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Question 1114577: The sum of three number is 13. The 1st number plus 3 times the second plus the third is 1. Twice the first minus the second plus the third is 15. Which of the numbers are odd numbers?
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
Note first that the question should say integers -- not just numbers.
The question that is asked makes it sound as if we can find the answer without actually finding the values of each of the numbers. However, there is not enough information to do that; so we have to do something with the system of equations to answer the question.
But the logical reasoning involved in trying to answer the question without solving the system of equations is good mental exercise.
The first condition is met if all three integers are odd, or if two are even and the other is odd; likewise for the second condition.
The third condition says twice the first, minus the second, plus the third, is odd. Since "twice the first" has to be even and the sum is odd, the second and third have to be one even and one odd.
So since we know either the second or third is even, the first must be even also.
So we know for sure that the first integer is even.
But there is no information given that allows us to determine whether the second or third integer is the other even integer. So we have to at least partially solve the system of equations to answer the question.
Comparing the first and second conditions, we easily see that twice the second number is -12, so the second number is -6.
And now we know that the first and second numbers are even and the third is odd.
Answer: the third integer, and only that one, is odd.
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