| Fractional System |
Solve the system of equations in positive real a,b,c:
a/b + b/a + a/c + c/a + b/c + c/b = 6
a/bc + b/ac + c/ab = 4
Please note: A complete solution must demonstrate that you have all possible solutions! |
Problem Moderated by: Sasha |
| Problem Solution |
First, we make note of the fact that:
a/b + b/a >= 2to prove this: assume a/b + b/a < 2
(a2 + b2)/ab < 2
a2 + b2 < 2ab
a2 + b2 - 2ab < 0
(a - b)2 < 0, which is not possible. Similarly, a/c + c/a >=2, and b/c + c/b >= 2
Since the first equation is the sum of three quantities which are all greater than or equal to two, and their sum is 6, we conclude that each of these quantities must be equal to two.
Thus a/b + b/a = 2(a2 + b2)/ab = 2
a2 + b2 = 2ab
a2 + b2 - 2ab = 0
(a - b)2 = 0
a = b Similar reasoning gives us b = c
Now we simply need to substutite a everywhere in the second equation:
a/a2 + a/a2 + a/a2 = 4
3/a = 4
a = 3/4
Solution: (3/4, 3/4, 3/4) |
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