$$ \begin{aligned} x+\frac{1}{3} &= 3x+1&& \text{multiply ALL terms by } \color{blue}{ 3 }. \\[1 em]3x+3\cdot\frac{1}{3} &= 3\cdot3x+3\cdot1&& \text{cancel out the denominators} \\[1 em]3x+1 &= 9x+3&& \text{move the $ \color{blue}{ 9x } $ to the left side and $ \color{blue}{ 1 }$ to the right} \\[1 em]3x-9x &= 3-1&& \text{simplify left and right hand side} \\[1 em]-6x &= 2&& \text{ divide both sides by $ -6 $ } \\[1 em]x &= -\frac{2}{6}&& \\[1 em]x &= -\frac{1}{3}&& \\[1 em] \end{aligned} $$

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