportional.

2

Substitute the point

(a, b)

into the second equation

ax + by = c

.

a^2 + b^2 = c

The graph of the equation passes through the point

(a, b)

, so the coordinates must satisfy the equation.

3

Substitute the parametric expressions of

a

,

b

, and

c

into the equation

a^2 + b^2 = c

.

25k^2 = 10k

This allows us to solve for the parameter

k

using a single variable quadratic equation.

4

Solve the equation

25k^2 = 10k

for

k

, given that

k > 0

.

k = 0.4

Since

c > 0

and

c = 10k

,

k

must be strictly positive, allowing us to divide both sides by

k

.

5

Compute the final value of

c

using the value of

k

.

c = 4

Substituting

k = 0.4

back into the expression

c = 10k

gives the value of

c

.

Anahtar Kavram

Systems of linear equations with infinitely many solutions and coordinate geometry constraints

Soru 135Soru →

If $\frac{2}{3}(6x - 9) - \frac{3}{4}(4x - 8) = \frac{1}{2}(x + 10)$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

The correct answer is 10.

Distributing the fractions on the left-hand side of the equation simplifies the expression to

x

. Setting this equal to the right-hand side yields the equation

x = \frac{1}{2}(x + 10)

. Multiplying both sides by 2 gives

2x = x + 10

, and subtracting

x

from both sides gives the correct value of 10.

Adım Adım Çözüm

1

Distribute the fraction

\frac{2}{3}

to the terms inside the first parentheses,

(6x - 9)

, and distribute the fraction

-\frac{3}{4}

to the terms inside the second parentheses,

(4x - 8)

.

4x - 6 - 3x + 6 = \frac{1}{2}(x + 10)

Applying the distributive property removes the parentheses so that like terms can be combined.

2

Combine the variable terms and constant terms on the left side of the equation.

x = \frac{1}{2}(x + 10)

Combining

4x

and

-3x

yields

x

, while

-6

and

6

cancel each other out.

3

Multiply both sides of the equation by 2 to eliminate the fraction.

2x = x + 10

Multiplying by the denominator simplifies the equation by removing the fraction.

4

Subtract

x

from both sides of the equation to isolate the variable.

x = 10

Subtracting

x

isolates the variable

x

on the left side of the equation.

Anahtar Kavram

Solving linear equations in one variable by applying the distributive property and combining like terms.

Soru 136Soru →

For all positive values of $x$ and $y$ , the expression $\frac{(3x^2 y^4)^3}{9x^4 y^7}$ can be written in the form $a x^b y^c$ , where $a$ , $b$ , and $c$ are constants. What is the value of $a + b + c$ ?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

The value of

a + b + c

is

10

.

To find the value of

a + b + c

, we simplify the given expression using the rules of exponents. First, apply the power of a product rule and power of a power rule to the numerator:

(3x^2 y^4)^3 = 3^3 (x^2)^3 (y^4)^3 = 27x^6y^{12}

. Next, divide this by the denominator:

\frac{27x^6y^{12}}{9x^4y^7}

. Divide the coefficients to get

\frac{27}{9} = 3

, and apply the quotient rule for exponents to the variable terms:

x^{6-4} = x^2

and

y^{12-7} = y^5

. The fully simplified expression is

3x^2y^5

, which matches the form

a x^b y^c

. Comparing coefficients and exponents, we find

a = 3

,

b = 2

, and

c = 5

. Summing these values gives

3 + 2 + 5 = 10

.

Adım Adım Çözüm

1

Simplify the numerator of the expression using exponent rules.

27x^6y^{12}

Applying the power of a product rule

(ab)^n = a^n b^n

and the power of a power rule

(a^m)^n = a^{mn}

to

(3x^2 y^4)^3

gives

3^3 \cdot x^{2 \times 3} \cdot y^{4 \times 3} = 27x^6y^{12}

.

2

Divide the simplified numerator by the denominator.

3x^2y^5

Divide the coefficients (

27 \div 9 = 3

) and subtract the exponents of the corresponding variables using the quotient of powers rule

\frac{a^m}{a^n} = a^{m-n}

(

x^{6-4} = x^2

and

y^{12-7} = y^5

).

3

Identify the values of the constants

a

,

b

, and

c

from the simplified expression

3x^2y^5

.

a = 3

,

b = 2

, and

c = 5

Comparing the simplified expression

3x^2y^5

with the template

a x^b y^c

yields

a = 3

,

b = 2

, and

c = 5

.

4

Calculate the sum of

a

,

b

, and

c

.

10

Adding the identified constant values together:

3 + 2 + 5 = 10

.

Anahtar Kavram

Simplifying exponential expressions using exponent rules

Soru 137Soru →

Consider the system of equations below:

\begin{aligned} \frac{1}{3}x + \frac{1}{4}y &= 5 \\ x - y &= 8 \end{aligned}

If

(x, y)

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

y

?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

y

is

4

.

Substituting

x = y + 8

from the second equation into the first equation yields

\frac{1}{3}(y + 8) + \frac{1}{4}y = 5

. Multiplying the entire equation by

12

to clear denominators gives

4(y + 8) + 3y = 60

, which simplifies to

7y + 32 = 60

. Solving for

y

gives

7y = 28

, or

y = 4

.

Adım Adım Çözüm

1

Express

x

in terms of

y

from the second equation.

x = y + 8

This allows for substitution into the first equation to solve for

y

directly.

2

Substitute

x = y + 8

into the first equation.

_\frac{1}{3}(y + 8) + \frac{1}{4}y = 5

This reduces the system to a single-variable linear equation in terms of

y

.

3

Multiply the entire equation by the least common multiple of the denominators, which is

12

.

4(y + 8) + 3y = 60

This clears the fractional coefficients and simplifies the arithmetic.

4

Distribute the

4

and combine like terms.

7y + 32 = 60

Simplifies the equation to prepare for isolating the variable

y

.

5

Isolate the variable term by subtracting

32

from both sides, then dividing by

7

.

y = 4

This gives the final value of

y

that satisfies the system.

Anahtar Kavram

Solving systems of linear equations using substitution and clearing fractional coefficients

Soru 138Soru →

A state department of transportation models the relationship between the age of a highway, in years, and its road quality index on a scale from $0$ to $100$ . The index decreases at a constant rate with respect to the age of the highway. At age $4$ years, the highway has a road quality index of $82$ . At age $12$ years, the highway has a road quality index of $66$ . According to the model, after how many years from its construction will the highway's road quality index reach $50$ ?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

According to the model, the highway's road quality index will reach

50

after

20

years.

Since the relationship is linear, the rate of change is constant. Using the points

(4, 82)

and

(12, 66)

, the slope is

\frac{66 - 82}{12 - 4} = -2

. The linear function can be written in point-slope form as

I - 82 = -2(t - 4)

, which simplifies to

I = -2t + 90

. Setting

I = 50

yields the equation

50 = -2t + 90

. Subtracting

90

from both sides gives

-40 = -2t

, and dividing by

-2

results in

t = 20

. Therefore, the index reaches

50

after

20

years.

Adım Adım Çözüm

1

Calculate the slope (rate of change) of the linear function.

The slope is

-2

.

Since the index decreases at a constant rate, the relationship is linear. The slope is calculated as the change in the road quality index divided by the change in the highway's age:

\frac{66 - 82}{12 - 4} = \frac{-16}{8} = -2

.

2

Determine the linear equation relating the road quality index,

I

, and the age,

t

.

I = -2t + 90

Using the point-slope formula with the point

(4, 82)

and slope

-2

:

I - 82 = -2(t - 4) \Rightarrow I - 82 = -2t + 8 \Rightarrow I = -2t + 90

.

3

Solve for the age,

t

, when the index

I

is

50

.

t = 20

Substitute

50

for

I

in the equation:

50 = -2t + 90

. Subtract

90

from both sides to get

-40 = -2t

. Divide by

-2

to find

t = 20

.

Anahtar Kavram

Writing and solving linear equations from two points

Soru 139Soru →

In the $xy$ -plane, a point $(x, y)$ is in the solution set of the system of inequalities below:

x - 2y \leq -6

x + y \leq 11

y \leq 5

What is the maximum possible value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The maximum possible value of

x

is

4

.

The maximum possible value of

x

is

4

. For all points in the solution set where

y \leq 5

, the active upper bound is

x \leq 2y - 6

. Since

2y - 6

increases as

y

increases, the maximum value of

x

occurs at the maximum boundary

y = 5

, giving

x = 2(5) - 6 = 4

.

Adım Adım Çözüm

1

Express

x

in terms of

y

using the first two inequalities.

x \leq 2y - 6

and

x \leq 11 - y

To isolate the variable

x

and analyze how its upper bound is constrained by

y

.

2

Apply the third inequality constraint,

y \leq 5

, to find the limits on these upper bounds.

2y - 6 \leq 4

and

11 - y \geq 6

Since

y

cannot exceed

5

, the value of

2y - 6

is maximized when

y = 5

, and

11 - y

is minimized when

y = 5

.

3

Compare the two upper bounds to determine the active constraint for the domain

y \leq 5

.

Since

2y - 6 \leq 4

and

11 - y \geq 6

, the active bound is

x \leq 2y - 6

.

For any point to satisfy the system,

x

must be less than or equal to both bounds, meaning it is restricted by the smaller of the two bounds.

4

Calculate the maximum value of

x

at the boundary point.

x = 4

at the point

(4, 5)

The function

2y - 6

is increasing with respect to

y

, so its maximum value occurs at the largest possible value of

y

, which is

5

.

Anahtar Kavram

Finding the maximum value of a coordinate within a system of linear inequalities.

Soru 140Soru →

In the $xy$ -plane, a point $(x, y)$ lies in the solution set of the system of inequalities below.

$y \leq -x + 8$
$y \leq 2x + 2$

What is the maximum possible value of $y$ ?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The maximum possible value of

y

is 6.

The solution set to the system of inequalities is the region in the coordinate plane that lies on or below both boundary lines,

y = -x + 8

and

y = 2x + 2

. The maximum

y

-value in this region occurs at the intersection of the two lines. Solving the system of equations by setting

-x + 8 = 2x + 2

gives

3x = 6

, which simplifies to

x = 2

. Substituting

x = 2

back into either equation yields

y = 6

. Therefore, the maximum possible value of

y

is 6.

Adım Adım Çözüm

1

Set the two boundary equations equal to each other to find the

x

-coordinate of the intersection point.

-x + 8 = 2x + 2 \implies 3x = 6 \implies x = 2

Since the solution region is bounded from above by both inequalities, the maximum value of

y

must occur at the intersection of the two boundary lines.

2

Substitute the

x

-value back into one of the equations to find the corresponding

y

-value.

y = 2(2) + 2 = 6

This determines the

y

-coordinate of the intersection point, which represents the maximum height of the shaded region.

Anahtar Kavram

Systems of Linear Inequalities in Two Variables

Tahmini Süre:45s

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