To solve the equation
\[ \frac{4x}{7} - 3 = x, \]
first, we'll isolate the term with \(x\). We can do this by adding 3 to both sides of the equation:
\[ \frac{4x}{7} = x + 3. \]
Next, to eliminate the fraction, multiply both sides by 7:
\[ 7 \cdot \frac{4x}{7} = 7(x + 3). \]
This simplifies to:
\[ 4x = 7x + 21. \]
Now, we want to isolate \(x\). First, we can subtract \(7x\) from both sides:
\[ 4x - 7x = 21, \]
which simplifies to:
\[ -3x = 21. \]
Now, divide both sides by -3:
\[ x = \frac{21}{-3} = -7. \]
Now, we have \(x = -7\). Let's check our answer by substituting \(x\) back into the original equation:
\[ \frac{4(-7)}{7} - 3 = -7. \]
Calculating the left side:
\[ \frac{-28}{7} - 3 = -4 - 3 = -7. \]
Since both sides are equal, the solution is verified. Therefore, the solution to the equation is:
\[ \boxed{-7}. \]