SOLUTION: Hockey teams get 2 points in the standing for a win, one point for a tie, and no points for a loss. If a team has 27 points, and has 4 more wins than ties, how many wins do they ha

Question 109664: Hockey teams get 2 points in the standing for a win, one point for a tie, and no points for a loss. If a team has 27 points, and has 4 more wins than ties, how many wins do they have ?
Found 2 solutions by stanbon, solver91311:
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
Hockey teams get 2 points in the standing for a win, one point for a tie, and no points for a loss. If a team has 27 points, and has 4 more wins than ties, how many wins do they have ?
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Let number of wins be "w"; let number of ties be "t".
EQUATION:
2w+t=27
w = t+4
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Rearrange:
t = -2w+27
t = w - 4
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Substitue to solve for "w":
-2w+27 = w-4
3w = 31
w = 31/3
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Comment: w should turn out to be a whole number. Check your
posting and/or the wording in your text.
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Cheers,
Stan H.

Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
The number of points is equal to two times the number of wins plus the number of ties, so you can write:
:

:
But you also know that there are four more wins than ties, so you can also write:
:

:
Putting these two expressions together by substituting for the in the first equation, you get:
:

:
Which doesn't make any sense because you can't have a fractional win. This means that the situation described is impossible, that is to say there is no combination of two integers having a difference of 4, where 2 times one of the integers plus the other integer equals 27. If the total points were, say, 26, or the difference were 3, then you would have a valid solution. Check the problem in your text: If you wrote it wrong, then make the appropriate adjustments to the solution above. On the other hand, if you did correctly copy the problem, then the correct answer is that there is no answer.
Here's another way to convince yourself that there is no correct answer. Since a team gets two points for a win, the number of points the team received for a win must be an even number. Since the total points (27) is an odd number and wins is even, then the number of ties must be odd. Wins even and ties odd means that the difference between wins and ties must also be odd, and 4 is not odd.
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