$$ \begin{aligned} \frac{1}{x}+\frac{1}{x}-8 &= \frac{1}{3}&& \text{multiply ALL terms by } \color{blue}{ x\cdot3 }. \\[1 em]x\cdot3\cdot\frac{1}{x}+x\cdot3\cdot\frac{1}{x}-x\cdot3\cdot8 &= x\cdot3\cdot\frac{1}{3}&& \text{cancel out the denominators} \\[1 em]3+3-24x &= x&& \text{simplify left side} \\[1 em]-24x+6 &= x&& \text{move the $ \color{blue}{ x } $ to the left side and $ \color{blue}{ 6 }$ to the right} \\[1 em]-24x-x &= -6&& \text{simplify left side} \\[1 em]-25x &= -6&& \text{ divide both sides by $ -25 $ } \\[1 em]x &= \frac{-6}{-25}&& \\[1 em]x &= \frac{6}{25}&& \\[1 em] \end{aligned} $$

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