$$ \begin{aligned} \frac{7}{x} &= \frac{9}{x+6}&& \text{multiply ALL terms by } \color{blue}{ x(x+6) }. \\[1 em]x(x+6)\cdot\frac{7}{x} &= x(x+6)\cdot\frac{9}{x+6}&& \text{cancel out the denominators} \\[1 em]7x+42 &= 9x&& \text{move the $ \color{blue}{ 9x } $ to the left side and $ \color{blue}{ 42 }$ to the right} \\[1 em]7x-9x &= -42&& \text{simplify left side} \\[1 em]-2x &= -42&& \text{ divide both sides by $ -2 $ } \\[1 em]x &= \frac{-42}{-2}&& \\[1 em]x &= 21&& \\[1 em] \end{aligned} $$

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