Question:The table gives the coordinates of two points on a line in the xy-plane. The x-intercept of the line is...

1. TRANSLATE the problem information

• Given information:
  • Two points on a line: \((5, \mathrm{p})\) and \((-3, \mathrm{p+4})\)
  • X-intercept: \((\mathrm{q}, \mathrm{p-8})\)
  • Need to find: q
• What this tells us: We have three collinear points, with one being the x-intercept

2. INFER the key insight about x-intercepts

• Since the x-intercept is where the line crosses the x-axis, its y-coordinate must be 0
• This means: \(\mathrm{p - 8} = 0\), so \(\mathrm{p} = 8\)

3. TRANSLATE the points with known p-value

• With \(\mathrm{p} = 8\), our points become:
  • Point 1: \((5, 8)\)
  • Point 2: \((-3, 12)\)
  • X-intercept: \((\mathrm{q}, 0)\)

4. SIMPLIFY to find the slope

• Calculate slope between the two given points:
  \(\mathrm{m} = \frac{12 - 8}{-3 - 5}\)
  \(\mathrm{m} = \frac{4}{-8}\)
  \(\mathrm{m} = -\frac{1}{2}\)

5. INFER using the collinear property

• Since all three points lie on the same line, the slope between any pair must be \(-\frac{1}{2}\)
• Using points \((5, 8)\) and \((\mathrm{q}, 0)\):
  \(\frac{0 - 8}{\mathrm{q} - 5} = -\frac{1}{2}\)

6. SIMPLIFY the equation to solve for q

• \(\frac{-8}{\mathrm{q} - 5} = -\frac{1}{2}\)
• Cross multiply: \(-8 = -\frac{1}{2}(\mathrm{q} - 5)\)
• \(-8 = -\frac{1}{2} \cdot \mathrm{q} + \frac{5}{2}\)
• \(-16 = -\mathrm{q} + 5\)
• \(-21 = -\mathrm{q}\)
• \(\mathrm{q} = 21\)

Answer: 21

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge about x-intercepts: Students may not recognize that the x-intercept has a y-coordinate of 0, failing to establish that \(\mathrm{p} - 8 = 0\).

Without this insight, they can't determine \(\mathrm{p} = 8\), making it impossible to work with concrete coordinate values. This leads to confusion about how to use the given information and often results in guessing.

Second Most Common Error:

Weak INFER skill: Students may correctly find \(\mathrm{p} = 8\) but fail to recognize that all three points must be collinear with the same slope.

Instead, they might try to use just two points to write an equation and then struggle with how to incorporate the third point. This incomplete understanding of the relationship between the points leads to getting stuck partway through the solution.

The Bottom Line:

This problem tests whether students understand what an x-intercept represents and can apply the fundamental property that points on the same line have consistent slope relationships. The key breakthrough is recognizing that the y-coordinate condition gives you the missing parameter immediately.