Let's solve the given equation step-by-step. The equation is:

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

First, let's simplify the equation. Notice that the equation seems to have some formatting issues. Let's break it down and solve it in parts.

1. Simplify the left-hand side:
\[ 7^2 \cdot x - 5 \]

2. Simplify the right-hand side:
\[ 7^7 \cdot 497 \cdot x - 5 = 7^7 \cdot 49 \]

Let's rewrite the equation more clearly:

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

Now, let's solve for \( x \).

First, simplify the constants:

\[ 7^2 = 49 \]

So, the left-hand side becomes:
\[ 49x - 5 \]

For the right-hand side, we need to simplify \( 7^7 \cdot 497 \cdot x - 5 \) and \( 7^7 \cdot 49 \).

Notice that \( 49 = 7^2 \), so we can rewrite \( 7^7 \cdot 49 \) as:
\[ 7^7 \cdot 7^2 = 7^{7+2} = 7^9 \]

Now, let's rewrite the equation with these simplifications:

\[ 49x - 5 = 7^7 \cdot 497 \cdot x - 5 = 7^9 \]

Since the equation is a bit complex, let's assume that the equation should be split into two separate equations:

1. \( 49x - 5 = 7^9 \)
2. \( 7^7 \cdot 497 \cdot x - 5 = 7^9 \)

Let's solve each equation separately.

### Equation 1: \( 49x - 5 = 7^9 \)

First, calculate \( 7^9 \):

\[ 7^9 = 40353607 \]

So, the equation becomes:

\[ 49x - 5 = 40353607 \]

Add 5 to both sides:

\[ 49x = 40353612 \]

Divide by 49:

\[ x = \frac{40353612}{49} \]

\[ x = 823544 \]

### Equation 2: \( 7^7 \cdot 497 \cdot x - 5 = 7^9 \)

First, calculate \( 7^7 \):

\[ 7^7 = 823543 \]

So, the equation becomes:

\[ 823543 \cdot 497 \cdot x - 5 = 40353607 \]

Add 5 to both sides:

\[ 823543 \cdot 497 \cdot x = 40353612 \]

Divide by \( 823543 \cdot 497 \):

\[ x = \frac{40353612}{823543 \cdot 497} \]

\[ x = \frac{40353612}{409657471} \]

\[ x = \frac{1}{10} \]

So, we have two possible solutions for \( x \):

1. \( x = 823544 \)
2. \( x = \frac{1}{10} \)

However, it's important to verify the context and ensure the equation was interpreted correctly. If there are any errors in the interpretation, please provide additional details or corrections.