A line ell is defined by the equation \(\mathrm{y - 8 = 4(x - 1)}\).A point P lies on ell...

1. TRANSLATE the problem information

• Given information:
  • Line ℓ has equation: \(\mathrm{y - 8 = 4(x - 1)}\)
  • Point P lies on line ℓ
  • Point P has y-coordinate 24
  • Need to find: x-coordinate of P

2. INFER the key strategy

• Since P lies on line ℓ, the coordinates of P must satisfy the line's equation
• We know \(\mathrm{y = 24}\) for point P, so we can substitute this value to find x

3. TRANSLATE and substitute

• Replace y with 24 in the equation:
  \(\mathrm{24 - 8 = 4(x - 1)}\)

4. SIMPLIFY through algebraic steps

• Simplify the left side:
  \(\mathrm{16 = 4(x - 1)}\)
• Divide both sides by 4:
  \(\mathrm{4 = x - 1}\)
• Add 1 to both sides:
  \(\mathrm{x = 5}\)

Answer: C (5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students don't connect that "P lies on ℓ" means P's coordinates must satisfy the line equation. They might try to work with the equation algebraically without using the given y-coordinate, or they might not understand what the problem is asking for.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students make computational mistakes when solving \(\mathrm{24 - 8 = 4(x - 1)}\). For example, they might incorrectly distribute the 4 or make arithmetic errors when isolating x.

Common algebraic mistakes could lead them to select Choice A (3) or Choice B (4) depending on the specific error made.

The Bottom Line:

This problem tests whether students understand the fundamental relationship between points and the lines they lie on, combined with basic algebraic manipulation skills.