$$ \begin{aligned} 6(\frac{1}{2}y-1) &= \frac{1}{3}y\cdot6&& \text{multiply ALL terms by } \color{blue}{ 6 }. \\[1 em]6\cdot6(\frac{1}{2}y-1) &= 6 \cdot \frac{1}{3}y\cdot6&& \text{cancel out the denominators} \\[1 em]6\cdot6(\frac{1}{2}y-1) &= 12y&& \text{simplify left side} \\[1 em]36(\frac{1}{2}y-1) &= 12y&& \\[1 em]36(\frac{y}{2}-1) &= 12y&& \\[1 em]36 \cdot \frac{y-2}{2} &= 12y&& \\[1 em]\frac{36y-72}{2} &= 12y&& \text{multiply ALL terms by } \color{blue}{ 2 }. \\[1 em]2 \cdot \frac{36y-72}{2} &= 2\cdot12y&& \text{cancel out the denominators} \\[1 em]36y-72 &= 24y&& \text{move the $ \color{blue}{ 24y } $ to the left side and $ \color{blue}{ -72 }$ to the right} \\[1 em]36y-24y &= 72&& \text{simplify left side} \\[1 em]12y &= 72&& \text{ divide both sides by $ 12 $ } \\[1 em]y &= \frac{72}{12}&& \\[1 em]y &= 6&& \\[1 em] \end{aligned} $$

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