$$ \begin{aligned} 10x-3 &= \frac{1}{2}(15x+8)&& \text{simplify right side} \\[1 em]10x-3 &= \frac{15x+8}{2}&& \text{multiply ALL terms by } \color{blue}{ 2 }. \\[1 em]2\cdot10x-2\cdot3 &= 2 \cdot \frac{15x+8}{2}&& \text{cancel out the denominators} \\[1 em]20x-6 &= 15x+8&& \text{move the $ \color{blue}{ 15x } $ to the left side and $ \color{blue}{ -6 }$ to the right} \\[1 em]20x-15x &= 8+6&& \text{simplify left and right hand side} \\[1 em]5x &= 14&& \text{ divide both sides by $ 5 $ } \\[1 em]x &= \frac{14}{5}&& \\[1 em] \end{aligned} $$

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