$$\frac{1}{x-1} = 1$$

$$ \begin{aligned} \frac{1}{x-1} &= 1&& \text{multiply ALL terms by } \color{blue}{ x-1 }. \\[1 em](x-1)\cdot\frac{1}{x-1} &= (x-1)\cdot1&& \text{cancel out the denominators} \\[1 em]1 &= x-1&& \text{ swap the left and right side of the equation } \\[1 em]x-1 &= 1&& \text{ move -1 to the right } \\[1 em]x &= 1+1&& \\[1 em]x &= 2&& \\[1 em]x &= 2&& \\[1 em] \end{aligned} $$

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