$$ \begin{aligned} \frac{3}{x-7}+3 &= 0&& \text{multiply ALL terms by } \color{blue}{ x-7 }. \\[1 em](x-7)\cdot\frac{3}{x-7}+(x-7)\cdot3 &= (x-7)\cdot0&& \text{cancel out the denominators} \\[1 em]3+3x-21 &= 0&& \text{simplify left side} \\[1 em]3x-18 &= 0&& \text{ move the constants to the right } \\[1 em]3x &= 18&& \text{ divide both sides by $ 3 $ } \\[1 em]x &= \frac{18}{3}&& \\[1 em]x &= 6&& \\[1 em] \end{aligned} $$

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