1. Stating the problem: Simplify and solve the equation $$\frac{0.3x^{0.7}y^{-0.7}}{0.7x^{-0.3}y^{0.3}}=4$$ for $x$ and $y$. 2. Use the property of exponents for division: $$\frac{a^m}{a^n} = a^{m-n}$$ and simplify constants. 3. Simplify the constants: $$\frac{0.3}{0.7} = \frac{3}{7}$$. 4. Simplify the powers of $x$: $$x^{0.7 - (-0.3)} = x^{0.7 + 0.3} = x^{1}$$. 5. Simplify the powers of $y$: $$y^{-0.7 - 0.3} = y^{-1}$$. 6. Substitute back: $$\frac{3}{7} x^{1} y^{-1} = 4$$. 7. Rewrite $y^{-1}$ as $\frac{1}{y}$: $$\frac{3}{7} \cdot x \cdot \frac{1}{y} = 4$$. 8. Multiply both sides by $y$ to isolate $x$: $$\frac{3}{7} x = 4y$$. 9. Multiply both sides by $\frac{7}{3}$ to isolate $x$: $$x = 4y \cdot \frac{7}{3}$$. 10. Simplify: $$x = \frac{28}{3} y$$. Final answer: $$x = \frac{28}{3} y$$.