1. **State the problem:** Solve for $b$ in the equation $$\sqrt{19b - 13} = \sqrt{19 + 11b}.$$\n\n2. **Formula and rules:** To solve equations involving square roots, we can square both sides to eliminate the square roots, but we must check for extraneous solutions afterwards.\n\n3. **Square both sides:**\n$$\left(\sqrt{19b - 13}\right)^2 = \left(\sqrt{19 + 11b}\right)^2$$\nwhich simplifies to\n$$19b - 13 = 19 + 11b.$$\n\n4. **Isolate $b$ terms:**\n$$19b - 11b = 19 + 13$$\n$$8b = 32.$$\n\n5. **Solve for $b$:**\n$$b = \frac{32}{8} = 4.$$\n\n6. **Check for extraneous solutions:** Substitute $b=4$ back into the original equation:\n$$\sqrt{19(4) - 13} = \sqrt{19 + 11(4)}$$\n$$\sqrt{76 - 13} = \sqrt{19 + 44}$$\n$$\sqrt{63} = \sqrt{63}$$\nBoth sides are equal, so $b=4$ is a valid solution.