$$ \begin{aligned} \frac{a+6}{a} &= \frac{10}{7}&& \text{multiply ALL terms by } \color{blue}{ a\cdot7 }. \\[1 em]a\cdot7 \cdot \frac{a+6}{a} &= a\cdot7\cdot\frac{10}{7}&& \text{cancel out the denominators} \\[1 em]7a+42 &= 10a&& \text{move the $ \color{blue}{ 10a } $ to the left side and $ \color{blue}{ 42 }$ to the right} \\[1 em]7a-10a &= -42&& \text{simplify left side} \\[1 em]-3a &= -42&& \text{ divide both sides by $ -3 $ } \\[1 em]a &= \frac{-42}{-3}&& \\[1 em]a &= 14&& \\[1 em] \end{aligned} $$

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