1. **State the problem:** Solve the equation $$\frac{10}{x+8} = \frac{5}{x-8}$$ for $x$. 2. **Use the cross-multiplication formula:** For an equation of the form $$\frac{a}{b} = \frac{c}{d},$$ cross-multiply to get $$a \cdot d = b \cdot c.$$ This eliminates the fractions. 3. **Apply cross-multiplication:** $$10 \cdot (x - 8) = 5 \cdot (x + 8)$$ 4. **Expand both sides:** $$10x - 80 = 5x + 40$$ 5. **Bring all terms involving $x$ to one side and constants to the other:** $$10x - 5x = 40 + 80$$ 6. **Simplify:** $$5x = 120$$ 7. **Divide both sides by 5 to solve for $x$:** $$x = \frac{120}{5}$$ $$x = \cancel{\frac{120}{5}} 24$$ 8. **Final answer:** $$x = 24$$ This value satisfies the original equation as long as it does not make any denominator zero (which it does not).