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Zorluk: Çok zorSimultaneous Equations and Systems

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 44 Configuration P flights and 33 Configuration C flights requires a total of 98,00098,000 gallons of fuel.
- Operating 22 Configuration P flights and 55 Configuration C flights requires a total of 112,000112,000 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.

  • Configuration P fuel consumption per flight (gallons)11,000
  • Configuration C fuel consumption per flight (gallons)18,000

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 4fP+3fC=98,0004 f_P + 3 f_C = 98,000 and 2fP+5fC=112,0002 f_P + 5 f_C = 112,000 gives fP=11,000f_P = 11,000 gallons and fC=18,000f_C = 18,000 gallons.

Adım Adım Çözüm

1
Formulate a system of simultaneous linear equations based on the problem statement.
Let fPf_P be fuel per Configuration P flight and fCf_C be fuel per Configuration C flight. Equation 1: 4fP+3fC=98,0004 f_P + 3 f_C = 98,000. Equation 2: 2fP+5fC=112,0002 f_P + 5 f_C = 112,000.
Translates the two total fuel conditions into explicit linear equations.
2
Eliminate variable fPf_P by multiplying Equation 2 by 2.
Multiplying Equation 2 by 2 yields 4fP+10fC=224,0004 f_P + 10 f_C = 224,000.
Equalizes the coefficients of fPf_P across both equations to enable cancellation.
3
Subtract Equation 1 from the modified equation to solve for fCf_C.
(4fP+10fC)−(4fP+3fC)=224,000−98,000  ⟹  7fC=126,000  ⟹  fC=18,000(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 fPf_P, yielding the fuel consumption per Configuration C flight.
4
Substitute fC=18,000f_C = 18,000 back into Equation 1 to find fPf_P.
4fP+3(18,000)=98,000  ⟹  4fP+54,000=98,000  ⟹  4fP=44,000  ⟹  fP=11,0004 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.
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Konular

Simultaneous Equations and Systems

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