Let's break down and solve the given equation step by step.

The equation given is:

\[ 7^2 \cdot x - 5 = 7^7 \cdot 497^2 \cdot x - 5 = 7^7 \cdot 49 \]

First, let's simplify each part of the equation.

1. Simplify \( 7^2 \cdot x - 5 \):
\[ 7^2 = 49 \]
So,
\[ 49 \cdot x - 5 \]

2. Simplify \( 7^7 \cdot 497^2 \cdot x - 5 \):
\[ 497^2 = 497 \cdot 497 \]
This is a large number, but we can leave it as \( 497^2 \) for now.

So,
\[ 7^7 \cdot 497^2 \cdot x - 5 \]

3. Simplify \( 7^7 \cdot 49 \):
\[ 7^7 \cdot 49 \]

Now, let's rewrite the equation with these simplified parts:

\[ 49 \cdot x - 5 = 7^7 \cdot 497^2 \cdot x - 5 = 7^7 \cdot 49 \]

Since the equation has two equalities, we can break it down into two separate equations and solve them individually:

### Equation 1:
\[ 49 \cdot x - 5 = 7^7 \cdot 497^2 \cdot x - 5 \]

Subtract \(-5\) from both sides:
\[ 49 \cdot x = 7^7 \cdot 497^2 \cdot x \]

Since \( x \) is common on both sides, we can divide both sides by \( x \) (assuming \( x \neq 0 \)):
\[ 49 = 7^7 \cdot 497^2 \]

This equation is not true because \( 49 \neq 7^7 \cdot 497^2 \). Therefore, there is no solution for \( x \) that satisfies this equation.

### Equation 2:
\[ 7^7 \cdot 497^2 \cdot x - 5 = 7^7 \cdot 49 \]

Subtract \(-5\) from both sides:
\[ 7^7 \cdot 497^2 \cdot x = 7^7 \cdot 49 \]

Divide both sides by \( 7^7 \):
\[ 497^2 \cdot x = 49 \]

Divide both sides by \( 497^2 \):
\[ x = \frac{49}{497^2} \]

So, the solution for \( x \) is:
\[ x = \frac{49}{497^2} \]

In conclusion, the only solution for \( x \) that satisfies the given equation is:
\[ x = \frac{49}{497^2} \]