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.
+130071 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
