$$ \begin{aligned} \frac{x}{x-7} &= \frac{x+6}{x}&& \text{multiply ALL terms by } \color{blue}{ (x-7)x }. \\[1 em](x-7)x \cdot \frac{x}{x-7} &= (x-7)x \cdot \frac{x+6}{x}&& \text{cancel out the denominators} \\[1 em]x^2 &= x^2-x-42&& \text{move all terms to the left hand side } \\[1 em]x^2-x^2+x+42 &= 0&& \text{simplify left side} \\[1 em]x^2-x^2+x+42 &= 0&& \\[1 em]x+42 &= 0&& \text{ move the constants to the right } \\[1 em]x &= -42&& \\[1 em]x &= -42&& \\[1 em] \end{aligned} $$

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