1. **State the problem:** Solve the equation $$\frac{x-2}{5} = \frac{x}{3}$$ for $x$. 2. **Use the cross-multiplication method:** When two fractions are equal, their cross products are equal. So, $$3(x-2) = 5x$$ 3. **Distribute the 3 on the left side:** $$3x - 6 = 5x$$ 4. **Bring all terms involving $x$ to one side:** $$3x - 5x = 6$$ 5. **Simplify the left side:** $$\cancel{3x} - 5x = 6 \Rightarrow -2x = 6$$ 6. **Divide both sides by $-2$ to solve for $x$:** $$x = \frac{6}{\cancel{-2}} \Rightarrow x = -3$$ **Final answer:** $$x = -3$$