$$\frac{x+5}{7}+\frac{x+3}{8} = 1$$

$$ \begin{aligned} \frac{x+5}{7}+\frac{x+3}{8} &= 1&& \text{multiply ALL terms by } \color{blue}{ 56 }. \\[1 em]56 \cdot \frac{x+5}{7}+56\frac{x+3}{8} &= 56\cdot1&& \text{cancel out the denominators} \\[1 em]8x+40+7x+21 &= 56&& \text{simplify left side} \\[1 em]15x+61 &= 56&& \text{ move 61 to the right } \\[1 em]15x &= 56-61&& \\[1 em]15x &= -5&& \text{ divide both sides by $ 15 $ } \\[1 em]x &= -\frac{5}{15}&& \\[1 em]x &= -\frac{1}{3}&& \\[1 em] \end{aligned} $$

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