$$ \begin{aligned} \frac{1}{2}(3x-1)+\frac{1}{3}(2x-5) &= 0&& \text{multiply ALL terms by } \color{blue}{ 6 }. \\[1 em]6 \cdot \frac{1}{2}(3x-1)+6\frac{1}{3}(2x-5) &= 6\cdot0&& \text{cancel out the denominators} \\[1 em]9x-3+4x-10 &= 0&& \text{simplify left side} \\[1 em]13x-13 &= 0&& \text{ move the constants to the right } \\[1 em]13x &= 13&& \text{ divide both sides by $ 13 $ } \\[1 em]x &= \frac{13}{13}&& \\[1 em]x &= 1&& \\[1 em] \end{aligned} $$

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