$$ \begin{aligned} \frac{x}{x-1} &= 2&& \text{multiply ALL terms by } \color{blue}{ x-1 }. \\[1 em](x-1)\frac{x}{x-1} &= (x-1)\cdot2&& \text{cancel out the denominators} \\[1 em]x &= 2x-2&& \text{move the $ \color{blue}{ 2x } $ to the left side} \\[1 em]x-2x &= -2&& \text{simplify left side} \\[1 em]-x &= -2&& \text{switch signs on both sides} \\[1 em]x &= 2&& \\[1 em] \end{aligned} $$

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