$$ \begin{aligned} \frac{x}{7}+\frac{x}{3} &= 1-x&& \text{multiply ALL terms by } \color{blue}{ 21 }. \\[1 em]21 \cdot \frac{x}{7}+21\frac{x}{3} &= 21\cdot1-21x&& \text{cancel out the denominators} \\[1 em]3x+7x &= 21-21x&& \text{simplify left and right hand side} \\[1 em]10x &= -21x+21&& \text{move the $ \color{blue}{ -21x } $ to the left side} \\[1 em]10x+21x &= 21&& \text{simplify left side} \\[1 em]31x &= 21&& \text{ divide both sides by $ 31 $ } \\[1 em]x &= \frac{21}{31}&& \\[1 em] \end{aligned} $$

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