$$ \begin{aligned} \frac{13}{x+2} &= \frac{8}{x-1}&& \text{multiply ALL terms by } \color{blue}{ (x+2)(x-1) }. \\[1 em](x+2)(x-1)\cdot\frac{13}{x+2} &= (x+2)(x-1)\cdot\frac{8}{x-1}&& \text{cancel out the denominators} \\[1 em]13x-13 &= 8x+16&& \text{move the $ \color{blue}{ 8x } $ to the left side and $ \color{blue}{ -13 }$ to the right} \\[1 em]13x-8x &= 16+13&& \text{simplify left and right hand side} \\[1 em]5x &= 29&& \text{ divide both sides by $ 5 $ } \\[1 em]x &= \frac{29}{5}&& \\[1 em] \end{aligned} $$

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