1. **State the problem:** Solve for $x$ in the equation $$\frac{6}{x+5} = \frac{7}{x-7}.$$\n\n2. **Use the cross-multiplication rule:** When two fractions are equal, their cross products are equal. So, $$6(x-7) = 7(x+5).$$\n\n3. **Expand both sides:** $$6x - 42 = 7x + 35.$$\n\n4. **Bring all terms involving $x$ to one side and constants to the other:** $$6x - 7x = 35 + 42.$$\n\n5. **Simplify both sides:** $$\cancel{6x} - 7x = 77 + \cancel{42} \implies -x = 77.$$\n\n6. **Divide both sides by $-1$ to solve for $x$:** $$x = \frac{-77}{-1} = 77.$$\n\n7. **Final answer:** $$x = 77.$$