1. **State the problem:** Solve for $x$ in the equation $$7^{-6x} = 49^{3 - 2x}.$$\n\n2. **Rewrite bases as powers of the same base:** Note that $49 = 7^2$, so rewrite the right side as $$49^{3 - 2x} = (7^2)^{3 - 2x} = 7^{2(3 - 2x)} = 7^{6 - 4x}.$$\n\n3. **Rewrite the equation with the same base:** $$7^{-6x} = 7^{6 - 4x}.$$\n\n4. **Set the exponents equal:** Since the bases are the same and nonzero, the exponents must be equal: $$-6x = 6 - 4x.$$\n\n5. **Solve for $x$:**\n$$-6x = 6 - 4x$$\nAdd $4x$ to both sides:\n$$-6x + 4x = 6$$\n$$\cancel{-6x} + \cancel{4x} = 6$$\n$$-2x = 6$$\nDivide both sides by $-2$:\n$$x = \frac{6}{-2} = -3.$$\n\n**Final answer:** $$x = -3.$$