Suppose that $\alpha$ is inversely proportional to $\beta$. If $\alpha = -3$ when $\beta = -6$, find $\alpha$ when $\beta = 18$. Express your answer as a fraction.
+580 \(\frac{-3}{6}=\frac{a}{18}\quad \)
\(\frac{a}{18}=-\frac{1}{2}\)
\(:\quad a=-9\)
-Vinculum
+9674 Since \(\alpha\) is inversely proportional to \(\beta\), \(\alpha = \dfrac k\beta\) for some nonzero real number \(k\), which we call the proportionality constant.
We find k by substituting a pair of values of \(\alpha\) and \(\beta\) into the equation.
Substituting \(\alpha = -3\) and \(\beta = -6\), \(-3 = \dfrac k{-6}\). Solving gives \(k = 18\).
Now, substitute \(\beta = 18\) and \(k = 18\) to get the corresponding value of \(\alpha\) when \(\beta = 18\).
\(\alpha = \dfrac{18}{18} = \boxed{1}\).
