<s>[INST] Solve the equation $\frac{x-13}{x-14}-\frac{x-15}{x-16}=-\frac{1}{12}$ Solve the equation $$\dfrac{x-13}{x-14}-\dfrac{x-15}{x-16}=-\dfrac{1}{12}.$$
For $x\ne14$ and $x\ne 16$ by multiplying the whole equation by $$12(x-14)(x-16)$$ we get: $$12(x-16)(x-13)-12(x-14)(x-15)=-(x-14)(x-16).$$ This doesn't look very nice. Can we do something else at the beginning? $$x-14=(x-13)-1\\x-15=(x-14)-1\\..?$$
 [/INST] $$\frac{x-13}{x-14} - \frac{x-15}{x-16} = -\frac{1}{12}$$
Let $x = t+ 15$
\begin{align}
   \frac{t+2}{t+1} - \frac{t}{t-1} &= -\frac{1}{12} \\
   \dfrac{(t^2+t-2) - (t^2+t)}{t^2-1} &= -\frac{1}{12} \\
   \dfrac{-2}{t^2-1} &= -\frac{1}{12} \\
   t^2-1 &= 24 \\
   t^2 &= 25 \\
   t &= \pm 5 \\
   x &= 20, 10
\end{align}
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