$$\frac{1}{2}x+\frac{1}{3} = 3(\frac{5}{6}x+3)$$

$$ \begin{aligned} \frac{1}{2}x+\frac{1}{3} &= 3(\frac{5}{6}x+3)&& \text{multiply ALL terms by } \color{blue}{ 6 }. \\[1 em]6 \cdot \frac{1}{2}x+6\cdot\frac{1}{3} &= 6\cdot3(\frac{5}{6}x+3)&& \text{cancel out the denominators} \\[1 em]3x+2 &= 6\cdot3(\frac{5}{6}x+3)&& \text{simplify right side} \\[1 em]3x+2 &= 18(\frac{5}{6}x+3)&& \\[1 em]3x+2 &= 18(\frac{5x}{6}+3)&& \\[1 em]3x+2 &= 18 \cdot \frac{5x+18}{6}&& \\[1 em]3x+2 &= \frac{90x+324}{6}&& \text{multiply ALL terms by } \color{blue}{ 6 }. \\[1 em]6\cdot3x+6\cdot2 &= 6 \cdot \frac{90x+324}{6}&& \text{cancel out the denominators} \\[1 em]18x+12 &= 90x+324&& \text{move the $ \color{blue}{ 90x } $ to the left side and $ \color{blue}{ 12 }$ to the right} \\[1 em]18x-90x &= 324-12&& \text{simplify left and right hand side} \\[1 em]-72x &= 312&& \text{ divide both sides by $ -72 $ } \\[1 em]x &= -\frac{312}{72}&& \\[1 em]x &= -\frac{13}{3}&& \\[1 em] \end{aligned} $$

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