$$ \begin{aligned} \frac{x}{3}+\frac{1}{3} &= 5-\frac{x}{7}+\frac{2}{7}&& \text{multiply ALL terms by } \color{blue}{ 21 }. \\[1 em]21 \cdot \frac{x}{3}+21\cdot\frac{1}{3} &= 21\cdot5-21 \cdot \frac{x}{7}+21\cdot\frac{2}{7}&& \text{cancel out the denominators} \\[1 em]7x+7 &= 105-3x+6&& \text{simplify right side} \\[1 em]7x+7 &= -3x+111&& \text{move the $ \color{blue}{ -3x } $ to the left side and $ \color{blue}{ 7 }$ to the right} \\[1 em]7x+3x &= 111-7&& \text{simplify left and right hand side} \\[1 em]10x &= 104&& \text{ divide both sides by $ 10 $ } \\[1 em]x &= \frac{104}{10}&& \\[1 em]x &= \frac{52}{5}&& \\[1 em] \end{aligned} $$

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