$$\frac{4}{x-7} = \frac{3}{x+1}$$

$$ \begin{aligned} \frac{4}{x-7} &= \frac{3}{x+1}&& \text{multiply ALL terms by } \color{blue}{ (x-7)(x+1) }. \\[1 em](x-7)(x+1)\cdot\frac{4}{x-7} &= (x-7)(x+1)\cdot\frac{3}{x+1}&& \text{cancel out the denominators} \\[1 em]4x+4 &= 3x-21&& \text{move the $ \color{blue}{ 3x } $ to the left side and $ \color{blue}{ 4 }$ to the right} \\[1 em]4x-3x &= -21-4&& \text{simplify left and right hand side} \\[1 em]x &= -25&& \\[1 em]x &= -25&& \\[1 em] \end{aligned} $$

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