Solving a Tailwind/Headwind Problem Using a System of Equations by Elimination
Problem: A private jet can fly 1,095 miles in 3 hours with a tailwind but only 987 miles in 3 hours into a headwind. Find the speed of the jet in still air and the speed of the wind.
Solution:
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Name Variables: Let be the speed of the jet in still air (mph) and be the speed of the wind (mph).
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Determine Effective Rates:
- Tailwind: The wind assists the jet, so the effective rate is .
- Headwind: The wind opposes the jet, so the effective rate is .
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Translate into a System of Equations: Using the uniform motion formula (Distance = Rate Time):
- Tailwind leg:
- Headwind leg:
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Solve by Elimination: Distribute the time to write both equations in standard form:
Since the travel times are equal, the -coefficients are additive opposites. Add the equations to eliminate :
Substitute into the first equation to solve for :
The speed of the jet in still air is 347 mph, and the speed of the wind is 18 mph.
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