Question 15508
Great Question!
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First of all, we must convert all the information given in the question into equations so we can look it from a simplified angle, and hence, solve it.
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So, let's start off by giving each of the unknown numbers a name. Let's call the first number a, the second number b, and the third number c.
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Therefore:
"The second of three numbers is 8 more than the first" gives us:
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b = 8 + a
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"... the third is 3 less than 3 times the first" gives us:
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c = 3a - 3
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and "the third number is 15 more than the second" gives us:
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c =  15 + b
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Now, we can substitute b = 8 + a into c = 15 + b, to get: 
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c = 15 + (8 + a), which simplified gives us:
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c = 23 + a
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With that done we can use elimination.
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We've got c = 23 + a, and c = 3a - 3. We want to eliminate 'a', so we should multiply both sides of c = 23 + a by (-3) to get: -3c = -69 - 3a.
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Now we can eliminate the 'a' variable:
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  c =  3a - 3
-3c = -3a - 69
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-2c = -72
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Therefore, we can solve for 'c' by dividing both sides by -2, which leaves us with:
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c = 36
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Now we can plug that value of 'c' into the equation c = 15 + b, to solve for b:
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36 = 15 + b          (Subtract 15 from both sides)
21 = b
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Now that we've got a value for 'b', we can plug it into the equation b = 8 + a to solve for 'a':
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21 = 8 + a           (Subtract 8 from both sides)
13 = a
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And there we have it, the three numbers in order are 13, 21, 36. It's a long process, but it's worth it. Again, Great Question!