Systems of Linear Equations Alıştırma Soruları — SAT — Sayfa 3

For each real number

k

except

2

, the system of equations below has a unique solution

(x, y)

.

\begin{aligned} kx + (k + 2)y &= 3k + 1 \\ (k - 1)x + ky &= 2k - 1 \end{aligned}

If the solution

(x, y)

to the system also satisfies the equation

x + 2y = 6

, what is the value of

k

?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

k

is 4.

Subtracting the second equation from the first equation yields

x + 2y = k + 2

. Since the solution must also satisfy

x + 2y = 6

, we set

k + 2 = 6

, which gives

k = 4

.

Adım Adım Çözüm

1

Subtract the second equation from the first equation.

x + 2y = k + 2

To find a direct linear combination of

x

and

y

that can be compared directly to the target equation.

2

Equate the resulting expression to the target equation

x + 2y = 6

.

k + 2 = 6

Since the solution

(x, y)

must satisfy

x + 2y = 6

, the value of the linear combination

x + 2y

from the system must equal

6

.

3

Solve for

k

by subtracting 2 from both sides.

k = 4

Isolating the variable

k

yields the final solution.

Anahtar Kavram

Solving systems of linear equations with parameter coefficients by identifying algebraic structure and linear combinations.

Alternatif Yöntem

Solve the system for

x

and

y

in terms of

k

using elimination. Multiplying the first equation by

(k-1)

and the second by

k

, and then subtracting them yields

y = \frac{k^2 - k - 1}{k - 2}

. Substituting this back gives

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

. Substituting these expressions into the equation

x + 2y = 6

results in

\frac{k^2 - 4}{k - 2} = 6

. For

k \neq 2

, factoring

k^2 - 4

as

(k-2)(k+2)

allows simplification to

k + 2 = 6

, which yields

k = 4

.

Tahmini Süre:2m 0s

Soru 42Soru →

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

Cevabı ve açıklamayı göster

Cevap: 600

Cevap

600

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

B

represent the number of Basic subscriptions in April and

P

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

1,200

, we have

B + P = 1,200

. The revenue increase from April to May is

\$2,700

. The increase in Basic subscriptions is

20\%

, which contributes

15 \times 0.20B = 3B

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

5\%

, which contributes

30 \times 0.05P = 1.5P

dollars to the revenue increase. Thus,

3B + 1.5P = 2,700

. Substituting

B = 1,200 - P

into this equation yields

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

. Solving for

P

gives

3,600 - 1.5P = 2,700

, which simplifies to

1.5P = 900

, or

P = 600

.

Adım Adım Çözüm

1

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

B + P = 1,200

, where

B

is the number of Basic subscriptions and

P

is the number of Premium subscriptions.

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

2

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

3B + 1.5P = 2,700

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

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

.

3

Solve the system of equations by substituting

B = 1,200 - P

into the revenue equation.

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

Substitution eliminates the variable

B

, allowing us to solve for

P

directly.

4

Isolate the variable

P

to find the number of Premium subscriptions in April.

-1.5P = -900 \implies P = 600

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

P

.

Anahtar Kavram

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

Soru 43Soru →

Consider the system of equations below.

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

If

(x, y)

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

x

?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The value of

x

is

5

.

Expanding and simplifying the given system yields

x + 5y = 20

and

5x - y = 22

. Multiplying the second equation by 5 results in

25x - 5y = 110

. Adding this to the first equation eliminates

y

, leaving

26x = 130

. Dividing both sides by 26 gives

x = 5

.

Adım Adım Çözüm

1

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

x + 5y = 20

and

5x - y = 22

Expanding the first equation gives

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

, which simplifies to

x + 5y = 20

. Expanding the second equation gives

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

, which simplifies to

5x - y = 22

.

2

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

25x - 5y = 110

Multiplying

5x - y = 22

by 5 allows the subtraction of

y

to cancel with the addition of

5y

in the first equation.

3

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

26x = 130

, which simplifies to

x = 5

.

Adding

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

yields

26x = 130

. Dividing by 26 gives the final value of

x

.

Anahtar Kavram

Solving systems of linear equations using expansion and elimination.

Alternatif Yöntem

Define substitution variables

u = x + y

and

v = x - y

. The system simplifies to

3u - 2v = 20

and

2u + 3v = 22

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

9u - 6v = 60

and

4u + 6v = 44

. Adding these equations yields

13u = 104 \implies u = 8

. Substituting

u = 8

back in gives

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

. Now, solve the system

x + y = 8

and

x - y = 2

. Adding these two equations gives

2x = 10 \implies x = 5

.

Tahmini Süre:1m 30s

Soru 44Soru →

The table below shows several values of $x$ and the corresponding values of two linear functions, $f$ and $g$ .

$x$	$f(x)$	$g(x)$
$-2$	$12$	$-3$
$0$	$6$	$1$
$2$	$0$	$5$
$4$	$-6$	$9$

If the system of equations $y = f(x)$ and $y = g(x)$ has solution $(x, y)$ , what is the value of $x + y$ ?

-20

4

10

\frac{16}{7}

Cevabı ve açıklamayı göster

Cevap: $4$

Cevap

The sum of the coordinates of the solution to the system is

4

.

To find the solution to the system

y = f(x)

and

y = g(x)

, we must first determine the equations of the linear functions

f

and

g

from the given table. For

f(x)

, using the points

(0, 6)

and

(2, 0)

, the slope is

\frac{0 - 6}{2 - 0} = -3

. Since the

y

-intercept is

6

, the equation is

f(x) = -3x + 6

. For

g(x)

, using the points

(0, 1)

and

(2, 5)

, the slope is

\frac{5 - 1}{2 - 0} = 2

. Since the

y

-intercept is

1

, the equation is

g(x) = 2x + 1

. Setting the two equations equal to find their intersection gives

-3x + 6 = 2x + 1

. Solving for

x

yields

5x = 5

, or

x = 1

. Substituting

x = 1

into

g(x)

gives

y = 2(1) + 1 = 3

. The sum of the coordinates of the solution is

x + y = 1 + 3 = 4

.

Adım Adım Çözüm

1

Determine the linear equation for

f(x)

using the table values.

f(x) = -3x + 6

The slope of

f

is calculated as

\frac{f(2) - f(0)}{2 - 0} = \frac{0 - 6}{2} = -3

. Since

f(0) = 6

, the

y

-intercept is

6

.

2

Determine the linear equation for

g(x)

using the table values.

g(x) = 2x + 1

The slope of

g

is calculated as

\frac{g(2) - g(0)}{2 - 0} = \frac{5 - 1}{2} = 2

. Since

g(0) = 1

, the

y

-intercept is

1

.

3

Set

f(x) = g(x)

to solve for the

x

-coordinate of the intersection.

x = 1

Equating the two expressions gives

-3x + 6 = 2x + 1

. Adding

3x

to both sides and subtracting

1

from both sides results in

5x = 5

, which simplifies to

x = 1

.

4

Substitute

x = 1

back into either equation to solve for

y

.

y = 3

Using the equation

g(x) = 2x + 1

, substituting

x = 1

gives

y = 2(1) + 1 = 3

.

5

Calculate the sum

x + y

.

4

Adding the coordinates of the solution

(1, 3)

gives

1 + 3 = 4

.

Anahtar Kavram

Solving a system of linear equations derived from a table of values.

Soru 45Soru →

Consider the system of equations below.

\begin{aligned} 2(x + 3y) - 5y &= 10 \\ 3(x - y) + 2x &= 3 \end{aligned}

If

(x, y)

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

x + 2y

?

9

10

11

17

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

The value of 11 is the correct answer.

The correct answer is 11. Simplifying the first equation gives

2x + y = 10

, which allows us to write

y = 10 - 2x

. The second equation simplifies to

5x - 3y = 3

. Substituting the expression for

y

into the second equation yields

5x - 3(10 - 2x) = 3

. Distributing the

-3

gives

5x - 30 + 6x = 3

, which simplifies to

11x = 33

, or

x = 3

. Substituting

x = 3

back into the equation for

y

gives

y = 10 - 2(3) = 4

. Evaluating the expression

x + 2y

with these values results in

3 + 2(4) = 11

.

Adım Adım Çözüm

1

Simplify both equations in the system by distributing coefficients and combining like terms.

The first equation becomes

2x + 6y - 5y = 10 \Rightarrow 2x + y = 10

.
The second equation becomes

3x - 3y + 2x = 3 \Rightarrow 5x - 3y = 3

.

Simplifying the equations makes it easier to use substitution or elimination methods.

2

Solve the simplified system using substitution.

From

2x + y = 10

, express

y

as

y = 10 - 2x

. Substitute this into the second equation:

5x - 3(10 - 2x) = 3 \Rightarrow 5x - 30 + 6x = 3 \Rightarrow 11x = 33 \Rightarrow x = 3

. Substituting

x = 3

back into the expression for

y

gives

y = 10 - 2(3) = 4

.

This determines the unique values of the variables

x

and

y

that satisfy both equations.

3

Evaluate the expression

x + 2y

using the solved values.

Substitute

x = 3

and

y = 4

into

x + 2y

to get

3 + 2(4) = 11

.

This finds the specific quantity requested by the question.

Anahtar Kavram

Solving systems of linear equations using algebraic simplification and substitution.

Soru 46Soru →

A distributor plans to mix two coffee blends, Blend A and Blend B, to create a custom mixture. The table below shows the distribution of Colombian and Ethiopian coffee beans in each blend by weight:

Coffee Blend	Colombian Beans	Ethiopian Beans
Blend A	60%	40%
Blend B	20%	80%

The distributor wants the final custom mixture to contain exactly 14 kilograms of Colombian coffee beans and 16 kilograms of Ethiopian coffee beans. How many kilograms of Blend A should the distributor use to create this mixture?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The distributor should use 20 kilograms of Blend A.

By setting up a system of linear equations based on the percentage of each bean type in Blend A (

A

) and Blend B (

B

), we get

0.60A + 0.20B = 14

and

0.40A + 0.80B = 16

. Solving this system yields

A = 20

and

B = 10

. Therefore, 20 kilograms of Blend A are required.

Adım Adım Çözüm

1

Define variables for the unknowns and write the system of equations.

Let

A

be the number of kilograms of Blend A, and let

B

be the number of kilograms of Blend B. The system of equations is:
For Colombian beans:

0.60A + 0.20B = 14

For Ethiopian beans:

0.40A + 0.80B = 16

This translates the verbal and tabular constraints of the problem into mathematical equations.

2

Simplify the system by multiplying both sides of each equation to eliminate decimals.

Multiply the first equation by 5:

3A + B = 70 \implies B = 70 - 3A

Multiply the second equation by 5:

2A + 4B = 80 \implies A + 2B = 40

Working with integers makes the algebraic manipulation easier and reduces arithmetic errors.

3

Substitute the expression for

B

into the simplified second equation to solve for

A

.

A + 2(70 - 3A) = 40

A + 140 - 6A = 40

-5A = -100

A = 20

This isolates the variable

A

, which represents the required kilograms of Blend A.

Anahtar Kavram

Systems of Linear Equations

Soru 47Soru →

A bakery sells blueberry muffins and bran muffins. The total cost of $6$ blueberry muffins and $4$ bran muffins is $\$22.00$ . The total cost of $5$ blueberry muffins and $8$ bran muffins is $\$26.50$ . What is the cost, in dollars, of $1$ blueberry muffin?

Cevabı ve açıklamayı göster

Cevap: 2.5

Cevap

The cost of one blueberry muffin is 2.5 dollars.

The correct answer is 2.5. By setting up the system of linear equations

6x + 4y = 22

and

5x + 8y = 26.5

, where

x

represents the cost of one blueberry muffin and

y

represents the cost of one bran muffin, we can solve for

x

. Multiplying the first equation by 2 gives

12x + 8y = 44

. Subtracting the second equation,

5x + 8y = 26.5

, from the modified first equation yields

7x = 17.5

. Dividing by 7 results in

x = 2.5

.

Adım Adım Çözüm

1

Set up a system of equations.

6x + 4y = 22

and

5x + 8y = 26.5

Translate the context into mathematical statements where

x

is the price of a blueberry muffin and

y

is the price of a bran muffin.

2

Multiply the first equation by 2.

12x + 8y = 44

Align the coefficients of

y

to facilitate elimination.

3

Subtract the second equation from the modified first equation.

7x = 17.5

Eliminate the variable

y

to solve for

x

.

4

Divide by 7.

x = 2.5

Isolate the variable

x

to find the cost of a blueberry muffin.

Anahtar Kavram

Solving systems of two linear equations using elimination.

Alternatif Yöntem

Alternatively, solve for

y

in terms of

x

using the first equation:

y = 5.5 - 1.5x

. Substitute this expression into the second equation:

5x + 8(5.5 - 1.5x) = 26.5

, which simplifies to

5x + 44 - 12x = 26.5 \\Rightarrow -7x = -17.5 \\Rightarrow x = 2.5

.

Tahmini Süre:1m 30s

Soru 48Soru →

Consider the system of equations below.

\begin{aligned} 2x - y &= 4 \\ x + 2y &= 7 \end{aligned}

If

(x, y)

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

x + y

?

1

5

8

11

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The value of

x + y

is 5.

To solve the system, we can express

y

in terms of

x

from the first equation:

y = 2x - 4

. Substituting this into the second equation gives

x + 2(2x - 4) = 7

. Distributing the 2 gives

x + 4x - 8 = 7

, which simplifies to

5x - 8 = 7

. Adding 8 to both sides yields

5x = 15

, which gives

x = 3

. Substituting

x = 3

back into the first equation yields

2(3) - y = 4 \implies 6 - y = 4 \implies y = 2

. Thus, the value of

x + y

is

3 + 2 = 5

.

Adım Adım Çözüm

1

Express

y

in terms of

x

using the first equation.

y = 2x - 4

This sets up the system for solving by substitution.

2

Substitute the expression for

y

into the second equation and solve for

x

.

x + 2(2x - 4) = 7 \implies 5x - 8 = 7 \implies 5x = 15 \implies x = 3

This isolates and solves for the variable

x

.

3

Substitute the value of

x

back into the expression for

y

, then compute

x + y

.

y = 2(3) - 4 = 2

, and

x + y = 3 + 2 = 5

This finds the value of

y

and computes the final required sum.

Anahtar Kavram

Solving a system of linear equations in two variables to find the value of a linear combination.

Soru 49Soru →

In the system of equations below, $k$ is a constant.

\begin{aligned} kx - 3y &= 12 \\ (k-5)x - 2y &= 10 \end{aligned}

If the system has no solution, what is the value of $k$ ?

-15

-10

5

15

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

15

The correct answer is

15

. For a system of two linear equations to have no solution, the equations must represent parallel lines. This occurs when the coefficients of the variables are proportional but the constant terms are not. In this system, the ratio of the

x

-coefficients,

\frac{k}{k-5}

, must equal the ratio of the

y

-coefficients,

\frac{-3}{-2} = \frac{3}{2}

. Solving the equation

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

by cross-multiplying gives

2k = 3(k-5)

, which simplifies to

2k = 3k - 15

. Subtracting

3k

from both sides gives

-k = -15

, so

k = 15

. Since the ratio of the constants is

\frac{12}{10} = 1.2

, which is not equal to

1.5

, this value of

k

ensures the lines are parallel and distinct.

Adım Adım Çözüm

1

Set up the condition for a system of linear equations to have no solution.

The coefficients of

x

and

y

must be proportional, but not equal to the ratio of the constant terms:

\frac{A_1}{A_2} = \frac{B_1}{B_2} \neq \frac{C_1}{C_2}

.

If the coefficients are proportional, the lines have the same slope. If the constant terms are not in the same proportion, the lines have different intercepts, meaning they are parallel and distinct, and thus never intersect.

2

Substitute the coefficients from the given equations into the proportion.

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

and verify that

\frac{-3}{-2} = 1.5 \neq \frac{12}{10} = 1.2

.

This establishes the relationship between the coefficients of

x

and

y

required to make the lines parallel, while ensuring they are not identical lines.

3

Solve the proportion for

k

.

\frac{k}{k-5} = \frac{3}{2} \Rightarrow 2k = 3(k-5) \Rightarrow 2k = 3k - 15 \Rightarrow k = 15

.

Cross-multiplying and isolating the variable

k

yields the value of the constant.

Anahtar Kavram

For a system of linear equations to have no solution, the lines represented by the equations must be parallel and distinct, meaning their slopes are equal but their y-intercepts are different.

Tahmini Süre:1m 30s

Soru 50Soru →

For the system of equations shown below,

(x, y)

is the unique solution:

\begin{aligned} \frac{3}{4}(2x - y) &= \frac{2}{3}(x + 2y) \\ \frac{1}{2}(x - y) + \frac{5}{6}(2x + y) &= 11.5 \end{aligned}

What is the value of

x + y

?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The value of

x + y

is 7.

To find the value of

x + y

, we first simplify the first equation by multiplying both sides by the least common multiple of

3

and

4

, which is

12

. This yields

9(2x - y) = 8(x + 2y)

, which simplifies to

18x - 9y = 8x + 16y

, and further simplifies to

10x = 25y

, or

x = 2.5y

. Next, we substitute

x = 2.5y

into the second equation:

\frac{1}{2}(2.5y - y) + \frac{5}{6}(2(2.5y) + y) = 11.5

. Simplifying the terms gives

0.75y + 5y = 11.5

, or

5.75y = 11.5

, which yields

y = 2

. Substituting

y = 2

back into

x = 2.5y

gives

x = 5

. Thus, the value of

x + y

is

5 + 2 = 7

.

Adım Adım Çözüm

1

Clear the fractions in the first equation by multiplying by the least common multiple of the denominators.

10x = 25y

, which simplifies to

x = 2.5y

.

To express one variable in terms of the other for substitution.

2

Substitute the expression for

x

into the second equation and solve for

y

.

y = 2

To find the numerical value of one of the variables.

3

Substitute the value of

y

back into the simplified first equation to find

x

.

x = 5

To find the numerical value of the remaining variable.

4

Add the values of

x

and

y

to find

x + y

.

7

To answer the specific question asked.

Anahtar Kavram

Solving systems of linear equations using algebraic simplification and substitution

Soru 51Soru →

In a certain video game, players earn points for completing quests and defeating bosses. Completing a quest earns $q$ points, and defeating a boss earns $b$ points. Leo completed $5$ quests and defeated $3$ bosses, earning a total of $250$ points. Maya completed $7$ quests and defeated $2$ bosses, earning a total of $240$ points. What is the value of $q$ ?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The value of

q

is

20

.

To find the value of

q

, we translate the given scenarios into a system of two linear equations:

5q + 3b = 250

and

7q + 2b = 240

. Multiplying the first equation by

2

gives

10q + 6b = 500

. Multiplying the second equation by

3

gives

21q + 6b = 720

. Subtracting the first equation from the second yields

(21q - 10q) + (6b - 6b) = 720 - 500

, which simplifies to

11q = 220

. Dividing both sides by

11

results in

q = 20

.

Adım Adım Çözüm

1

Set up the system of linear equations from the given information.

5q + 3b = 250

and

7q + 2b = 240

To represent the points earned by Leo and Maya mathematically.

2

Multiply the first equation by

2

and the second equation by

3

.

10q + 6b = 500

and

21q + 6b = 720

To make the coefficients of

b

equal so they can be eliminated.

3

Subtract the first modified equation from the second modified equation.

11q = 220

To eliminate

b

and solve for

q

directly.

4

Divide both sides of the equation by

11

.

q = 20

To find the number of points earned per completed quest.

Anahtar Kavram

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

Soru 52Soru →

In the $xy$ -plane, the system of equations below has infinitely many solutions:

\begin{aligned} \frac{1}{2}(ax - by) &= 7 \\ 3x - 2y &= c \end{aligned}

where $a$ , $b$ , and $c$ are constants. If the line representing the first equation passes through the point $(4, 1)$ , what is the value of $c$ ?

5

8

10

20

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

10

The correct answer is 10. For a system of linear equations in two variables to have infinitely many solutions, the two equations must describe the exact same line, meaning their coefficients and constant terms are proportional. Multiplying the first equation by

2

yields

ax - by = 14

. Comparing this with

3x - 2y = c

shows that

\frac{a}{3} = \frac{b}{2} = \frac{14}{c}

, which simplifies to

b = \frac{2}{3}a

and

c = \frac{42}{a}

. Since the first line passes through

(4, 1)

, we substitute

x = 4

and

y = 1

to get

4a - b = 14

. Substituting

b = \frac{2}{3}a

gives

4a - \frac{2}{3}a = 14

, which simplifies to

\frac{10}{3}a = 14

, or

a = 4.2

. Finally, solving for

c

gives

c = \frac{42}{4.2} = 10

.

Adım Adım Çözüm

1

Clear the fraction in the first equation by multiplying both sides by

2

.

ax - by = 14

This puts the first equation into standard form, making it easier to compare with the second equation.

2

Set up the proportionality of the coefficients for the two equations to represent the same line.

\frac{a}{3} = \frac{-b}{-2} = \frac{14}{c} \implies b = \frac{2}{3}a

and

c = \frac{42}{a}

For a system of two linear equations to have infinitely many solutions, the equations must be equivalent, meaning their corresponding coefficients and constants must be proportional.

3

Substitute the given point

(4, 1)

into the first equation.

a(4) - b(1) = 14 \implies 4a - b = 14

Since the line passes through the point

(4, 1)

, the coordinates must satisfy the equation of the line.

4

Substitute

b = \frac{2}{3}a

into

4a - b = 14

and solve for

a

.

4a - \frac{2}{3}a = 14 \implies \frac{10}{3}a = 14 \implies a = 4.2

This reduces the equation to a single variable, allowing us to find the value of the parameter

a

.

5

Substitute

a = 4.2

into the expression for

c

to find its value.

c = \frac{42}{4.2} = 10

This uses the coefficient proportionality relation from step 2 to determine the constant term of the second equation.

Anahtar Kavram

Systems of linear equations with infinitely many solutions require the equations to represent the same line, meaning their coefficients and constant terms are proportional.

Soru 53Soru →

A chemist needs to mix a $10\%$ acid solution with a $30\%$ acid solution to create a $100$ -milliliter mixture. If the final mixture must be $18\%$ acid, how many milliliters of the $10\%$ acid solution should the chemist use?

40

60

70

80

Cevabı ve açıklamayı göster

Cevap: 60

Cevap

60 milliliters

The correct answer is 60 milliliters. Defining

x

as the volume of the

10\%

solution and

y

as the volume of the

30\%

solution gives the system of equations

x + y = 100

and

0.10x + 0.30y = 18

. Substituting

y = 100 - x

into the second equation yields

0.10x + 30 - 0.30x = 18

. Simplifying this equation gives

-0.20x = -12

, which simplifies to

x = 60

.

Adım Adım Çözüm

1

Define variables for the volume of each solution and set up the system of equations representing the total volume and the total amount of pure acid.

Let

x

be the number of milliliters of the

10\%

acid solution, and let

y

be the number of milliliters of the

30\%

acid solution. The system is:

\begin{aligned} x + y &= 100 \\ 0.10x + 0.30y &= 18 \end{aligned}

To represent the physical relationships between the two solutions mathematically.

2

Solve the first equation for

y

in terms of

x

and substitute this expression into the second equation.

y = 100 - x

0.10x + 0.30(100 - x) = 18

To reduce the system to a single linear equation in terms of

x

.

3

Distribute the coefficients, combine like terms, and solve for

x

.

0.10x + 30 - 0.30x = 18

-0.20x + 30 = 18

-0.20x = -12

x = 60

To isolate

x

and determine the volume of the

10\%

acid solution.

Anahtar Kavram

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

Soru 54Soru →

A landscape designer purchased a total of

35

plants, consisting of boxwood shrubs and fern plants, for a total of \

510$. Each boxwood shrub cost \

18

, and each fern plant cost \$

12

. If the designer purchased

b

boxwood shrubs and

f

fern plants, what is the value of

f - b$?

15

5

27

29

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The correct answer is 5, representing the difference between the 20 fern plants and 15 boxwood shrubs.

To find the value of

f - b

, we can set up a system of linear equations based on the given information. The total number of plants purchased is

35

, which gives the equation

b + f = 35

. The total cost of the plants is \

510$, with boxwood shrubs costing \

18

each and fern plants costing \$

12

each, giving the equation

18b + 12f = 510

. Solving the first equation for

f

gives

f = 35 - b

. Substituting this into the second equation yields

18b + 12(35 - b) = 510

. Distributing and simplifying gives

18b + 420 - 12b = 510

, which simplifies to

6b + 420 = 510

. Subtracting

420

from both sides gives

6b = 90

, so

b = 15

. Substituting

b = 15

back into

b + f = 35

gives

15 + f = 35

, so

f = 20

. The value of

f - b

is

20 - 15 = 5$.

Adım Adım Çözüm

1

Define variables and write the system of linear equations representing the problem.

Let

b

be the number of boxwood shrubs and

f

be the number of fern plants. The system of equations is:

\begin{aligned} b + f &= 35 \\ 18b + 12f &= 510 \end{aligned}

We need to translate the word problem into a mathematical system of two linear equations with two variables.

2

Express one variable in terms of the other from the first equation and substitute it into the second equation.

From

b + f = 35

, we get

f = 35 - b

.
Substituting this into the second equation gives:

18b + 12(35 - b) = 510

Substitution is a standard method to reduce a system of two equations to a single equation in one variable.

3

Solve the resulting single-variable equation for

b

.

18b + 420 - 12b = 510

6b + 420 = 510

6b = 90

b = 15

By distributing the coefficient and combining like terms, we isolate and solve for the variable b.

4

Substitute the value of

b

back into the first equation to solve for

f

, and then calculate

f - b

.

15 + f = 35 \Rightarrow f = 20

f - b = 20 - 15 = 5

Finding the individual values of both variables allows us to calculate the required difference.

Anahtar Kavram

Solving systems of linear equations in two variables using substitution or elimination, and evaluating a linear combination of the solution.

Alternatif Yöntem

Instead of substitution, the system can be solved using elimination. Multiply the first equation,

b + f = 35

, by

12

to get

12b + 12f = 420

. Subtract this from the second equation,

18b + 12f = 510

, to eliminate

f

:

(18b - 12b) + (12f - 12f) = 510 - 420

, which simplifies to

6b = 90

, giving

b = 15

. Then, find

f = 20

and compute

f - b = 5

.

Tahmini Süre:1m 30s

Soru 55Soru →

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

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The weight of one large box is 20 pounds.

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

8s + 5L = 180

and

6s + 10L = 260

), we can eliminate

L

by multiplying the first equation by 2, resulting in

16s + 10L = 360

. Subtracting the second equation from this gives

10s = 100

, which solves to

s = 10

. Substituting

s = 10

back into

8s + 5L = 180

yields

80 + 5L = 180

, which simplifies to

5L = 100

, meaning

L = 20

.

Adım Adım Çözüm

1

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

Let

s

be the weight of a small box and

L

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

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

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

2

Multiply the first equation by 2 to align the coefficients of

L

.

16s + 10L = 360

This sets up the variable

L

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

3

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

s

.

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

10s = 100

s = 10

Subtracting eliminates the variable

L

, leaving a single-variable linear equation to solve.

4

Substitute

s = 10

back into the first equation to solve for

L

.

8(10) + 5L = 180

80 + 5L = 180

5L = 100

L = 20

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

Anahtar Kavram

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

Soru 56Soru →

An athlete participates in a charity event where they earn money by swimming laps in a pool and running laps on a track. The athlete earns $s$ dollars for each lap swum and $r$ dollars for each lap run. Completing 10 laps in the pool and 15 laps on the track raises a total of $135. Completing 12 laps in the pool and 20 laps on the track raises a total of$ 176. What is the value of $r$ ?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The value of

r

is 7.

The correct answer is 7. To find this, we translate the word problem into a system of two linear equations:

10s + 15r = 135

and

12s + 20r = 176

. Simplifying these equations by dividing by their common factors yields

2s + 3r = 27

and

3s + 5r = 44

. Multiplying the first equation by 3 gives

6s + 9r = 81

, and multiplying the second equation by 2 gives

6s + 10r = 88

. Subtracting the two equations eliminates

s

and gives

r = 7

.

Adım Adım Çözüm

1

Write the system of linear equations representing the athlete's earnings.

The system is:

10s + 15r = 135

12s + 20r = 176

To represent the relationships between the number of laps completed and the total amount raised.

2

Simplify both equations to work with smaller coefficients.

Divide the first equation by 5:

2s + 3r = 27

Divide the second equation by 4:

3s + 5r = 44

Simplifying equations reduces the likelihood of arithmetic errors.

3

Eliminate the variable

s

to solve for

r

.

Multiply the first simplified equation by 3 and the second by 2:

3(2s + 3r) = 3(27) \implies 6s + 9r = 81

2(3s + 5r) = 2(44) \implies 6s + 10r = 88

Subtract the first equation from the second:

(6s + 10r) - (6s + 9r) = 88 - 81 \implies r = 7

Using the elimination method allows us to solve directly for the target variable,

r

.

Anahtar Kavram

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

Soru 57Soru →

An online store sells two types of monthly subscription plans: a basic plan for $12 per month and a premium plan for$ 20 per month. In April, the store had a total of 150 active subscribers for these two plans and collected a total of $2,280 in subscription fees. How many of the active subscribers in April were on the premium plan?

15

50

60

90

Cevabı ve açıklamayı göster

Cevap: 60

Cevap

60

By representing the number of basic subscribers as

x

and premium subscribers as

y

, we set up the system of equations

x + y = 150

and

12x + 20y = 2280

. Solving for

x

in the first equation gives

x = 150 - y

. Substituting this into the second equation yields

12(150 - y) + 20y = 2280

. Distributing the 12 and combining like terms gives

1800 + 8y = 2280

. Subtracting 1800 from both sides gives

8y = 480

, and dividing by 8 results in

y = 60

. Therefore, there were 60 premium subscribers.

Adım Adım Çözüm

1

Define variables for the unknown quantities: let

x

represent the number of basic subscribers and let

y

represent the number of premium subscribers.

The system of equations is:

x + y = 150

and

12x + 20y = 2280

.

Translate the given information about total subscribers and total revenue into algebraic equations.

2

Express

x

in terms of

y

using the first equation (

x = 150 - y

) and substitute it into the second equation.

The substituted equation is:

12(150 - y) + 20y = 2280

, which expands to

1800 - 12y + 20y = 2280

.

Eliminate one variable to create a single equation with only one variable,

y

.

3

Simplify the equation and isolate

y

.

1800 + 8y = 2280 \implies 8y = 480 \implies y = 60

.

Combine like terms and use basic operations to solve for the number of premium subscribers.

Anahtar Kavram

Solving a system of two linear equations using substitution or elimination.

Tahmini Süre:1m 30s

Soru 58Soru →

A coffee shop sells two sizes of bagged coffee: a standard bag and a deluxe bag. On Monday, the shop sold $8$ standard bags and $5$ deluxe bags for a total of $\$108$ . On Tuesday, the shop sold $6$ standard bags and $10$ deluxe bags for a total of $\$156$ . What is the price, in dollars, of one deluxe bag of coffee?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The price of one deluxe bag of coffee is

12

dollars.

By setting up a system of equations, we represent the total cost on Monday as

8s + 5d = 108

and on Tuesday as

6s + 10d = 156

. Multiplying the first equation by

2

gives

16s + 10d = 216

. Subtracting the second equation yields

10s = 60

, so

s = 6

. Substituting

s = 6

into the first equation yields

8(6) + 5d = 108

, which simplifies to

48 + 5d = 108

, then

5d = 60

, and finally

d = 12

.

Adım Adım Çözüm

1

Define the variables and write the system of equations.

Let

s

be the price of a standard bag and

d

be the price of a deluxe bag. The system is:

8s + 5d = 108

6s + 10d = 156

To translate the word problem into mathematical equations.

2

Multiply the first equation by

2

to align the coefficients of

d

.

16s + 10d = 216

To prepare for elimination of the variable

d

.

3

Subtract the second equation from the new equation to solve for

s

.

10s = 60

, which simplifies to

s = 6

.

To eliminate

d

and find the price of a standard bag.

4

Substitute the value of

s

back into one of the original equations to solve for

d

.

8(6) + 5d = 108 \implies 48 + 5d = 108 \implies 5d = 60 \implies d = 12

.

To find the price of a deluxe bag.

Anahtar Kavram

Solving systems of linear equations using the elimination method.

Tahmini Süre:1m 30s

Soru 59Soru →

A community theater sold adult tickets and student tickets for its opening weekend performances. On Friday, the theater sold $20$ adult tickets and $40$ student tickets for a total of $\$760$ . On Saturday, the theater sold $35$ adult tickets and $30$ student tickets for a total of $\$930$ . What is the price, in dollars, of one adult ticket?

10

16

18

30

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

The price of one adult ticket is 18 dollars.

The price of one adult ticket is

18

. By setting up the system of linear equations representing the total sales on Friday (

20x + 40y = 760

) and Saturday (

35x + 30y = 930

), where

x

is the adult ticket price and

y

is the student ticket price, we can simplify them to

x + 2y = 38

and

7x + 6y = 186

. Multiplying the first simplified equation by

3

gives

3x + 6y = 114

. Subtracting this equation from

7x + 6y = 186

eliminates

y

and yields

4x = 72

, which simplifies to

x = 18

.

Adım Adım Çözüm

1

Define variables and write the system of linear equations based on the given information.

Let

x

represent the price of an adult ticket and

y

represent the price of a student ticket. The system of equations is:

20x + 40y = 760

35x + 30y = 930

Variables represent the unknown quantities, and the equations relate these quantities to the total sales on Friday and Saturday.

2

Simplify both equations by dividing by their greatest common divisors.

Divide the first equation by

20

:

x + 2y = 38

Divide the second equation by

5

:

7x + 6y = 186

Simplifying equations reduces the coefficient values and minimizes the chance of arithmetic errors in later steps.

3

Use the elimination method to solve for

x

. Multiply the first simplified equation by

3

to align the

y

-coefficients.

3(x + 2y) = 3(38) \Rightarrow 3x + 6y = 114

Matching the coefficients of

y

allows us to eliminate the variable

y

by subtracting the equations.

4

Subtract the equation obtained in Step 3 from the second simplified equation.

(7x + 6y) - (3x + 6y) = 186 - 114 \Rightarrow 4x = 72

Subtracting the equations eliminates

y

, leaving a single linear equation in terms of

x

.

5

Solve for

x

by dividing both sides of the equation by

4

.

x = 18

Isolating

x

provides the price of one adult ticket.

Anahtar Kavram

Solving a system of linear equations using substitution or elimination.

Alternatif Yöntem

Instead of using elimination on the simplified equations, you can use substitution. Express

x

in terms of

y

using the first simplified equation:

x = 38 - 2y

. Substitute this expression into the second equation:

7(38 - 2y) + 6y = 186

. Expanding this gives

266 - 14y + 6y = 186

, which simplifies to

266 - 8y = 186

. Subtracting

266

from both sides gives

-8y = -80

, which solves to

y = 10

. Finally, substitute

y = 10

back into

x = 38 - 2y

to find

x = 38 - 2(10) = 18

.

Tahmini Süre:1m 30s

Soru 60Soru →

A scientist is monitoring the population of two species of bacteria, Species A and Species B, in a controlled environment. Initially, the combined population of the two species is $12,000$ . Over the next week, the population of Species A doubles, and the population of Species B triples. If the total combined population of both species is $29,000$ at the end of the week, what was the initial population of Species A?

Cevabı ve açıklamayı göster

Cevap: 7000

Cevap

The initial population of Species A was 7000.

To find the initial population of Species A, we set up a system of linear equations representing the total initial population and the population after one week. Let

x

represent the initial population of Species A and

y

represent the initial population of Species B. The initial combined population is given by

x + y = 12,000

. After one week, the population of Species A doubles (

2x

) and Species B triples (

3y

), so the new combined population is

2x + 3y = 29,000

. We can express

y

in terms of

x

as

y = 12,000 - x

. Substituting this expression into the second equation gives

2x + 3(12,000 - x) = 29,000

. Distributing and simplifying yields

-x + 36,000 = 29,000

, which simplifies to

x = 7,000

. Therefore, the initial population of Species A was 7,000.

Adım Adım Çözüm

1

Define the variables and set up the first equation.

x + y = 12,000

Let

x

be the initial population of Species A and

y

be the initial population of Species B. Their combined initial population is

12,000

.

2

Set up the second equation based on the growth after one week.

2x + 3y = 29,000

The population of Species A doubles to

2x

and Species B triples to

3y

, summing to a total of

29,000

.

3

Solve for

x

using substitution.

x = 7,000

From the first equation,

y = 12,000 - x

. Substituting this into the second equation gives

2x + 3(12,000 - x) = 29,000

. Distributing the

3

yields

2x + 36,000 - 3x = 29,000

. Combining like terms results in

-x + 36,000 = 29,000

. Subtracting

36,000

from both sides gives

-x = -7,000

, which simplifies to

x = 7,000

.

Anahtar Kavram

Solving systems of linear equations using substitution or elimination methods

Systems of Linear Equations

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