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

1. TRANSLATE the problem information

• Given information:
  • Line s passes through point \((0, 0)\)
  • Line s is parallel to the line \(\mathrm{y = 18x + 2}\)
  • Line s also passes through point \((4, \mathrm{d})\)
  • Need to find the value of d

2. INFER what parallel lines tell us

• Since line s is parallel to \(\mathrm{y = 18x + 2}\), they have the same slope
• The slope of \(\mathrm{y = 18x + 2}\) is 18, so line s also has \(\mathrm{slope = 18}\)

3. INFER the equation of line s

• Line s has \(\mathrm{slope = 18}\) and passes through \((0, 0)\)
• Since it passes through the origin, the y-intercept is 0
• Therefore: \(\mathrm{y = 18x + 0}\), which simplifies to \(\mathrm{y = 18x}\)

4. INFER how to find d using the second point

• Since line s passes through \((4, \mathrm{d})\), this point must satisfy the equation \(\mathrm{y = 18x}\)
• Substitute \(\mathrm{x = 4}\) and \(\mathrm{y = d}\): \(\mathrm{d = 18(4)}\)

5. SIMPLIFY to find the final answer

• \(\mathrm{d = 18(4) = 72}\)

Answer: C. 72

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't realize they need to derive a new equation for line s. Instead, they think "parallel to \(\mathrm{y = 18x + 2}\)" means they can use that equation directly.

They substitute \((4, \mathrm{d})\) into \(\mathrm{y = 18x + 2}\):
\(\mathrm{d = 18(4) + 2}\)
\(\mathrm{= 72 + 2}\)
\(\mathrm{= 74}\)

This leads them to select Choice D (74).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what "passes through \((0, 0)\)" means for the equation structure, or they don't fully grasp that parallel lines have identical slopes but potentially different y-intercepts.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students can distinguish between "parallel to a line" (same slope, potentially different y-intercept) versus "identical to a line" (same equation). The key insight is building the new equation \(\mathrm{y = 18x}\) specifically for line s, rather than borrowing the given equation \(\mathrm{y = 18x + 2}\).