To solve the equation \(\frac{x-7}{x} = \frac{49}{x(x-7)} + \frac{3x}{x-7}\), follow these steps:

1. Find a common denominator for all the fractions. The common denominator is \(x(x-7)\).

2. Rewrite each term with the common denominator:
\[
\frac{x-7}{x} = \frac{(x-7)^2}{x(x-7)}
\]
\[
\frac{49}{x(x-7)} = \frac{49}{x(x-7)}
\]
\[
\frac{3x}{x-7} = \frac{3x^2}{x(x-7)}
\]

3. Combine all the fractions:
\[
\frac{(x-7)^2}{x(x-7)} = \frac{49 + 3x^2}{x(x-7)}
\]

4. Since the denominators are the same, set the numerators equal to each other:
\[
(x-7)^2 = 49 + 3x^2
\]

5. Expand and simplify the equation:
\[
x^2 - 14x + 49 = 49 + 3x^2
\]

6. Subtract \(49\) from both sides:
\[
x^2 - 14x = 3x^2
\]

7. Move all terms to one side to set the equation to zero:
\[
x^2 - 14x - 3x^2 = 0
\]
\[
-2x^2 - 14x = 0
\]

8. Factor out the common term \(-2x\):
\[
-2x(x + 7) = 0
\]

9. Set each factor to zero and solve for \(x\):
\[
-2x = 0 \quad \Rightarrow \quad x = 0
\]
\[
x + 7 = 0 \quad \Rightarrow \quad x = -7
\]

10. Check for any extraneous solutions by substituting back into the original equation. Since \(x = 0\) would make the original equation undefined (division by zero), it is not a valid solution.

Therefore, the solution is:
\[
x = -7
\]