$$ \begin{aligned} 5x-\frac{9}{5}x+1 &= 0&& \text{multiply ALL terms by } \color{blue}{ 5 }. \\[1 em]5\cdot5x-5 \cdot \frac{9}{5}x+5\cdot1 &= 5\cdot0&& \text{cancel out the denominators} \\[1 em]25x-9x+5 &= 0&& \text{simplify left side} \\[1 em]16x+5 &= 0&& \text{ move the constants to the right } \\[1 em]16x &= -5&& \text{ divide both sides by $ 16 $ } \\[1 em]x &= -\frac{5}{16}&& \\[1 em] \end{aligned} $$

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