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If a + b = 7 and a^3 + b^3 = 42 - 7ab, what is the value of the sum 1/a + 1/b? Express your answer as a common fraction.

 Apr 23, 2022
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Note that    1/a  +  1/b =  (a + b)   /  (ab )

 

a + b =  7          square both sides

 

a^2 + b^2  + 2ab   = 49

 

a^2  + b^2  =  49 - 2ab

 

Factoring

 

a^3  + b^3 =    (a + b) ( a^2 - ab + b^2)  =  (7) ( 49 -2ab -ab)   =  (7) (49 - 3ab)  =  42 - 7ab

 

So

 

(49 -3ab)  =  (42 - 7ab)  / 7

 

49 -3ab =  6 - ab

 

49 - 6  =  2ab

 

43 / 2  = ab

 

So

 

1/a + 1/b   =   ( a + b)  / (ab) =    7 / ( 43/2) =  14  / 43

 

 

cool cool cool

 Apr 23, 2022

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