1. **State the problem:** Solve the equation $$\frac{x - 5}{4} = \frac{x - 9}{10}$$ for $x$. 2. **Formula and rules:** To solve an equation with fractions, we can cross-multiply to eliminate the denominators. Cross-multiplication rule: $$\frac{a}{b} = \frac{c}{d} \implies ad = bc$$. 3. **Apply cross-multiplication:** $$ (x - 5) \times 10 = (x - 9) \times 4 $$ 4. **Expand both sides:** $$ 10x - 50 = 4x - 36 $$ 5. **Isolate $x$ terms on one side:** Subtract $4x$ from both sides: $$ 10x - 4x - 50 = -36 $$ $$ 6x - 50 = -36 $$ 6. **Isolate the constant term:** Add $50$ to both sides: $$ 6x = -36 + 50 $$ $$ 6x = 14 $$ 7. **Solve for $x$:** Divide both sides by $6$: $$ x = \frac{14}{6} = \frac{7}{3} $$ **Final answer:** $$x = \frac{7}{3}$$