$$ \begin{aligned} -\frac{1}{5}x\cdot2+\frac{1}{2}x+\frac{1}{10} &= \frac{1}{10}&& \text{multiply ALL terms by } \color{blue}{ 10 }. \\[1 em]-10 \cdot \frac{1}{5}x\cdot2+10\frac{1}{2}x+10\cdot\frac{1}{10} &= 10\cdot\frac{1}{10}&& \text{cancel out the denominators} \\[1 em]-4x+5x+1 &= 1&& \text{simplify left side} \\[1 em]x+1 &= 1&& \text{ move 1 to the right } \\[1 em]x &= 1-1&& \\[1 em]x &= 0&& \text{ divide both sides by $ 1 $ } \\[1 em]x &= \frac{0}{1}&& \\[1 em]x &= 0&& \\[1 em] \end{aligned} $$

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