$$ \begin{aligned} 8 \cdot \frac{x}{5}+5\frac{x}{8} &= 0&& \text{multiply ALL terms by } \color{blue}{ 40 }. \\[1 em]40\cdot8 \cdot \frac{x}{5}+40\cdot5\frac{x}{8} &= 40\cdot0&& \text{cancel out the denominators} \\[1 em]64x+25x &= 0&& \text{simplify left side} \\[1 em]89x &= 0&& \text{ divide both sides by $ 89 $ } \\[1 em]x &= \frac{0}{89}&& \\[1 em]x &= 0&& \\[1 em] \end{aligned} $$

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