$$ \begin{aligned} \frac{y+3}{5} &= \frac{y-3}{3}&& \text{multiply ALL terms by } \color{blue}{ 15 }. \\[1 em]15 \cdot \frac{y+3}{5} &= 15 \cdot \frac{y-3}{3}&& \text{cancel out the denominators} \\[1 em]3y+9 &= 5y-15&& \text{move the $ \color{blue}{ 5y } $ to the left side and $ \color{blue}{ 9 }$ to the right} \\[1 em]3y-5y &= -15-9&& \text{simplify left and right hand side} \\[1 em]-2y &= -24&& \text{ divide both sides by $ -2 $ } \\[1 em]y &= \frac{-24}{-2}&& \\[1 em]y &= 12&& \\[1 em] \end{aligned} $$

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