Find the Line Equation: Parallel to 2(y+1)=16x Through Point (0,1)

To solve for the equation of the line parallel to $2(y+1)=16x$ that passes through the point $(0,1)$, follow these steps:

• Step 1: Convert the given line's equation to slope-intercept form. Start with $2(y+1) = 16x$.
• Step 2: Distribute and simplify to get $2y + 2 = 16x$, which rearranges to $2y = 16x - 2$.
• Step 3: Solve for $y$ by dividing everything by 2: $y = 8x - 1$. Hence, the slope $m$ is 8.
• Step 4: Use the slope of 8 and the point $(0,1)$ in the point-slope form of a line equation:
  $\quad y - y_1 = m(x - x_1)$
• Step 5: Substitute the point $(0,1)$ and the slope $8$:
  $\quad y - 1 = 8(x - 0)$
• Step 6: Simplify the expression
  $\quad y - 1 = 8x$
• Step 7: Rewrite the equation to achieve the desired form by moving 1 to the other side:
  $\quad y = 8x + 1$
• Step 8: To match the expected form, rewrite
  $y - 8x = 1$

Therefore, the equation of the line that meets the criteria is given by $y - 8x = 1$, corresponding to choice 3.