To solve the equation \( 5^{3x} = 5^{7x - 2} \), we can use the property of exponents that states if \( a^m = a^n \), then \( m = n \) when the base \( a \) is the same (and \( a \neq 0 \)).
In this case, the base is \( 5 \). Therefore, we can set the exponents equal to each other:
\[ 3x = 7x - 2 \]
Now, let's solve for \( x \):
- Subtract \( 7x \) from both sides:
\[ 3x - 7x = -2 \]
- This simplifies to:
\[ -4x = -2 \]
- Now, divide both sides by \( -4 \):
\[ x = \frac{-2}{-4} = \frac{1}{2} \]
Thus, the solution is:
\[ \boxed{\frac{1}{2}} \]