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Let $a$, $b$, and $c$, be nonzero real numbers such that $a+b+c=0$. Compute the value of\[a\left(\frac 1b + \frac1c\right) + b\left(\frac1a + \frac1c\right) + c\left(\frac1a+\frac1b\right).\]

 Aug 15, 2021
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Let \(a\), \(b\), and \(c\), be nonzero real numbers such that \(a+b+c=0\).
Compute the value of

\(a\left(\dfrac 1b + \dfrac1c\right) + b\left(\dfrac1a + \dfrac1c\right) + c\left(\dfrac1a+\dfrac1b\right)\).

 

\(\begin{array}{|rcll|} \hline a+b+c &=& 0 \\ a+b &=& -c \qquad (1) \\ a+c &=& -b \qquad (2) \\ b+c &=& -a \qquad (3) \\ \hline \end{array}\)

 

\(\begin{array}{|rcll|} \hline && \mathbf{a\left(\dfrac 1b + \dfrac1c\right) + b\left(\dfrac1a + \dfrac1c\right) + c\left(\dfrac1a+\dfrac1b\right)} \\\\ &=& \dfrac ab + \dfrac ac + \dfrac ba + \dfrac bc + \dfrac ca+\dfrac cb \\\\ &=& \dfrac ba + \dfrac ca + \dfrac ab + \dfrac cb + \dfrac ac+\dfrac bc \\\\ &=& \dfrac {b+c}a + \dfrac {a+c}b + \dfrac {a+b}c \\\\ &=& \dfrac {-a}a + \dfrac {-b}b + \dfrac {-c}c \\\\ &=& -1-1-1 \\\\ &=& \mathbf{-3} \\ \hline \end{array}\)

 

\(a\left(\dfrac 1b + \dfrac1c\right) + b\left(\dfrac1a + \dfrac1c\right) + c\left(\dfrac1a+\dfrac1b\right)=-3 \)

 

laugh

 Aug 16, 2021

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