1. **State the problem:** Solve for $x$ in the equation $$5^{3x - 7} = 3^{3x - 7}.$$\n\n2. **Understand the equation:** Both sides have the same exponent $3x - 7$, but different bases 5 and 3. Since 5 and 3 are different and positive numbers, the only way for the equation to hold true is if the exponents are equal or the bases are equal. The bases are not equal, so we consider the exponent.\n\n3. **Set the exponents equal:** Since the bases are different and the equation holds, the only way is if the exponent is zero, because any number to the zero power is 1. So, set $$3x - 7 = 0.$$\n\n4. **Solve for $x$:**\n$$3x - 7 = 0$$\n$$3x = 7$$\n$$x = \frac{7}{3}.$$\n\n5. **Check the solution:** Substitute $x = \frac{7}{3}$ back into the original equation:\n$$5^{3(\frac{7}{3}) - 7} = 5^{7 - 7} = 5^0 = 1,$$\n$$3^{3(\frac{7}{3}) - 7} = 3^{7 - 7} = 3^0 = 1.$$\nBoth sides equal 1, so the solution is correct.\n\n**Final answer:** $$x = \frac{7}{3}.$$