Systems of Linear Equations Alıştırma Soruları — GRE General Test — Sayfa 2

Soru 21Soru →

An investor allocates a total of $\$24,000$ among three accounts: Account A, which earns $3\%$ annual simple interest; Account B, which earns $5\%$ annual simple interest; and Account C, which earns $7\%$ annual simple interest. The total annual interest earned from all three accounts combined at the end of one year is $\$1,260$ . If the amount invested in Account C is $\$2,000$ more than twice the amount invested in Account A, what is the amount invested in Account B?

\$1,000

\$4,000

\$11,000

\$15,000

\$19,000

Cevabı ve açıklamayı göster

Cevap: $\$19,000$

Cevap

The amount invested in Account B is

\$19,000

.

The system of linear equations representing the total investment, interest earned, and relative account values yields

A = \$1,000

,

C = \$4,000

, and

B = \$19,000

. Thus, the amount invested in Account B is

\$19,000

.

Adım Adım Çözüm

1

Define variables and write the system of three linear equations based on total investment, total annual interest, and account relationship.

Let

A

,

B

, and

C

represent the dollars invested in Accounts A, B, and C respectively.
Equation (1):

A + B + C = 24,000

Equation (2):

0.03A + 0.05B + 0.07C = 1,260

, which simplifies to

3A + 5B + 7C = 126,000

Equation (3):

C = 2A + 2,000

, or

2A - C = -2,000

Translate the verbal conditions into an explicit

3 \times 3

system of linear equations.

2

Substitute

C = 2A + 2,000

into Equations (1) and (2) to reduce the system to two variables (

A

and

B

).

From Equation (1):

A + B + (2A + 2,000) = 24,000 \implies 3A + B = 22,000 \implies B = 22,000 - 3A

.
From Equation (2):

3A + 5B + 7(2A + 2,000) = 126,000 \implies 17A + 5B + 14,000 = 126,000 \implies 17A + 5B = 112,000

.

Eliminate variable

C

to simplify solving the linear system.

3

Substitute

B = 22,000 - 3A

into

17A + 5B = 112,000

to solve for

A

.

17A + 5(22,000 - 3A) = 112,000 \implies 17A + 110,000 - 15A = 112,000 \implies 2A = 2,000 \implies A = 1,000

.

Solve for the single variable

A

.

4

Determine the values of

C

and

B

.

C = 2(1,000) + 2,000 = 4,000

.

B = 22,000 - 3(1,000) = 19,000

.

Substitute

A = 1,000

back into the expressions for

C

and

B

to find the targeted investment amount.

Anahtar Kavram

Setting up and solving a system of three linear equations in three variables by substitution and elimination.

Tahmini Süre:2m 0s

Soru 22Soru →

If $3x + y = 14$ and $x - 2y = -7$ , what is the value of $x + y$ ?

3

5

8

11

2

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

8

Solving the linear system yields

x = 3

and

y = 5

. Adding these values together gives

3 + 5 = 8

, which is the correct answer.

Adım Adım Çözüm

1

Express

y

in terms of

x

using the first equation

y = 14 - 3x

Isolating one variable allows for direct substitution into the second equation.

2

Substitute

y = 14 - 3x

into the second equation

x - 2y = -7

x - 2(14 - 3x) = -7 \implies x - 28 + 6x = -7 \implies 7x = 21 \implies x = 3

Solving the single-variable equation gives the exact value for

x

.

3

Calculate

y

using

x = 3

y = 14 - 3(3) = 5

Substituting

x = 3

back into

y = 14 - 3x

gives the value for

y

.

4

Compute the sum

x + y

x + y = 3 + 5 = 8

The question specifically asks for the sum of both variables.

Anahtar Kavram

Solving a 2x2 System of Linear Equations by Substitution or Elimination

Tahmini Süre:45s

Soru 23Soru →

A specialty coffee shop prepares two custom coffee bean blends using Arabica and Robusta beans. The first 10-pound blend consists of 4 pounds of Arabica beans and 6 pounds of Robusta beans and costs $\$52$ . The second 10-pound blend consists of 7 pounds of Arabica beans and 3 pounds of Robusta beans and costs $\$61$ . What is the cost, in dollars, of 1 pound of Arabica coffee beans?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The cost of 1 pound of Arabica coffee beans is 7 dollars.

By setting up the system

4A + 6R = 52

and

7A + 3R = 61

, we can simplify the first equation to

2A + 3R = 26

. Subtracting this simplified equation from the second equation eliminates

R

, giving

5A = 35

, which simplifies directly to

A = 7

.

Adım Adım Çözüm

1

Define variables and set up the system of linear equations.

Let

A

be the price per pound of Arabica beans and

R

be the price per pound of Robusta beans.
Equation 1:

4A + 6R = 52

Equation 2:

7A + 3R = 61

Translating the quantitative relationship given in the problem statement into algebraic equations.

2

Simplify Equation 1 and eliminate variable

R

by subtraction.

Dividing Equation 1 by 2 gives

2A + 3R = 26

. Subtracting this from Equation 2 yields

(7A + 3R) - (2A + 3R) = 61 - 26

, which reduces to

5A = 35

.

Matching coefficients of

R

allows for straightforward elimination of

R

.

3

Solve for variable

A

.

A = 7

Dividing 35 by 5 yields the price per pound of Arabica beans.

Anahtar Kavram

Solving a system of 2x2 linear equations using substitution or elimination.

Soru 24Soru →

A venue offers two types of event packages: Standard and Deluxe. The total cost of 3 Standard packages and 2 Deluxe packages is $410. The total cost of 2 Standard packages and 5 Deluxe packages is$ 640. What is the cost, in dollars, of 1 Deluxe package?

Cevabı ve açıklamayı göster

Cevap: 100

Cevap

100

Setting up the linear system

3S + 2D = 410

and

2S + 5D = 640

allows us to multiply the equations by 2 and 3 respectively, obtaining

6S + 4D = 820

and

6S + 15D = 1920

. Subtracting the two equations eliminates

S

and gives

11D = 1100

, leading to

D = 100

.

Adım Adım Çözüm

1

Define variables and establish the system of linear equations.

Let

S

represent the cost of a Standard package and

D

represent the cost of a Deluxe package.
Equation 1:

3S + 2D = 410

Equation 2:

2S + 5D = 640

Translating the scenario into mathematical equations forms a 2x2 system of linear equations.

2

Use elimination to eliminate variable

S

.

Multiply Equation 1 by 2:

6S + 4D = 820

Multiply Equation 2 by 3:

6S + 15D = 1920

Creating matching coefficients for

S

allows elimination by subtraction.

3

Subtract the transformed equations and solve for

D

.

(6S + 15D) - (6S + 4D) = 1920 - 820 \implies 11D = 1100 \implies D = 100

Subtracting cancels out

S

, leaving a single linear equation in terms of

D

.

Anahtar Kavram

Solving 2x2 Systems of Linear Equations via Elimination

Alternatif Yöntem

Use the substitution method: Solve for

S

in terms of

D

from the first equation (

S = \frac{410 - 2D}{3}

) and substitute this into the second equation (

2\left(\frac{410 - 2D}{3}\right) + 5D = 640

). Multiplying both sides by 3 yields

820 - 4D + 15D = 1920

, which simplifies to

11D = 1100

, so

D = 100

.

Tahmini Süre:1m 30s

Soru 25Soru →

In the $xy$ -plane, line $L_1$ passes through the points $(1, 1)$ and $(3, 5)$ . Line $L_2$ is defined by the equation $\frac{x}{4} + \frac{y}{7} = 2$ . If $(x, y)$ is the point of intersection of lines $L_1$ and $L_2$ , what is the value of $x + y$ ?

11

22

4

9.5

15

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

The value of

x + y

is 11.

The correct solution first determines the equation of the first line,

y = 2x - 1

, from its given points. Substituting this into the second line's equation

\frac{x}{4} + \frac{y}{7} = 2

and clearing fractions yields

x = 4

and

y = 7

. Adding these coordinates gives

4 + 7 = 11

.

Adım Adım Çözüm

1

Determine the slope and equation of line

L_1

.

The slope

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

. Using point-slope form with

(1, 1)

,

y - 1 = 2(x - 1)

, which simplifies to

y = 2x - 1

.

Two points uniquely define a line, allowing us to express

y

in terms of

x

.

2

Substitute the expression for

y

into the equation for line

L_2

.

\frac{x}{4} + \frac{2x - 1}{7} = 2

.

At the intersection point, both equations share the exact same

(x, y)

values.

3

Clear the denominators by multiplying the equation by the least common multiple, 28.

7x + 4(2x - 1) = 56 \implies 7x + 8x - 4 = 56 \implies 15x = 60 \implies x = 4

.

Eliminating fractions simplifies solving for

x

.

4

Calculate the value of

y

and find

x + y

.

y = 2(4) - 1 = 7

, so

x + y = 4 + 7 = 11

.

The question specifically asks for the sum of the intersection coordinates.

Anahtar Kavram

Systems of Linear Equations and Line Intersections

Tahmini Süre:1m 30s

Soru 26Soru →

A startup hired two freelance software developers, Developer A and Developer B, for a combined total of $45$ hours on a single project. Developer A charges $\$65$ per hour, and Developer B charges $\$80$ per hour. If the total amount paid to both developers was $\$3{}225$ , how many hours did Developer A work on the project?

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

Developer A worked on the project for 25 hours.

Let

x

be the hours Developer A worked and

y

be the hours Developer B worked. From

x + y = 45

, we get

y = 45 - x

. Substituting into

65x + 80y = 3225

yields

65x + 80(45 - x) = 3225

. Expanding gives

65x + 3600 - 80x = 3225

, which simplifies to

-15x = -375

, so

x = 25

.

Adım Adım Çözüm

1

Set up a system of linear equations representing the total hours and total cost.

x + y = 45

and

65x + 80y = 3225

, where

x

is Developer A's hours and

y

is Developer B's hours.

Translating the verbal conditions into mathematical equations.

2

Express

y

in terms of

x

from the hours equation.

y = 45 - x

Prepares the linear system for substitution.

3

Substitute

y = 45 - x

into the total cost equation and solve for

x

.

65x + 80(45 - x) = 3225 \implies -15x = -375 \implies x = 25

Solves for the requested variable

x

directly.

Anahtar Kavram

Solving word problems using 2x2 systems of linear equations via substitution or elimination.

Soru 27Soru →

A bakery packages two types of gift baskets containing gourmet croissants and blueberry muffins. Basket X contains $4$ croissants and $3$ muffins, with a total production cost of $\$19.00$ . Basket Y contains $2$ croissants and $5$ muffins, with a total production cost of $\$16.50$ . Assuming the cost per croissant and the cost per muffin are constant across all baskets, what is the production cost, in dollars, of a single croissant?

Cevabı ve açıklamayı göster

Cevap: 3.25

Cevap

The production cost of a single croissant is 3.25 dollars.

Let

c

represent the cost of a croissant and

m

represent the cost of a muffin. The given situation translates to the system of equations

4c + 3m = 19.00

and

2c + 5m = 16.50

. Multiplying the second equation by

2

gives

4c + 10m = 33.00

. Subtracting

4c + 3m = 19.00

from

4c + 10m = 33.00

results in

7m = 14.00

, which gives

m = 2.00

. Substituting

m = 2.00

into

2c + 5(2.00) = 16.50

yields

2c + 10 = 16.50

, so

2c = 6.50

and

c = 3.25

. Thus, a single croissant costs $3.25 dollars.

Adım Adım Çözüm

1

Set up a system of linear equations

4c + 3m = 19.00

and

2c + 5m = 16.50

Translate the contents and costs of Basket X and Basket Y into algebraic equations where

c

is the price of a croissant and

m

is the price of a muffin.

2

Eliminate variable c

4c + 10m = 33.00

, then subtracting

4c + 3m = 19.00

yields

7m = 14.00

, so

m = 2.00

Multiplying the second equation by 2 aligns the coefficients of

c

, allowing elimination by subtraction.

3

Solve for variable c

2c + 5(2.00) = 16.50 \implies 2c = 6.50 \implies c = 3.25

Substitute the value found for

m

back into one of the original linear equations to calculate the cost of a croissant.

Anahtar Kavram

Solving Systems of Linear Equations via Elimination

Alternatif Yöntem

Express

c

in terms of

m

using the second equation:

c = 8.25 - 2.5m

. Substitute this expression into the first equation:

4(8.25 - 2.5m) + 3m = 19.00 \implies 33 - 10m + 3m = 19.00 \implies -7m = -14.00 \implies m = 2.00

. Finally, calculate

c = 8.25 - 2.5(2.00) = 3.25

.

Tahmini Süre:1m 30s

Soru 28Soru →

Consider the system of linear equations in two variables

x

and

y

shown below, where

k

is a constant:

\begin{aligned} 2x + ky &= 10 \\ 3x - 6y &= 15 \end{aligned}

Which of the following statements must be true? Select all such statements.

Geçerli olan tümünü seçin

If

k = -4

, the system has infinitely many solutions.

If

k = 0

, the system has a unique solution

(x, y) = (5, 0)

.

If

k = 4

, the system has no solution.

If

k = 2

, the solution to the system satisfies

x + y = 7

.

There is no real value of

k

for which the system has a solution with

y > 0

.

Cevabı ve açıklamayı göster

Cevap: If $k = -4$ , the system has infinitely many solutions.; If $k = 0$ , the system has a unique solution $(x, y) = (5, 0)$ .

Cevap

The correct statements are the ones stating that if

k = -4

, the system has infinitely many solutions, and if

k = 0

, the system has a unique solution

(x, y) = (5, 0)

.

Analyzing the simplified relation

(k + 4)y = 0

demonstrates that setting

k = -4

makes the equation identity

0y = 0

, yielding infinitely many solutions. For any other value of

k

, including

k = 0

,

y

must equal

0

, which gives

x = 5

, establishing a unique solution at

(5, 0)

.

Adım Adım Çözüm

1

Simplify the second equation to express

x

in terms of

y

.

3x - 6y = 15 \implies x - 2y = 5 \implies x = 2y + 5

.

Expressing

x

explicitly allows direct substitution into the first linear equation.

2

Substitute

x = 2y + 5

into the first equation

2x + ky = 10

.

2(2y + 5) + ky = 10 \implies 4y + 10 + ky = 10 \implies (k + 4)y = 0

.

This reduces the 2x2 system to a single linear equation in

y

parameterized by

k

.

3

Analyze the conditions for

y

based on the parameter

k

.

If

k = -4

, the equation becomes

0y = 0

, which is true for all real

y

(infinitely many solutions). If

k \neq -4

, then

y = 0

and

x = 5

(a unique solution).

Determines system consistency and solution multiplicity across all values of

k

.

Anahtar Kavram

Parametric Analysis of 2x2 Linear Systems

Soru 29Soru →

A research laboratory operates two types of automated centrifuges, Model X and Model Y. A single operating cycle of Model X processes $40$ biological samples and consumes $10$ kilowatt-hours (kWh) of electricity. A single operating cycle of Model Y processes $25$ biological samples and consumes $15$ kWh of electricity. On a given day, the laboratory processed a total of $775$ biological samples and consumed $325$ kWh of electricity using only these two models. What is the total number of operating cycles completed by Model X and Model Y combined on that day?

10

15

25

30

35

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

The total number of combined operating cycles completed by Model X and Model Y is 25.

The correct answer is 25. Setting up the linear system

40x + 25y = 775

and

10x + 15y = 325

and solving via elimination gives

x = 10

cycles for Model X and

y = 15

cycles for Model Y. Adding these together gives

10 + 15 = 25

total cycles.

Adım Adım Çözüm

1

Define variables and set up the system of linear equations based on total samples and total electricity consumption.

Let

x

be the number of cycles for Model X and

y

be the number of cycles for Model Y.
System of equations:
1)

40x + 25y = 775

(Sample constraint)
2)

10x + 15y = 325

(Energy constraint)

Modeling the word problem as two linear equations in two variables allows for algebraic elimination.

2

Simplify both equations by dividing by their greatest common factors.

Divide Equation 1 by 5:

8x + 5y = 155

Divide Equation 2 by 5:

2x + 3y = 65

Simplifying coefficients reduces computational complexity and minimizes arithmetic errors.

3

Eliminate variable

x

by multiplying the simplified second equation by 4 and subtracting the simplified first equation.

4 \times (2x + 3y = 65) \Rightarrow 8x + 12y = 260

Subtract

(8x + 5y = 155)

:

(8x - 8x) + (12y - 5y) = 260 - 155

7y = 105 \Rightarrow y = 15

Equalizing the coefficients of

x

enables solving directly for

y

.

4

Substitute

y = 15

back into

2x + 3y = 65

to find

x

.

2x + 3(15) = 65 \Rightarrow 2x + 45 = 65 \Rightarrow 2x = 20 \Rightarrow x = 10

Evaluating

x

completes the solution for individual cycle counts.

5

Calculate the requested combined total

x + y

.

x + y = 10 + 15 = 25

The question specifically requests the sum of operating cycles of both models combined.

Anahtar Kavram

Systems of Linear Equations in Two Variables

Tahmini Süre:1m 45s

Soru 30Soru →

A community theater sold a total of $250$ tickets for a performance, consisting of adult tickets for $\$15$ each and student tickets for $\$10$ each. The total revenue collected from ticket sales was $\$3,100$ . Let $a$ represent the number of adult tickets sold and $s$ represent the number of student tickets sold. Which of the following statements must be true? Select all such statements.

Geçerli olan tümünü seçin

The number of student tickets sold was

130

.

The number of adult tickets sold was

130

.

The total revenue generated from adult ticket sales was

\$1,800

.

The ratio of adult tickets sold to student tickets sold was

13

to

12

.

More adult tickets were sold than student tickets.

Cevabı ve açıklamayı göster

Cevap: The number of student tickets sold was $130$ .; The total revenue generated from adult ticket sales was $\$1,800$ .

Cevap

The correct statements are that the number of student tickets sold was 130, and the total revenue generated from adult ticket sales was $1,800.

The system of equations

a + s = 250

and

15a + 10s = 3100

uniquely yields

a = 120

adult tickets and

s = 130

student tickets. Therefore, the statement that 130 student tickets were sold is true. Furthermore, multiplying 120 adult tickets by their price of

15 per ticket gives a total adult ticket revenue of

1,800, making that statement true as well.

Adım Adım Çözüm

1

Set up a system of two linear equations in terms of

a

and

s

.

Equation 1 (total tickets):

a + s = 250

; Equation 2 (total revenue):

15a + 10s = 3100

.

The sum of the ticket quantities equals total tickets, and the sum of monetary contributions equals total revenue.

2

Express

s

in terms of

a

using Equation 1 and substitute into Equation 2.

s = 250 - a \implies 15a + 10(250 - a) = 3100 \implies 15a + 2500 - 10a = 3100

.

Using substitution eliminates variable

s

to solve for

a

.

3

Solve for

a

and then find

s

.

5a = 600 \implies a = 120

;

s = 250 - 120 = 130

.

Dividing

600

by

5

gives

a = 120

, and subtracting

120

from

250

gives

s = 130

.

4

Evaluate the statements against the values

a = 120

and

s = 130

.

s = 130

is true. Revenue from adult tickets is

120 \times 15 = \$1,800

, which is also true.

Comparing calculated values directly confirms which options state accurate quantitative properties.

Anahtar Kavram

Solving 2x2 Systems of Linear Equations using Substitution or Elimination in Word Problems

Soru 31Soru →

Consider the system of linear equations in two variables

x

and

y

shown below:

\begin{aligned} 3x + 2y &= 16 \\ x - 4y &= -4 \end{aligned}

If

(x, y)

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

\frac{y}{x}

?

\frac{1}{2}

2

4

\frac{2}{5}

6

Cevabı ve açıklamayı göster

Cevap: $\frac{1}{2}$

Cevap

The value of

\frac{y}{x}

is

\frac{1}{2}

.

Solving the system by substitution gives

x = 4

and

y = 2

. Dividing

y

by

x

gives

\frac{2}{4} = \frac{1}{2}

, which is the target ratio.

Adım Adım Çözüm

1

Isolate

x

in the second equation.

x = 4y - 4

Expressing

x

in terms of

y

allows for substitution into the first equation.

2

Substitute

x = 4y - 4

into the first equation.

3(4y - 4) + 2y = 16 \implies 12y - 12 + 2y = 16 \implies 14y = 28 \implies y = 2

Solving the single-variable equation determines the value of

y

.

3

Substitute

y = 2

back into the isolated expression for

x

.

x = 4(2) - 4 = 4

Determines the corresponding value of

x

.

4

Compute the required ratio

\frac{y}{x}

.

\frac{y}{x} = \frac{2}{4} = \frac{1}{2}

Evaluates the final expression requested by the prompt.

Anahtar Kavram

Solving systems of linear equations using substitution or elimination to evaluate a combined expression.

Soru 32Soru →

A commercial print shop operates two types of high-speed printers, Printer M and Printer N. Operating simultaneously for $5$ hours, $2$ units of Printer M and $3$ units of Printer N print a total of $5,500$ pages. Operating simultaneously for $4$ hours, $5$ units of Printer M and $2$ units of Printer N print a total of $6,600$ pages. What is the hourly page output of a single Printer M?

Cevabı ve açıklamayı göster

Cevap: 250

Cevap

The hourly page output of a single Printer M is 250 pages per hour.

Dividing each total page output by the corresponding number of hours produces the simplified linear system:

2m + 3n = 1100

and

5m + 2n = 1650

, where

m

and

n

are the hourly rates of Printer M and Printer N. Multiplying the first equation by

2

gives

4m + 6n = 2200

, and multiplying the second equation by

3

gives

15m + 6n = 4950

. Subtracting the two equations eliminates

n

, giving

11m = 2750

, which simplifies to

m = 250

pages per hour.

Adım Adım Çözüm

1

Define variables and write initial algebraic equations based on time and rate.

Let

m

represent the hourly page output of Printer M and

n

represent the hourly page output of Printer N. The total outputs give

5(2m + 3n) = 5500

and

4(5m + 2n) = 6600

.

Total page output equals total operating time multiplied by the combined hourly output rate.

2

Simplify the system by dividing each equation by its respective number of hours.

First equation:

2m + 3n = 1100

. Second equation:

5m + 2n = 1650

.

Simplifying yields a standard system of linear equations representing the combined hourly rate.

3

Eliminate variable

n

to solve for

m

.

Multiply

2m + 3n = 1100

by

2

to get

4m + 6n = 2200

. Multiply

5m + 2n = 1650

by

3

to get

15m + 6n = 4950

. Subtracting the first modified equation from the second yields

11m = 2750

, so

m = 250

.

Equating the coefficients of

n

allows direct elimination of

n

when subtracting the equations.

Anahtar Kavram

Solving a 2x2 system of linear equations using the method of elimination

Tahmini Süre:1m 30s

Systems of Linear Equations

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

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Sık Yapılan Hatalar

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Sık Yapılan Hatalar

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Cevap

Adım Adım Çözüm

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Cevap

Adım Adım Çözüm

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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

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar