Question 117848
For starters, you don't know whether the plane is being helped by the wind (tailwind) or
is being slowed down by the wind (headwind). 
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Ask yourself, "If there were no wind, how far would the plane go in the 11 hours that it is
flying?" To get that answer you multiply its speed in still air (no wind) by 11 as follows:
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Distance = 268 miles per hour times 11 hours = 268*11 = 2948 miles
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But since the plane only flies 629 miles in 11 hours, you know that the wind really has to
be slowing it down. The plane is flying into a pretty strong headwind because it is really being
slowed down. 
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If the plane is going directly into a headwind, its true rate of speed is its speed in still
air minus the speed of the wind that it is flying into. If W is the speed of the headwind, the
true speed of the plane (268 mph) minus W or (268 - W).
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Since that is the actual or true speed of the airplane, in 11 hours the distance that the
plane flies (you are told 629 miles) can be represented by the following equation:
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Distance = speed * time = (268 - W)*11
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Multiplying out the right side of this equation results in two products as follows:
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Distance = 268*11 - 11*W
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And since the distance is 629 miles, substituting 629 for distance results in the equation
becoming:
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629 = 268*11 - 11*W
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The product of 268 and 11 is 268*11 = 2948. So substitute 2948 into the equation  in place of 268*11
to get:
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629 = 2948 - 11*W
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Get rid of the 2948 on the right side by subtracting 2948 from both sides to make the equation:
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629 - 2948 = -11*W
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and combining the numbers on the left side results in:
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-2319 = - 11*W
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Solve this equation for W by dividing both sides by -11, the multiplier of W to get:
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{{{-2319/-11 = W}}}
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And performing the division on the left side (dividing -2319 by -11) reduces the equation to:
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210.8181818 = W
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This tells you that the headwind that the plane flies into is 210.8181818 miles per hour.
(Round this off to 210.8 miles per hour.) 
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If all the numbers you gave in the problem are correct, this plane is flying in a category 5+ 
hurricane or something similar ... headwinds over 210 miles per hour ... and probably shouldn't 
be up in the air at all.  Its actual speed is its speed in still air (268 miles per hour)
minus the speed of the wind (210.8 miles per hour) which results in an actual speed of
57.2 miles per hour relative to somebody on the ground ... That's a major slow down, isn't it?
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Hope this helps you to understand the problem and how to work your way through it. 
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