$$ \begin{aligned} \frac{2}{3}q-3 &= 3 \cdot \frac{q}{5}+1&& \text{multiply ALL terms by } \color{blue}{ 15 }. \\[1 em]15 \cdot \frac{2}{3}q-15\cdot3 &= 15\cdot3 \cdot \frac{q}{5}+15\cdot1&& \text{cancel out the denominators} \\[1 em]10q-45 &= 9q+15&& \text{move the $ \color{blue}{ 9q } $ to the left side and $ \color{blue}{ -45 }$ to the right} \\[1 em]10q-9q &= 15+45&& \text{simplify left and right hand side} \\[1 em]q &= 60&& \\[1 em]q &= 60&& \\[1 em] \end{aligned} $$

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