A research laboratory operates two types of automated centrifuges, Model X and Model Y. A … — GRE General Test

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?

A
10
B
15
25Cevap
D
30
E
35

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

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.

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.

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

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.

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

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