If a + b = 7 and a^3 + b^3 = 42 + ab, what is the value of the sum 1/a + 1/b? Express you answer as a common fraction.
+26367 If
\(a + b = 7\) and \(a^3 + b^3 = 42 + ab\),
what is the value of the sum \(\dfrac{1}{a} + \dfrac{1}{b}\)?
\(\begin{array}{|rcll|} \hline \left(a + b\right)^3 =&=& a^3+3a^2b+3ab^2+b^3 \\ \left(a + b\right)^3 =&=& a^3+b^3+3ab(a+b) \\ 7^3 =&=& 42+ab+3ab*7 \\ 343 =&=& 42+22ab \\ 22ab=&=& 343-42\\ ab &=& \dfrac{301}{22} \\ \hline \end{array} \)
\(\begin{array}{|rcll|} \hline \dfrac{1}{a} + \dfrac{1}{b} &=& \dfrac{a+b}{ab} \\\\ \dfrac{1}{a} + \dfrac{1}{b} &=& \dfrac{7}{\frac{301}{22}} \\\\ \dfrac{1}{a} + \dfrac{1}{b} &=& \dfrac{7*22}{301} \\\\ \dfrac{1}{a} + \dfrac{1}{b} &=& \dfrac{22}{43} \\ \hline \end{array}\)
