1. **Stating the problem:** Simplify the expression $$\frac{a^2 - ab}{ab - b^2}$$ and solve the equation $$\frac{1}{r}x - \frac{1}{m} = \frac{\omega}{y}$$. 2. **Simplify the fraction:** Start with $$\frac{a^2 - ab}{ab - b^2}$$. 3. **Factor numerator and denominator:** Numerator: $$a^2 - ab = a(a - b)$$. Denominator: $$ab - b^2 = b(a - b)$$. 4. **Rewrite the fraction:** $$\frac{a(a - b)}{b(a - b)}$$. 5. **Cancel common factor:** Since $$a - b \neq 0$$, cancel $$a - b$$: $$\frac{a}{b}$$. 6. **Final simplified form:** $$\frac{a}{b}$$. 7. **Solve the equation:** Given $$\frac{1}{r}x - \frac{1}{m} = \frac{\omega}{y}$$, solve for $$x$$. 8. **Isolate $$x$$:** Add $$\frac{1}{m}$$ to both sides: $$\frac{1}{r}x = \frac{\omega}{y} + \frac{1}{m}$$. 9. **Multiply both sides by $$r$$:** $$x = r\left(\frac{\omega}{y} + \frac{1}{m}\right) = \frac{r\omega}{y} + \frac{r}{m}$$. **Final answers:** - Simplified fraction: $$\frac{a}{b}$$ - Expression for $$x$$: $$x = \frac{r\omega}{y} + \frac{r}{m}$$