Simultaneous Equations and Systems Alıştırma Soruları — GMAT

Soru 1Soru →

An investor allocates a sum of money between two financial products, Account A and Account B. Account A yields an annual simple interest rate of $6\%$ , while Account B yields an annual simple interest rate of $10\%$ . After one year, the total interest earned from both accounts combined is $\$2,600$ . If the amount invested in Account B is exactly twice the amount invested in Account A, select the value that corresponds to the amount invested in Account A and the value that corresponds to the amount invested in Account B.

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Cevap

The amount invested in Account A is

\$10,000

and the amount invested in Account B is

\$20,000

.

Solving the simultaneous system

0.06x + 0.10y = 2,600

and

y = 2x

yields

0.26x = 2,600

, resulting in

x = \$10,000

for Account A and

y = \$20,000

for Account B.

Adım Adım Çözüm

1

Define variables and formulate the system of equations based on interest earned and relative investment ratio.

Let

x

be the amount in Account A and

y

be the amount in Account B. The problem statements give: (1)

0.06x + 0.10y = 2,600

and (2)

y = 2x

.

Translating word constraints into algebraic representations sets up a two-variable system.

2

Substitute equation (2) into equation (1) to solve for

x

.

0.06x + 0.10(2x) = 2,600 \implies 0.06x + 0.20x = 2,600 \implies 0.26x = 2,600 \implies x = 10,000

.

Substitution simplifies the simultaneous system into a single linear equation in terms of

x

.

3

Calculate

y

using the relationship

y = 2x

.

y = 2(10,000) = 20,000

.

Multiplying the value of

x

by 2 gives the value for Account B.

Anahtar Kavram

Simultaneous Linear Equations by Substitution

Soru 2Soru →

A logistics company packages cargo into two types of containers: Standard containers and Heavy-duty containers.

A shipment of $3$ Standard containers and $2$ Heavy-duty containers has a total mass of $1,400$ kilograms. A second shipment of $2$ Standard containers and $4$ Heavy-duty containers has a total mass of $2,000$ kilograms.

Based on this information, select the value in kilograms for the mass of a single Standard container and for the mass of a single Heavy-duty container so that the selections are consistent with the given information.

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Cevap

The mass of a single Standard container is 200 kg, and the mass of a single Heavy-duty container is 400 kg.

Setting up the simultaneous linear equations

3s + 2h = 1400

and

2s + 4h = 2000

yields

s = 200

and

h = 400

. Thus, selecting 200 for the Standard container column and 400 for the Heavy-duty container column correctly matches the calculated values.

Adım Adım Çözüm

1

Define variables and set up the system of equations from the problem stem.

Let

s

be the mass of a Standard container in kg and

h

be the mass of a Heavy-duty container in kg. The given shipments yield:
Equation 1:

3s + 2h = 1400

Equation 2:

2s + 4h = 2000

Translate word problem conditions into algebraic equations.

2

Simplify Equation 2 by dividing all terms by 2.

s + 2h = 1000 \implies s = 1000 - 2h

Isolate variable

s

to use the substitution method.

3

Substitute

s = 1000 - 2h

into Equation 1 and solve for

h

.

3(1000 - 2h) + 2h = 1400 \implies 3000 - 6h + 2h = 1400 \implies 3000 - 4h = 1400 \implies 4h = 1600 \implies h = 400

Determine the value for the mass of a Heavy-duty container.

4

Substitute

h = 400

back into

s = 1000 - 2h

to solve for

s

.

s = 1000 - 2(400) = 1000 - 800 = 200

Determine the value for the mass of a Standard container.

Anahtar Kavram

Solving two-variable linear systems using substitution or elimination in a Two-Part Analysis format

Soru 3Soru →

A theater sold a total of $100$ tickets for a performance, consisting of standard tickets ( $s$ ) at $\$25$ each and VIP tickets ( $v$ ) at $\$50$ each. The total revenue from ticket sales was $\$3,500$ . Match each ticket category to the correct number of tickets sold.

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Cevap

Standard Tickets matches to 60; VIP Tickets matches to 40.

The correct matches are Standard Tickets to 60 and VIP Tickets to 40 because solving the system

s + v = 100

and

25s + 50v = 3500

gives

s = 60

and

v = 40

.

Adım Adım Çözüm

1

Set up the simultaneous linear equations from the text.

s + v = 100

and

25s + 50v = 3500

The total count equation represents total tickets, and the price equation represents total revenue.

2

Divide the revenue equation by 25 to simplify terms.

s + 2v = 140

Simplifying the coefficients reduces calculation complexity.

3

Subtract the first equation (

s + v = 100

) from the simplified second equation (

s + 2v = 140

).

v = 40

Subtracting eliminates the variable

s

directly.

4

Substitute

v = 40

back into

s + v = 100

.

s = 60

Yields the value for the remaining variable

s

.

Anahtar Kavram

Solving simultaneous linear equations in two variables.

Soru 4Soru →

A chemist mixes $x$ liters of a $20\%$ acid solution with $y$ liters of a $50\%$ acid solution to produce $30$ liters of a $40\%$ acid solution. How many liters of the $20\%$ acid solution were used in the mixture?

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Cevap: 10

Cevap

10 liters of the 20% acid solution were used.

Solving the system of simultaneous linear equations

x + y = 30

and

0.20x + 0.50y = 12

yields

x = 10

liters.

Adım Adım Çözüm

1

Formulate the linear system of equations representing total volume and total pure acid content.

x + y = 30

and

0.20x + 0.50y = 12

.

The sum of component volumes equals total volume, and the sum of pure acid contents equals total pure acid.

2

Substitute

y = 30 - x

into the pure acid equation.

0.20x + 0.50(30 - x) = 12

Reduces the two-variable system to a single linear equation in terms of

x

.

3

Expand and solve for

x

.

15 - 0.30x = 12 \implies -0.30x = -3 \implies x = 10

Isolates the target variable

x

representing the volume of the 20% acid solution.

Anahtar Kavram

Simultaneous Linear Equations in Mixture Problems

Tahmini Süre:1m 0s

Soru 5Soru →

A manufacturing plant produces two types of solar panels: Model X and Model Y. Each Model X panel requires $2$ hours of assembly and $1$ hour of quality testing. Each Model Y panel requires $3$ hours of assembly and $2$ hours of quality testing. During a single week, the assembly line logged a total of $140$ hours and the quality testing line logged $80$ hours, with both departments operating at full capacity. How many Model X solar panels were produced during the week?

20

30

40

50

60

Cevabı ve açıklamayı göster

Cevap: $40$

Cevap

40

Model X solar panels were produced during the week.

Setting up the simultaneous linear equations for assembly time (

2x + 3y = 140

) and testing time (

x + 2y = 80

) and solving for

x

gives

x = 40

, which accurately satisfies both constraints.

Adım Adım Çözüm

1

Define variables and set up the system of linear equations based on resource constraints.

Let

x

be the number of Model X panels and

y

be the number of Model Y panels.
Assembly constraint:

2x + 3y = 140

Quality testing constraint:

x + 2y = 80

Each department's total logged hours must equal the sum of hours required by both panel types.

2

Express

x

in terms of

y

using the testing constraint equation.

x = 80 - 2y

Isolating

x

allows substitution into the assembly equation.

3

Substitute

x = 80 - 2y

into the assembly equation and solve for

y

.

2(80 - 2y) + 3y = 140 160 - 4y + 3y = 140 160 - y = 140 y = 20

Solving a single-variable equation determines the quantity of Model Y panels.

4

Substitute

y = 20

back to find

x

.

x = 80 - 2(20) = 40

Evaluating

x

determines the quantity of Model X panels requested in the question stem.

Anahtar Kavram

Solving simultaneous linear equations by substitution or elimination

Soru 6Soru →

A commercial airline manages its long-haul fleet using two aircraft configurations: Configuration P (Passenger-focused) and Configuration C (Cargo-focused).

On a standard transcontinental route:
- Operating $4$ Configuration P flights and $3$ Configuration C flights requires a total of $98,000$ gallons of fuel.
- Operating $2$ Configuration P flights and $5$ Configuration C flights requires a total of $112,000$ gallons of fuel.

Assuming fuel consumption per flight depends solely on the aircraft configuration, select the fuel consumption (in gallons) per flight for Configuration P and for Configuration C.

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Cevap

Configuration P fuel consumption per flight is 11,000 gallons; Configuration C fuel consumption per flight is 18,000 gallons.

Solving the linear system

4 f_P + 3 f_C = 98,000

and

2 f_P + 5 f_C = 112,000

gives

f_P = 11,000

gallons and

f_C = 18,000

gallons.

Adım Adım Çözüm

1

Formulate a system of simultaneous linear equations based on the problem statement.

Let

f_P

be fuel per Configuration P flight and

f_C

be fuel per Configuration C flight. Equation 1:

4 f_P + 3 f_C = 98,000

. Equation 2:

2 f_P + 5 f_C = 112,000

.

Translates the two total fuel conditions into explicit linear equations.

2

Eliminate variable

f_P

by multiplying Equation 2 by 2.

Multiplying Equation 2 by 2 yields

4 f_P + 10 f_C = 224,000

.

Equalizes the coefficients of

f_P

across both equations to enable cancellation.

3

Subtract Equation 1 from the modified equation to solve for

f_C

.

(4 f_P + 10 f_C) - (4 f_P + 3 f_C) = 224,000 - 98,000 \implies 7 f_C = 126,000 \implies f_C = 18,000

.

Subtracting cancels

f_P

, yielding the fuel consumption per Configuration C flight.

4

Substitute

f_C = 18,000

back into Equation 1 to find

f_P

.

4 f_P + 3(18,000) = 98,000 \implies 4 f_P + 54,000 = 98,000 \implies 4 f_P = 44,000 \implies f_P = 11,000

.

Determines the fuel consumption per Configuration P flight.

Anahtar Kavram

Solving simultaneous linear equations with two unknowns in a Two-Part Analysis format.

Soru 7Soru →

Two server clusters, Alpha and Beta, process data jobs of Type 1 and Type 2. Cluster Alpha processes $15$ Type 1 jobs and $25$ Type 2 jobs in a total time of $210$ minutes. Cluster Beta processes $25$ Type 1 jobs and $15$ Type 2 jobs in a total time of $190$ minutes. Assuming constant processing rates per job type across both clusters, how many total minutes will it take to process a workload consisting of $30$ Type 1 jobs and $10$ Type 2 jobs?

Cevabı ve açıklamayı göster

Cevap: 180

Cevap

The total processing time required is 180 minutes.

Let

x

be the processing time in minutes for a Type 1 job, and

y

be the processing time in minutes for a Type 2 job. Translating the given data gives the simultaneous equations

15x + 25y = 210

and

25x + 15y = 190

. Dividing both equations by

5

yields

3x + 5y = 42

and

5x + 3y = 38

. Adding these simplified equations yields

8x + 8y = 80

, so

x + y = 10

. Subtracting the first from the second gives

2x - 2y = -4

, so

x - y = -2

. Solving

x + y = 10

and

x - y = -2

gives

x = 4

and

y = 6

. The required workload duration is

30(4) + 10(6) = 120 + 60 = 180

minutes.

Adım Adım Çözüm

1

Set up a linear system of simultaneous equations based on the rates given.

15x + 25y = 210

and

25x + 15y = 190

, where

x

and

y

are the processing times per Type 1 and Type 2 job.

Establishing accurate algebraic expressions translates the real-world scenario into a solvable model.

2

Simplify the system by dividing through by common factors.

3x + 5y = 42

and

5x + 3y = 38

.

Simplification reduces arithmetic complexity and minimizes computational errors.

3

Solve for variables x and y using symmetric combinations (adding/subtracting equations).

x = 4

and

y = 6

.

Symmetric linear systems can be solved efficiently by taking the sum and difference of the equations.

4

Evaluate the target expression

30x + 10y

.

30(4) + 10(6) = 180

.

Multiplying individual job times by requested quantities yields the total workload time.

Anahtar Kavram

Simultaneous Linear Systems and Symmetric Reduction

Soru 8Soru →

A specialty coffee roaster produces two custom coffee blends, Blend Alpha and Blend Beta, using Arabica and Robusta beans. Each bag of Blend Alpha requires $300\text{ grams}$ of Arabica beans and $100\text{ grams}$ of Robusta beans. Each bag of Blend Beta requires $200\text{ grams}$ of Arabica beans and $200\text{ grams}$ of Robusta beans. On a given day, the roaster used a total of $14,000\text{ grams}$ of Arabica beans and $8,000\text{ grams}$ of Robusta beans to produce bags of these two blends, using all available inventory.

In the table below, select the number of bags of Blend Alpha produced and the number of bags of Blend Beta produced that are consistent with this information.

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Cevap

Bags of Blend Alpha produced = 30; Bags of Blend Beta produced = 25

By representing the number of bags of Blend Alpha as

x

and Blend Beta as

y

, the constraints form the system

3x + 2y = 140

and

x + 2y = 80

. Subtracting the equations yields

2x = 60

, so

x = 30

. Substituting

x = 30

gives

y = 25

. Thus, Blend Alpha produced is 30 and Blend Beta produced is 25.

Adım Adım Çözüm

1

Define variables for the unknowns.

Let

x

be the number of bags of Blend Alpha produced, and

y

be the number of bags of Blend Beta produced.

Setting up explicit variables allows formulation of linear algebraic equations.

2

Set up equations based on total bean usage.

For Arabica beans:

300x + 200y = 14,000

, which simplifies to

3x + 2y = 140

.
For Robusta beans:

100x + 200y = 8,000

, which simplifies to

x + 2y = 80

.

Translating the physical constraints into two linear equations in two variables.

3

Solve the system by subtracting the second equation from the first.

(3x + 2y) - (x + 2y) = 140 - 80 \implies 2x = 60 \implies x = 30

.

Subtracting eliminates

y

directly because both equations have identical

2y

terms.

4

Substitute

x = 30

into the simplified Robusta equation to find

y

.

30 + 2y = 80 \implies 2y = 50 \implies y = 25

.

Determines the remaining unknown variable.

Anahtar Kavram

Simultaneous Linear Equations in Two Variables

Soru 9Soru →

A microgrid power station utilizes two types of energy storage units: Lithium-ion batteries and Flow batteries. A primary grid configuration consisting of $4$ Lithium-ion batteries and $3$ Flow batteries provides a total continuous storage capacity of $140\text{ MWh}$ . An alternative configuration consisting of $2$ Lithium-ion batteries and $5$ Flow batteries provides a total continuous storage capacity of $126\text{ MWh}$ . Assuming that each battery of a given type contributes a constant storage capacity, what is the individual storage capacity, in MWh, of one Lithium-ion battery and one Flow battery, respectively?

23\text{ MWh}

for a Lithium-ion battery and

16\text{ MWh}

for a Flow battery

16\text{ MWh}

for a Lithium-ion battery and

23\text{ MWh}

for a Flow battery

20\text{ MWh}

for a Lithium-ion battery and

18\text{ MWh}

for a Flow battery

25\text{ MWh}

for a Lithium-ion battery and

14\text{ MWh}

for a Flow battery

21\text{ MWh}

for a Lithium-ion battery and

19\text{ MWh}

for a Flow battery

Cevabı ve açıklamayı göster

Cevap: $23\text{ MWh}$ for a Lithium-ion battery and $16\text{ MWh}$ for a Flow battery

Cevap

The storage capacity is

23\text{ MWh}

for a Lithium-ion battery and

16\text{ MWh}

for a Flow battery.

The system of equations

4L + 3F = 140

and

2L + 5F = 126

uniquely determines

L = 23\text{ MWh}

and

F = 16\text{ MWh}

. The option specifying

23\text{ MWh}

for Lithium-ion and

16\text{ MWh}

for Flow correctly matches the values of

L

and

F

in their respective order.

Adım Adım Çözüm

1

Set up a system of two linear equations in two variables.

Let

L

be the capacity of one Lithium-ion battery and

F

be the capacity of one Flow battery:
Equation 1:

4L + 3F = 140

Equation 2:

2L + 5F = 126

Translate the physical battery configurations and total capacities into algebraic equations.

2

Eliminate variable

L

by multiplying Equation 2 by

2

.

Equation 2 scaled:

2(2L + 5F) = 2(126) \Rightarrow 4L + 10F = 252

.

Align the coefficients of

L

to subtract Equation 1 from the scaled Equation 2.

3

Subtract Equation 1 from the scaled equation to solve for

F

.

(4L + 10F) - (4L + 3F) = 252 - 140 \Rightarrow 7F = 112 \Rightarrow F = 16

.

Eliminating

L

isolate the single-variable linear equation for

F

.

4

Substitute

F = 16

back into Equation 2 to solve for

L

.

2L + 5(16) = 126 \Rightarrow 2L + 80 = 126 \Rightarrow 2L = 46 \Rightarrow L = 23

.

Determine the value of

L

once

F

is known.

Anahtar Kavram

Simultaneous Linear Equations (System of 2 Equations with 2 Variables)

Tahmini Süre:2m 0s

Soru 10Soru →

A biotechnology laboratory operates two automated synthesis workflows, System X and System Y, to manufacture two compounds, Alpha ( $A$ ) and Beta ( $B$ ). System X operates for $x$ hours, producing $A$ at a rate of $3\text{ mg/hr}$ and $B$ at a rate of $5\text{ mg/hr}$ . System Y operates for $y$ hours, producing $A$ at a rate of $15\text{ mg/hr}$ and $B$ at a rate of $8\text{ mg/hr}$ . The combined mass of compounds $A$ and $B$ produced across both workflows is exactly $340\text{ mg}$ . Furthermore, the system's operating efficiency requires that the ratio of operating hours $\frac{x}{y}$ equals the ratio of total mass produced $\frac{B}{A}$ . What is the total operating time, $x + y$ , in hours?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The total operating time of both workflows is 20 hours.

Representing the total production of compounds A and B as functions of hours x and y yields the linear total mass equation 8x + 23y = 340 and the ratio relation x/y = B/A, which simplifies to the homogeneous quadratic 3x^2 + 10xy - 8y^2 = 0. Factoring gives (3x - 2y)(x + 4y) = 0. Since operating hours are strictly positive, x = (2/3)y. Substituting this into the linear equation yields (85/3)y = 340, giving y = 12 hours and x = 8 hours. The total operating time is 8 + 12 = 20 hours.

Adım Adım Çözüm

1

Set up algebraic expressions for total mass of each compound produced.

Compound A mass =

3x + 15y

mg; Compound B mass =

5x + 8y

mg.

Mass equals production rate multiplied by total operating hours.

2

Formulate the total combined production equation.

8x + 23y = 340

.

The sum of all Compound A and Compound B produced across both systems is given as 340 mg.

3

Set up and simplify the ratio equality

\frac{x}{y} = \frac{B}{A}

.

3x^2 + 10xy - 8y^2 = 0

.

Cross-multiplying

\frac{x}{y} = \frac{5x + 8y}{3x + 15y}

gives

x(3x + 15y) = y(5x + 8y)

.

4

Factor the homogeneous quadratic equation.

(3x - 2y)(x + 4y) = 0 \implies x = \frac{2}{3}y

.

Because operating hours

x

and

y

must be positive,

x + 4y > 0

, forcing

3x - 2y = 0

.

5

Substitute

x = \frac{2}{3}y

into

8x + 23y = 340

.

y = 12

hours and

x = 8

hours.

Solving

\frac{85}{3}y = 340

gives

y = 12

, and substituting back into

x = \frac{2}{3}y

gives

x = 8

.

6

Compute total operating time

x + y

.

8 + 12 = 20

hours.

The question asks for the combined total operating time of both systems.

Anahtar Kavram

Solving non-linear systems of simultaneous equations involving ratio constraints and quadratic factorization

Tahmini Süre:2m 30s

Soru 11Soru →

An event management firm hires two types of security personnel: Senior Officers and Junior Officers. Contract A requires 4 Senior Officers and 6 Junior Officers for a total daily cost of $2,180. Contract B requires 7 Senior Officers and 3 Junior Officers for a total daily cost of$ 2,690. Based on this system of equations, match each quantity on the left with its correct dollar value on the right.

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Cevabı ve açıklamayı göster

Cevap

Senior Officer rate =

320; Junior Officer rate =

150; Difference in rates =

170; Combined rate =

470.

Solving the linear system yields

s = 320

for Senior Officers and

j = 150

for Junior Officers. The difference between their rates is

320 - 150 = 170, and their combined daily rate is

320 + 150 = 470.

Adım Adım Çözüm

1

Formulate a system of linear equations from the given contract conditions.

Let

s

be the daily rate of a Senior Officer and

j

be the daily rate of a Junior Officer. Equation 1:

4s + 6j = 2180

. Equation 2:

7s + 3j = 2690

.

Translating context into algebraic form allows systematic elimination.

2

Eliminate the variable

j

to solve for

s

.

Divide Equation 1 by 2:

2s + 3j = 1090

. Subtract this from Equation 2 (

7s + 3j = 2690

):

(7s + 3j) - (2s + 3j) = 2690 - 1090 \implies 5s = 1600 \implies s = 320

.

Matching coefficients of

j

isolates variable

s

cleanly.

3

Substitute

s = 320

into

2s + 3j = 1090

to calculate

j

.

2(320) + 3j = 1090 \implies 640 + 3j = 1090 \implies 3j = 450 \implies j = 150

.

Determines the second variable value in the system.

4

Compute the difference (

s - j

) and sum (

s + j

) of the two rates.

Difference:

320 - 150 = 170

. Combined rate:

320 + 150 = 470

.

Provides the exact target values for all items in the matching task.

Anahtar Kavram

Solving simultaneous linear equations by variable elimination and substitution

Tahmini Süre:2m 0s

Soru 12Soru →

A chemical processing facility operates two purification units, Unit A and Unit B.

Unit A processes a liquid feed stream consisting of $x$ liters of Compound X and $y$ liters of Compound Y per hour. During an 8-hour operational cycle, Unit A processes a total of $360$ liters of feed stream.

Unit B processes Compound X at twice the hourly rate of Unit A ( $2x$ liters per hour) and Compound Y at three times the hourly rate of Unit A ( $3y$ liters per hour). During a 5-hour operational cycle, Unit B processes a total of $510$ liters of feed stream.

Based on the information provided, select for Unit A Compound X Rate the number of liters of Compound X processed by Unit A per hour, and select for Unit A Compound Y Rate the number of liters of Compound Y processed by Unit A per hour. Make exactly one selection in each column.

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Cevabı ve açıklamayı göster

Cevap

The hourly processing rate of Compound X for Unit A is 33 liters per hour, and the hourly processing rate of Compound Y for Unit A is 12 liters per hour.

Formulating equations from the operational cycles yields

8(x + y) = 360 \implies x + y = 45

and

5(2x + 3y) = 510 \implies 2x + 3y = 102

. Expressing

y

as

45 - x

and substituting into the second equation gives

2x + 3(45 - x) = 102

, which simplifies to

-x + 135 = 102 \implies x = 33

. Substituting

x = 33

back gives

y = 12

. Therefore, Unit A processes 33 liters of Compound X per hour and 12 liters of Compound Y per hour.

Adım Adım Çözüm

1

Set up the first linear equation using the total volume processed by Unit A during its 8-hour operational cycle.

8(x + y) = 360 \implies x + y = 45

Unit A processes

x + y

liters per hour, so over 8 hours it processes

8(x + y) = 360

liters.

2

Set up the second linear equation using the total volume processed by Unit B during its 5-hour operational cycle.

5(2x + 3y) = 510 \implies 2x + 3y = 102

Unit B processes

2x + 3y

liters per hour, so over 5 hours it processes

5(2x + 3y) = 510

liters.

3

Solve the system of two simultaneous linear equations for

x

and

y

.

x = 33

and

y = 12

From equation 1,

y = 45 - x

. Substituting into equation 2 gives

2x + 3(45 - x) = 102 \implies 2x + 135 - 3x = 102 \implies -x = -33 \implies x = 33

. Thus

y = 45 - 33 = 12

.

Anahtar Kavram

Solving systems of simultaneous linear equations by substitution or elimination in a Two-Part Analysis framework.

Soru 13Soru →

An agricultural cooperative prepares custom soil treatments using two standard nutrient formulations: Formulation X and Formulation Y. Each bag of Formulation X contains $4\text{ kg}$ of Nitrogen and $2\text{ kg}$ of Phosphate. Each bag of Formulation Y contains $2\text{ kg}$ of Nitrogen and $5\text{ kg}$ of Phosphate. A commercial farm requires a total of exactly $140\text{ kg}$ of Nitrogen and $150\text{ kg}$ of Phosphate for its crops. Which of the following pairs $(X, Y)$ represents the exact number of bags of Formulation X and Formulation Y, respectively, needed to fulfill these requirements?

25 bags of Formulation X and 20 bags of Formulation Y

20 bags of Formulation X and 25 bags of Formulation Y

30 bags of Formulation X and 10 bags of Formulation Y

15 bags of Formulation X and 30 bags of Formulation Y

35 bags of Formulation X and 0 bags of Formulation Y

Cevabı ve açıklamayı göster

Cevap: 25 bags of Formulation X and 20 bags of Formulation Y

Cevap

25 bags of Formulation X and 20 bags of Formulation Y

The correct answer is 25 bags of Formulation X and 20 bags of Formulation Y because substituting

x = 25

and

y = 20

into the system of equations satisfies both constraints:

4(25) + 2(20) = 140\text{ kg}

Nitrogen and

2(25) + 5(20) = 150\text{ kg}

Phosphate.

Adım Adım Çözüm

1

Set up the simultaneous equations for Nitrogen and Phosphate requirements.

Let

x

be the number of bags of Formulation X and

y

be the number of bags of Formulation Y.
Nitrogen Equation:

4x + 2y = 140

Phosphate Equation:

2x + 5y = 150

Each bag contributes a specific amount of nutrient toward the target total.

2

Simplify the first equation and express

y

in terms of

x

.

2x + y = 70 \implies y = 70 - 2x

Dividing the Nitrogen equation by 2 isolates

y

conveniently for substitution.

3

Substitute

y = 70 - 2x

into the Phosphate equation.

2x + 5(70 - 2x) = 150 \implies 2x + 350 - 10x = 150 \implies -8x = -200 \implies x = 25

Solving the single-variable linear equation yields the required bags of Formulation X.

4

Calculate

y

using the simplified expression.

y = 70 - 2(25) = 20

Determines the required bags of Formulation Y.

Anahtar Kavram

Solving Systems of Simultaneous Linear Equations

Soru 14Soru →

A logistics terminal at a seaport utilizes two types of automated cranes, Model X and Model Y, to unload container ships. During the morning shift, a fleet of 4 Model X cranes and 5 Model Y cranes unloaded a total of 330 containers. During the afternoon shift, a fleet of 6 Model X cranes and 2 Model Y cranes unloaded the same total of 330 containers. Assuming all cranes of a given model operate at a constant unloading rate per shift, select the value in the table that represents the number of containers unloaded per shift by a single Model X crane and the value that represents the number of containers unloaded per shift by a single Model Y crane.

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Cevabı ve açıklamayı göster

Cevap

Model X Crane unloads 45 containers per shift, and Model Y Crane unloads 30 containers per shift.

Setting up the system

4x + 5y = 330

and

6x + 2y = 330

yields

x = 45

for Model X and

y = 30

for Model Y. Substituting these values into both equations confirms accuracy:

4(45) + 5(30) = 180 + 150 = 330

and

6(45) + 2(30) = 270 + 60 = 330

.

Adım Adım Çözüm

1

Define variables and construct the system of linear equations from the given problem statement.

Let

x

be the number of containers unloaded per shift by a Model X crane, and

y

be the number of containers unloaded per shift by a Model Y crane. Morning shift equation:

4x + 5y = 330

. Afternoon shift equation:

6x + 2y = 330

.

Translating word problem constraints into explicit algebraic equations is required to solve for both unknowns.

2

Express

y

in terms of

x

using the afternoon shift equation.

2y = 330 - 6x \implies y = 165 - 3x

.

Using substitution allows reducing the two-variable system to a single equation in one variable.

3

Substitute the expression for

y

into the morning shift equation and solve for

x

.

4x + 5(165 - 3x) = 330 \implies 4x + 825 - 15x = 330 \implies -11x = -495 \implies x = 45

.

Solving the single-variable equation determines the exact unloading capacity of a Model X crane.

4

Substitute

x = 45

back into the expression for

y

to find the capacity of a Model Y crane.

y = 165 - 3(45) = 165 - 135 = 30

.

Substituting the value of

x

gives the specific unloading rate for Model Y.

Anahtar Kavram

Simultaneous Linear Equations in Two Variables

Soru 15Soru →

A telecommunications satellite utilizes two distinct signal amplifier modules, Module A and Module B, to process high-frequency and low-frequency data streams. Each Module A processes $18\text{ Gbps}$ of high-frequency data and $12\text{ Gbps}$ of low-frequency data, consuming $150\text{ watts}$ of operational power. Each Module B processes $30\text{ Gbps}$ of high-frequency data and $6\text{ Gbps}$ of low-frequency data, consuming $180\text{ watts}$ of operational power. During a peak transmission test, the active modules processed a combined total of $354\text{ Gbps}$ of high-frequency data and $138\text{ Gbps}$ of low-frequency data. What is the total operational power, in watts, consumed by all the active modules during this test?

Cevabı ve açıklamayı göster

Cevap: 2460

Cevap

The total operational power consumed by all active modules during the test is 2,460 watts.

By setting up a system of simultaneous equations representing the total high-frequency (

18a + 30b = 354

) and low-frequency (

12a + 6b = 138

) bandwidths, solving yields

a = 8

active Module A units and

b = 7

active Module B units. Substituting these module counts into the total power equation

150(8) + 180(7)

gives exactly 2,460 watts.

Adım Adım Çözüm

1

Define variables for the unknown counts of modules.

Let

a

represent the number of Module A units and

b

represent the number of Module B units.

Establishing explicit variables allows representing the data throughput and power constraints as algebraic equations.

2

Formulate equations for high-frequency and low-frequency data streams based on total throughput.

High-frequency equation:

18a + 30b = 354

. Low-frequency equation:

12a + 6b = 138

.

Summing the data contributions from each module type gives the total processed bandwidth for each frequency band.

3

Solve the system of linear equations.

Simplifying the low-frequency equation yields

2a + b = 23 \implies b = 23 - 2a

. Substituting into the simplified high-frequency equation (

3a + 5b = 59

) gives

3a + 5(23 - 2a) = 59 \implies -7a = -56 \implies a = 8

. Substituting

a = 8

back into

b = 23 - 2(8)

gives

b = 7

.

Determining the exact number of active modules of each type is required before calculating overall energy usage.

4

Calculate the total power consumption.

Total Power =

150a + 180b = 150(8) + 180(7) = 1,200 + 1,260 = 2,460\text{ watts}

.

Multiplying the module counts by their respective wattage ratings yields the total operational power.

Anahtar Kavram

Solving systems of simultaneous linear equations with two unknowns and applying the solution to calculate a weighted linear total.

Tahmini Süre:2m 30s

Soru 16Soru →

An urban courier service dispatches two types of electric cargo bikes, Cargo-X and Cargo-Y, for package deliveries. On Monday, a fleet of 3 Cargo-X bikes and 4 Cargo-Y bikes completed a total of 38 package deliveries. On Tuesday, a fleet of 5 Cargo-X bikes and 2 Cargo-Y bikes completed a total of 40 package deliveries. Assuming each bike of a given type completes a constant number of deliveries per day, how many total package deliveries will a combined fleet of 4 Cargo-X bikes and 3 Cargo-Y bikes complete in one day?

Cevabı ve açıklamayı göster

Cevap: 39

Cevap

A combined fleet of 4 Cargo-X bikes and 3 Cargo-Y bikes will complete 39 package deliveries in one day.

Solving the linear system

3x + 4y = 38

and

5x + 2y = 40

gives

x = 6

deliveries per Cargo-X bike and

y = 5

deliveries per Cargo-Y bike. Substituting these rates into

4x + 3y

yields

4(6) + 3(5) = 39

total deliveries.

Adım Adım Çözüm

1

Formulate a system of linear equations representing the delivery totals.

System established:

3x + 4y = 38

and

5x + 2y = 40

, where

x

and

y

represent the daily delivery capacities of Cargo-X and Cargo-Y bikes.

Translates contextual problem statements into quantitative algebraic equations.

2

Solve the system using elimination or substitution.

x = 6

deliveries per Cargo-X bike and

y = 5

deliveries per Cargo-Y bike.

Determines the exact rate per vehicle type.

3

Evaluate the target linear combination

4x + 3y

.

4(6) + 3(5) = 39

.

Computes the total capacity for the requested fleet combination.

Anahtar Kavram

Solving systems of simultaneous linear equations in two variables

Soru 17Soru →

A luxury boutique hotel offers two types of guest accommodations: Deluxe Suites and Executive Suites. On a busy Saturday, housekeeping spends 45 minutes cleaning each Deluxe Suite and 75 minutes cleaning each Executive Suite, accumulating a total of 38 hours of cleaning time. On the same day, laundry services supply 4 sets of premium towels for each Deluxe Suite and 7 sets of premium towels for each Executive Suite, delivering a total of 208 towel sets. Which of the following correctly pairs the number of Deluxe Suites cleaned with the number of Executive Suites cleaned on that day?

Deluxe Suites = 24; Executive Suites = 16

Deluxe Suites = 16; Executive Suites = 24

Deluxe Suites = 28; Executive Suites = 12

Deluxe Suites = 20; Executive Suites = 20

Deluxe Suites = 32; Executive Suites = 8

Cevabı ve açıklamayı göster

Cevap: Deluxe Suites = 24; Executive Suites = 16

Cevap

Deluxe Suites = 24; Executive Suites = 16

Solving the simultaneous system of linear equations

3d + 5e = 152

and

4d + 7e = 208

yields

d = 24

Deluxe Suites and

e = 16

Executive Suites. The choice indicating 24 Deluxe Suites and 16 Executive Suites correctly matches these calculated values to their respective variables.

Adım Adım Çözüm

1

Define variables and convert cleaning time to minutes.

Let

d

be the number of Deluxe Suites and

e

be the number of Executive Suites. Total cleaning time =

38 \text{ hours} \times 60 \text{ minutes/hour} = 2280 \text{ minutes}

.

Units must be consistent across variables before setting up algebraic equations.

2

Formulate the linear equations system.

Equation (1) [Cleaning time]:

45d + 75e = 2280

Equation (2) [Towel sets]:

4d + 7e = 208

Translates the real-world operational constraints into simultaneous linear algebraic equations.

3

Simplify Equation (1) by dividing all terms by 15.

3d + 5e = 152

Simplifying coefficients makes simultaneous elimination computationally easier.

4

Solve the system using elimination.

Multiply

3d + 5e = 152

by

4

:

12d + 20e = 608

.
Multiply

4d + 7e = 208

by

3

:

12d + 21e = 624

.
Subtracting the first equation from the second yields:

(12d + 21e) - (12d + 20e) = 624 - 608 \Rightarrow e = 16

.

Eliminates

d

to isolate variable

e

.

5

Substitute

e = 16

back into the simplified Equation (1) to solve for

d

.

3d + 5(16) = 152 \Rightarrow 3d + 80 = 152 \Rightarrow 3d = 72 \Rightarrow d = 24

.

Determines the value of the second variable

d

.

Anahtar Kavram

Setting up and solving a system of two linear equations in two variables

Tahmini Süre:2m 0s

Soru 18Soru →

An agricultural research station operates two types of automated irrigation pumps: Type Alpha and Type Beta. Operating 4 Type Alpha pumps and 3 Type Beta pumps simultaneously for 5 hours consumes a total of $215\text{ kilowatt-hours (kWh)}$ of electricity. Operating 2 Type Alpha pumps and 5 Type Beta pumps simultaneously for 4 hours consumes a total of $156\text{ kWh}$ of electricity. Based on the information provided, match each given pump metric on the left with its correct hourly electricity consumption value on the right.

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Cevabı ve açıklamayı göster

Cevap

The hourly consumption rate of one Type Alpha pump is

7\text{ kWh per hour}

, one Type Beta pump is

5\text{ kWh per hour}

, and their combined hourly rate is

12\text{ kWh per hour}

.

Solving the system of simultaneous linear equations

4a + 3b = 43

and

2a + 5b = 39

yields

a = 7

for Type Alpha pumps and

b = 5

for Type Beta pumps. The sum

a + b

equals

12

.

Adım Adım Çözüm

1

Set up equations for the total hourly consumption from the given scenario data.

Let

a

be the hourly consumption of Type Alpha in kWh and

b

be the hourly consumption of Type Beta in kWh. From 5 hours of operation:

5(4a + 3b) = 215 \implies 4a + 3b = 43

. From 4 hours of operation:

4(2a + 5b) = 156 \implies 2a + 5b = 39

.

Dividing total energy by total hours converts total consumption into hourly rates for each operating group.

2

Solve the system of simultaneous linear equations for

a

and

b

.

Multiply

2a + 5b = 39

by 2 to get

4a + 10b = 78

. Subtract

4a + 3b = 43

from

4a + 10b = 78

:

(4a + 10b) - (4a + 3b) = 78 - 43 \implies 7b = 35 \implies b = 5

.

Eliminating variable

a

allows direct solution for variable

b

.

3

Substitute

b = 5

back into one of the linear equations to find

a

.

2a + 5(5) = 39 \implies 2a + 25 = 39 \implies 2a = 14 \implies a = 7

.

Determines the single unit rate for Type Alpha pumps.

4

Calculate the combined hourly rate for one Type Alpha and one Type Beta pump.

a + b = 7 + 5 = 12\text{ kWh per hour}

.

Calculates the sum of both individual unit rates.

Anahtar Kavram

Formulating and solving a system of two linear equations in two variables from rates and total work/consumption.

Tahmini Süre:2m 0s

Soru 19Soru →

A cloud computing infrastructure employs two types of processing nodes: Server-Type X and Server-Type Y. A configuration consisting of 3 Server-Type X nodes and 5 Server-Type Y nodes processes 4,100 data transactions per second. A second configuration consisting of 5 Server-Type X nodes and 2 Server-Type Y nodes processes 4,300 data transactions per second. Assuming each node of a given type operates at a constant rate, what is the processing capacity, in data transactions per second, of a single Server-Type X node?

Cevabı ve açıklamayı göster

Cevap: 700

Cevap

700 data transactions per second

Let

x

be the capacity of Server-Type X and

y

be the capacity of Server-Type Y. The system

3x + 5y = 4100

and

5x + 2y = 4300

simplifies to

19x = 13300

after eliminating

y

, yielding

x = 700

.

Adım Adım Çözüm

1

Set up the linear equations representing the two node configurations.

3x + 5y = 4100

and

5x + 2y = 4300

, where

x

and

y

represent the processing rates of Server-Type X and Server-Type Y, respectively.

Translating word problems into algebraic equations allows systematic solving for unknown rates.

2

Eliminate the variable

y

by multiplying the equations by appropriate constants.

6x + 10y = 8200

and

25x + 10y = 21500

.

Aligning the coefficients of

y

enables elimination through subtraction.

3

Subtract the first modified equation from the second and solve for

x

.

19x = 13300 \implies x = 700

.

Solving the single-variable equation yields the processing capacity for Server-Type X.

Anahtar Kavram

Solving simultaneous linear equations in two variables using elimination

Soru 20Soru →

An automated municipal water treatment facility utilizes two primary filtration systems, System Alpha and System Beta, to treat urban wastewater. Let $x$ represent the daily processing rate of System Alpha in megaliters (ML) per day, and let $y$ represent the daily processing rate of System Beta in megaliters (ML) per day, where both $x$ and $y$ are positive values.

Operating System Alpha for 2 days and System Beta for 3 days yields a combined total throughput of 85 megaliters. Additionally, the difference between the square of System Alpha's daily processing rate and the square of System Beta's daily processing rate is equal to 175.

In the table below, select the value for the daily processing rate of System Alpha and the value for the daily processing rate of System Beta that are consistent with the information provided.

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Cevabı ve açıklamayı göster

Cevap

The daily processing rate for System Alpha is 20 ML and the daily processing rate for System Beta is 15 ML.

Solving the system of simultaneous equations

2x + 3y = 85

and

x^2 - y^2 = 175

yields

x = 20

ML for System Alpha and

y = 15

ML for System Beta. This satisfies both throughput and squared difference conditions while maintaining positive rates.

Adım Adım Çözüm

1

Formulate the system of simultaneous equations from the problem statement.

Equation 1:

2x + 3y = 85

; Equation 2:

x^2 - y^2 = 175

, with constraints

x > 0

and

y > 0

.

Operating Alpha for 2 days and Beta for 3 days totals 85 ML (

2x + 3y = 85

), and the difference of their squared rates is 175 (

x^2 - y^2 = 175

).

2

Express

x

in terms of

y

using Equation 1.

x = \frac{85 - 3y}{2}

Isolating

x

allows substitution into the non-linear equation.

3

Substitute

x = \frac{85 - 3y}{2}

into Equation 2 and simplify the quadratic equation.

\left(\frac{85 - 3y}{2}\right)^2 - y^2 = 175 \implies \frac{7225 - 510y + 9y^2}{4} - y^2 = 175 \implies 5y^2 - 510y + 6525 = 0 \implies y^2 - 102y + 1305 = 0

.

Expanding and clearing denominators yields a standard single-variable quadratic equation in terms of

y

.

4

Solve the quadratic equation

y^2 - 102y + 1305 = 0

for

y

.

(y - 15)(y - 87) = 0 \implies y = 15

or

y = 87

.

Factoring the quadratic gives the two mathematical roots for

y

.

5

Evaluate the corresponding values of

x

for each root of

y

to verify positivity constraints.

If

y = 87

,

x = \frac{85 - 3(87)}{2} = -88

(rejected since

x > 0

). If

y = 15

,

x = \frac{85 - 3(15)}{2} = 20

(valid since

x > 0

).

Physical processing rates must be positive, making

x = 20

and

y = 15

the unique valid solution.

Anahtar Kavram

Solving non-linear systems of simultaneous equations using algebraic substitution and quadratic factoring under real-world domain constraints.

Tahmini Süre:3m 0s

Simultaneous Equations and Systems

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