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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:

  1. Name Variables: Let jj be the speed of the jet in still air (mph) and ww be the speed of the wind (mph).

  2. Determine Effective Rates:

    • Tailwind: The wind assists the jet, so the effective rate is j+wj + w.
    • Headwind: The wind opposes the jet, so the effective rate is jwj - w.
  3. Translate into a System of Equations: Using the uniform motion formula (Distance = Rate ×\times Time):

    • Tailwind leg: 3(j+w)=1,0953(j + w) = 1{,}095
    • Headwind leg: 3(jw)=9873(j - w) = 987
  4. Solve by Elimination: Distribute the time to write both equations in standard form:

    • 3j+3w=1,0953j + 3w = 1{,}095
    • 3j3w=9873j - 3w = 987

    Since the travel times are equal, the ww-coefficients are additive opposites. Add the equations to eliminate ww: 6j=2,0826j = 2{,}082 j=347j = 347

    Substitute j=347j = 347 into the first equation to solve for ww: 3(347)+3w=1,0953(347) + 3w = 1{,}095 1,041+3w=1,0951{,}041 + 3w = 1{,}095 3w=543w = 54 w=18w = 18

The speed of the jet in still air is 347 mph, and the speed of the wind is 18 mph.

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Updated 2026-06-27

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