$$ \begin{aligned} \frac{5}{6}(2x-3) &= \frac{2}{3}(3x+1)&& \text{simplify left and right hand side} \\[1 em]\frac{10x-15}{6} &= \frac{6x+2}{3}&& \text{multiply ALL terms by } \color{blue}{ 6 }. \\[1 em]6 \cdot \frac{10x-15}{6} &= 6 \cdot \frac{6x+2}{3}&& \text{cancel out the denominators} \\[1 em]10x-15 &= 12x+4&& \text{move the $ \color{blue}{ 12x } $ to the left side and $ \color{blue}{ -15 }$ to the right} \\[1 em]10x-12x &= 4+15&& \text{simplify left and right hand side} \\[1 em]-2x &= 19&& \text{ divide both sides by $ -2 $ } \\[1 em]x &= -\frac{19}{2}&& \\[1 em] \end{aligned} $$

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