1. **State the problem:** Solve the equation $$\frac{10x}{25} = \frac{19}{15}$$ for $x$. 2. **Use the cross-multiplication method:** When two fractions are equal, their cross products are equal. So, $$10x \times 15 = 19 \times 25$$ 3. **Calculate the products:** $$150x = 475$$ 4. **Isolate $x$ by dividing both sides by 150:** $$x = \frac{475}{150}$$ 5. **Simplify the fraction:** $$x = \frac{\cancel{475}^{\times 19}}{\cancel{150}^{\times 6 \frac{1}{4}}}$$ Since 475 and 150 share a common factor 25, $$x = \frac{19}{6 \frac{1}{4}} = \frac{19}{6.25}$$ 6. **Convert to improper fraction or decimal:** $$6 \frac{1}{4} = \frac{25}{4}$$ So, $$x = \frac{19}{\frac{25}{4}} = 19 \times \frac{4}{25} = \frac{76}{25}$$ 7. **Final answer:** $$x = \frac{76}{25}$$ or approximately $$3.04$$.