$$ \begin{aligned} \frac{8}{x}+\frac{1}{5} &= \frac{3}{x}&& \text{multiply ALL terms by } \color{blue}{ x\cdot5 }. \\[1 em]x\cdot5\cdot\frac{8}{x}+x\cdot5\cdot\frac{1}{5} &= x\cdot5\cdot\frac{3}{x}&& \text{cancel out the denominators} \\[1 em]40+x &= 15&& \text{ move 40 to the right } \\[1 em]x &= 15-40&& \\[1 em]x &= -25&& \\[1 em]x &= -25&& \\[1 em] \end{aligned} $$

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