$$ \begin{aligned} \frac{5}{3}+\frac{2}{3}x &= \frac{31}{12}+\frac{7}{4}x+\frac{1}{4}&& \text{multiply ALL terms by } \color{blue}{ 12 }. \\[1 em]12\cdot\frac{5}{3}+12\frac{2}{3}x &= 12\cdot\frac{31}{12}+12\frac{7}{4}x+12\cdot\frac{1}{4}&& \text{cancel out the denominators} \\[1 em]20+8x &= 31+21x+3&& \text{simplify left and right hand side} \\[1 em]8x+20 &= 21x+34&& \text{move the $ \color{blue}{ 21x } $ to the left side and $ \color{blue}{ 20 }$ to the right} \\[1 em]8x-21x &= 34-20&& \text{simplify left and right hand side} \\[1 em]-13x &= 14&& \text{ divide both sides by $ -13 $ } \\[1 em]x &= -\frac{14}{13}&& \\[1 em] \end{aligned} $$

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