+184 If $a+b=7$ and $a^3+b^3=42$, what is the value of the sum $\dfrac{1}{a}+\dfrac{1}{b}$? Express you answer as a common fraction.
+128448 Note that 1/a + 1/b = (a + b) / (ab)
a^3 + b^3 =
(a + b) (a^2 - ab + b^2) = 42
(7) (a^2 - ab + b^2) = 42(1)
And
(a + b)^2 = 7^2
a^2 + 2ab + b^2 = 49
a^2 + b^2 = 49 - 2ab (2)
Sub (2) into (1)
( 7) ( 49 - 2ab - ab) = 42
49 - 3ab = 6
49 - 6 = 3ab
43 = 3ab
ab = 43/3
So 1/a +1/b = (a+ b) / ab = 7 / (43/3) = 21 / 43
