If a + b = 7 and a^3 + b^3 = 44, what is the value of the sum 1/a + 1/b? Express your answer as a common fraction.
+26367 If \(a + b = 7\) and \(a^3 + b^3 = 44\),
what is the value of the sum \(\dfrac{1}{a} + \dfrac{1}{b}\)?
Express your answer as a common fraction.
\(\begin{array}{|rcll|} \hline (a+b)^3 &=& a^3+b^3+3ab(a+b) \quad | \quad a + b = 7,~a^3 + b^3 = 44 \\ 7^3 &=& 44 +21ab \\ 21ab &=& 343 - 44 \\ \mathbf{ ab } &=& \mathbf{ \dfrac{299}{21} } \\ \hline \dfrac{1}{a} + \dfrac{1}{b} &=& \dfrac{a+b}{ab} \quad | \quad a + b = 7,~ ab=\dfrac{299}{21} \\\\ &=& \dfrac{7*21}{299} \\\\ \mathbf{ \dfrac{1}{a} + \dfrac{1}{b} } &=& \mathbf{ \dfrac{147}{299} } \\ \hline \end{array}\)
