1. **State the problem:** Solve for $x$ in the proportion $\frac{7x+1}{2x+3} = \frac{10}{3}$. 2. **Formula and rule:** When two ratios are equal, their cross products are equal. So, $$(7x+1) \times 3 = (2x+3) \times 10$$ 3. **Cross multiply:** $$3(7x+1) = 10(2x+3)$$ 4. **Expand both sides:** $$21x + 3 = 20x + 30$$ 5. **Isolate $x$:** Subtract $20x$ from both sides: $$21x - 20x + 3 = 30$$ $$x + 3 = 30$$ 6. **Solve for $x$:** Subtract 3 from both sides: $$x = 30 - 3$$ $$x = 27$$ **Final answer:** $x = 27$