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Rational Resolution (Posted on 2016-11-18) Difficulty: 3 of 5
Determine all possible rational solutions to this system of equations:

A + 1/B = 2, and:
B + C*A = 9.7, and:
C + A/B = 5.14

Prove that there are no others.

No Solution Yet Submitted by K Sengupta    
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Solution Direct solution Comment 2 of 2 |
(i) A + 1/B = 2
(ii) B + C*A = 9.7
(iii) C + A/B = 5.14
by inspection B cannot equal 0
from (i)
(iv) A = 2 - 1/B
from (ii)
(v) B = 9.7 - C*A
from (iii)
(vi) C= 5.14 - A/B
substitute (iv) into (v)
(vii) B = 9.7 - C*(2 - 1/B)
Substitute (iv) into (vi)
(viii) C = 5.14 - (2 - 1/B)/B
Substitute (viii) into (vii)
(ix) B = 9.7 - (5.14 - (2 - 1/B))*(2 - 1/B)
Multiply both sides by B^3 and rearrange 
(x) B^4 + .58B^3 - 9.14B^2 + 4B -1 = 0

This quartic has one rational solution: B=2.5 
(I used my trusty graphing calculator)
There is another real around -3.52 and complex approximately .22 +/- .25i (I used WolframAlpha)

Solution A=1.6, B=2.5, C=4.5

  Posted by Jer on 2016-11-19 22:39:14
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