To solve the equation
\[ \frac{b}{19} - \frac{6b}{19} = \frac{10}{19}, \]
we can simplify the left-hand side. Since both fractions have the same denominator, we can combine them:
\[ \frac{b - 6b}{19} = \frac{10}{19}. \]
This simplifies to:
\[ \frac{-5b}{19} = \frac{10}{19}. \]
Next, we can eliminate the denominator by multiplying both sides of the equation by 19:
\[ -5b = 10. \]
Now, to solve for \( b \), we divide both sides by -5:
\[ b = \frac{10}{-5} = -2. \]
Now that we have found \( b = -2 \), let's check the solution by substituting \( b \) back into the original equation:
\[ \frac{-2}{19} - \frac{6(-2)}{19} = \frac{10}{19}. \]
Calculating the left-hand side:
\[ \frac{-2}{19} + \frac{12}{19} = \frac{-2 + 12}{19} = \frac{10}{19}. \]
Since the left-hand side equals the right-hand side, our solution checks out.
Thus, the final answer is
\[ \boxed{-2}. \]