Help me with this system of equations a + b = 3 &#x2212;<!-- − --> c

syaoronsangelhwc17

syaoronsangelhwc17

Answered question

2022-05-16

Help me with this system of equations
a + b = 3 c
1 a + 1 b = 5 12 1 c
a 3 + b 3 = 45 c 3

Answer & Explanation

Gerardo Barry

Gerardo Barry

Beginner2022-05-17Added 17 answers

We have
. a + b + c = 3 1 a + 1 b + 1 c = 5 12 a 3 + b 3 + c 3 = 45
The second equation tells us that a b + b c + c a a b c = 5 12 , or that 12 ( a b + b c + c a ) = 5 ( a b c ).
Using
( a + b + c ) 3 a 3 b 3 c 3 = 3 ( a + b ) ( b + c ) ( c + a ) = 3 ( 3 a ) ( 3 b ) ( 3 c ), we get that
27 45 = 3 [ 27 9 ( a + b + c ) + 3 ( a b + b c + c a ) a b c ], or that
6 = 3 ( a b + b c + c a ) a b c .
Hence, we get that a b + b c + c a = 10, a b c = 24. by solving the simultaneous equations.
Hence, using Vieta, we get that a , b , c are (permutation of) roots to the cubic
x 3 3 x 2 10 x + 24 = 0.
This cubic is ( x 2 ) ( x 4 ) ( x + 3 ) = 0, and thus we have { a , b , c } = { 2 , 4 , 3 }.

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