In the xy-plane, line ℓ passes through the point \(\mathrm{(0, 0)}\) and is parallel to the line represented by the...

1. TRANSLATE the problem information

• Given information:
  • Line \(\ell\) passes through \((0, 0)\)
  • Line \(\ell\) is parallel to \(\mathrm{y = 8x + 2}\)
  • Line \(\ell\) also passes through \((3, d)\)
  • Need to find \(\mathrm{d}\)

2. INFER the equation of line \(\ell\)

• Since parallel lines have the same slope, line \(\ell\) has slope \(\mathrm{m = 8}\)
• Since line \(\ell\) passes through \((0, 0)\), the y-intercept is \(\mathrm{b = 0}\)
• Therefore: \(\mathrm{y = 8x + 0 = 8x}\)

3. SIMPLIFY by substituting the known point

• Line \(\ell\) passes through \((3, d)\), so when \(\mathrm{x = 3}\), \(\mathrm{y = d}\)
• Substitute into \(\mathrm{y = 8x}\):
  \(\mathrm{d = 8(3) = 24}\)

Answer: 24

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might confuse "parallel to \(\mathrm{y = 8x + 2}\)" and think line \(\ell\) has the same equation, forgetting that it passes through different points.

They might incorrectly use \(\mathrm{y = 8x + 2}\) for line \(\ell\), then substitute \((3, d)\):
\(\mathrm{d = 8(3) + 2 = 26}\)

This leads them to an incorrect answer of 26 instead of 24.

Second Most Common Error:

Missing conceptual knowledge about y-intercept: Students might not recognize that passing through \((0, 0)\) means the y-intercept is 0.

They might try to find the y-intercept by substituting both points into \(\mathrm{y = 8x + b}\), creating unnecessary complexity and potential calculation errors.

This causes confusion and may lead to guessing or abandoning systematic solution.

The Bottom Line:

This problem tests understanding of parallel lines and the connection between points and linear equations. The key insight is recognizing that "parallel" means same slope, while "passes through" determines the specific equation.