1. **State the problem:** Solve for $x$ in the proportion $\frac{x}{1} = \frac{4}{7}$. 2. **Formula used:** In a proportion $\frac{a}{b} = \frac{c}{d}$, the cross products are equal, so $a \times d = b \times c$. 3. **Apply the cross multiplication:** Here, $x \times 7 = 1 \times 4$. 4. **Simplify the equation:** $$7x = 4$$ 5. **Solve for $x$ by dividing both sides by 7:** $$x = \frac{4}{7}$$ 6. **Show cancellation step:** $$x = \frac{\cancel{4}}{\cancel{7}}$$ (No common factors to cancel, so fraction remains $\frac{4}{7}$.) 7. **Final answer:** $x = \frac{4}{7}$