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Difficulty: MediumSimultaneous Equations and Systems

A logistics company operates two types of delivery drones: Model X and Model Y. Working continuously at their respective constant hourly rates, a fleet of 3 Model X drones and 5 Model Y drones delivers a total of 190 packages per hour. Alternatively, a fleet of 4 Model X drones and 2 Model Y drones delivers a total of 160 packages per hour. What is the hourly package delivery capacity of a single Model X drone?

Answer: 30 packages per hour

Answer

The hourly package delivery capacity of a single Model X drone is 30 packages per hour.
By translating the problem into simultaneous linear equations, 3x+5y=1903x + 5y = 190 and 4x+2y=1604x + 2y = 160, simplifying the second equation gives 2x+y=802x + y = 80, so y=80−2xy = 80 - 2x. Substituting this into the first equation yields 3x+5(80−2x)=1903x + 5(80 - 2x) = 190, which reduces to −7x=−210-7x = -210, giving x=30x = 30 packages per hour.

Step-by-Step Solution

1
Set up a system of two linear equations based on the rates of Model X (xx) and Model Y (yy).
System of equations:
(1) 3x+5y=1903x + 5y = 190
(2) 4x+2y=1604x + 2y = 160
Each statement provides a linear combination of the hourly output of the two drone types.
2
Express yy in terms of xx using the second equation.
2x+y=80  ⟹  y=80−2x2x + y = 80 \implies y = 80 - 2x
Simplifying the second equation isolates yy easily for substitution.
3
Substitute the expression for yy into the first equation and solve for xx.
3x+5(80−2x)=190  ⟹  3x+400−10x=190  ⟹  −7x=−210  ⟹  x=303x + 5(80 - 2x) = 190 \implies 3x + 400 - 10x = 190 \implies -7x = -210 \implies x = 30
Solving the single-variable linear equation yields the precise capacity of Model X.

Key Concept

Solving Systems of Two Linear Equations with Two Unknowns
Estimated Time:1m 30s
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Topics

Simultaneous Equations and Systems

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