$$\frac{f+3}{f+1} = \frac{f+9}{f+6}$$

$$ \begin{aligned} \frac{f+3}{f+1} &= \frac{f+9}{f+6}&& \text{multiply ALL terms by } \color{blue}{ (f+1)(f+6) }. \\[1 em](f+1)(f+6)\frac{f+3}{f+1} &= (f+1)(f+6)\frac{f+9}{f+6}&& \text{cancel out the denominators} \\[1 em]f^2+9f+18 &= f^2+10f+9&& \text{move all terms to the left hand side } \\[1 em]f^2+9f+18-f^2-10f-9 &= 0&& \text{simplify left side} \\[1 em]f^2+9f+18-f^2-10f-9 &= 0&& \\[1 em]-f+9 &= 0&& \text{ move the constants to the right } \\[1 em]-f &= -9&& \text{switch signs on both sides} \\[1 em]f &= 9&& \\[1 em] \end{aligned} $$

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