9. \( \frac{7}{12 v}+\frac{1}{6 v}=\frac{1}{v-1} \)

To solve the equation \( \frac{7}{12 v}+\frac{1}{6 v}=\frac{1}{v-1} \), we first find a common denominator for the left side. The common denominator between \(12v\) and \(6v\) is \(12v\). Rewriting the left side: \[ \frac{7}{12v} + \frac{2}{12v} = \frac{9}{12v} = \frac{3}{4v} \] Now the equation becomes: \[ \frac{3}{4v} = \frac{1}{v-1} \] Cross multiplying gives us: \[ 3(v - 1) = 4v \] Expanding that: \[ 3v - 3 = 4v \] Rearranging the equation leads us to: \[ 3v - 4v = 3 \implies -v = 3 \implies v = -3 \] Thus, the solution to the equation is \( v = -3 \).