Solution: We are given a point $ (3, 11) $ lies on the line $ y = 2x + b $. Substitute $ x = 3 $ and $ y = 11 $:

["### How to Find the Value of $ b $ in the Line $ y = 2x + b $ Using the Point $ (3, 11) $", "When working with linear equations, one key skill is determining the missing constant $ b $ in the equation when a point lies on the line. This is especially useful in algebra, geometry, and real-world applications. Let’s explore how to solve for $ b $ when the point $ (3, 11) $ lies on the line $ y = 2x + b $.", "#### Step-by-Step Solution", "1. Start with the equation of the line\n We are given the line:\n $$ y = 2x + b $$\n Here, $ 2x $ reflects the slope, and $ b $ is the y-intercept we need to find.", "2. Substitute the known point $ (x = 3,\ y = 11) $ into the equation\n Replace $ x $ with 3 and $ y $ with 11:\n $$ 11 = 2(3) + b $$", "3. Simplify the equation\n Multiply:\n $$ 11 = 6 + b $$", "4. Solve for $ b $\n Subtract 6 from both sides:\n $$ b = 11 - 6 $$\n $$ b = 5 $$", "#### Final Result\nThe value of $ b $ that makes the line $ y = 2x + b $ pass through $ (3, 11) $ is $ 5 $.", "Thus, the complete line is:\n$$ y = 2x + 5 $$", "#### Why This Matters\nUnderstanding how to derive the y-intercept from a point and slope equation is essential for graphing lines, solving systems of equations, and modeling linear relationships in science and finance. By substituting known values and isolating $ b $, you build a strong foundation in algebraic problem-solving.", "Key Takeaways:\n- Replace $ x $ and $ y $ with the given point.\n- Simplify the equation by substitution and arithmetic.\n- Solve algebraically for the unknown constant.", "Use this method anytime you need to determine $ b $ in a linear equation given a point on the line!"]