$$ \begin{aligned} \frac{x}{3}+1 &= 5-x+\frac{2}{7}&& \text{multiply ALL terms by } \color{blue}{ 21 }. \\[1 em]21 \cdot \frac{x}{3}+21\cdot1 &= 21\cdot5-21x+21\cdot\frac{2}{7}&& \text{cancel out the denominators} \\[1 em]7x+21 &= 105-21x+6&& \text{simplify right side} \\[1 em]7x+21 &= -21x+111&& \text{move the $ \color{blue}{ -21x } $ to the left side and $ \color{blue}{ 21 }$ to the right} \\[1 em]7x+21x &= 111-21&& \text{simplify left and right hand side} \\[1 em]28x &= 90&& \text{ divide both sides by $ 28 $ } \\[1 em]x &= \frac{90}{28}&& \\[1 em]x &= \frac{45}{14}&& \\[1 em] \end{aligned} $$

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