$$ \begin{aligned} \frac{x}{3}-3 &= \frac{5}{3}x+2&& \text{multiply ALL terms by } \color{blue}{ 3 }. \\[1 em]3 \cdot \frac{x}{3}-3\cdot3 &= 3 \cdot \frac{5}{3}x+3\cdot2&& \text{cancel out the denominators} \\[1 em]x-9 &= 5x+6&& \text{move the $ \color{blue}{ 5x } $ to the left side and $ \color{blue}{ -9 }$ to the right} \\[1 em]x-5x &= 6+9&& \text{simplify left and right hand side} \\[1 em]-4x &= 15&& \text{ divide both sides by $ -4 $ } \\[1 em]x &= -\frac{15}{4}&& \\[1 em] \end{aligned} $$

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