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: \(\mathrm{(m, 15)}\) and \(\mathrm{(m + 6, -3)}\)
  • The x-intercept has form \(\mathrm{(b, m - 25)}\)
  • Need to find the value of b

2. INFER the solution approach

• To find b, I need to determine the actual x-intercept of the line
• This requires: find slope → write line equation → set \(\mathrm{y = 0}\) → solve for x-intercept
• Then match the x-intercept with the given form \(\mathrm{(b, m - 25)}\)

3. SIMPLIFY to find the slope

• Using slope formula with points \(\mathrm{(m, 15)}\) and \(\mathrm{(m + 6, -3)}\):
  \(\mathrm{slope = \frac{-3 - 15}{(m + 6) - m}}\)
  \(\mathrm{= \frac{-18}{6}}\)
  \(\mathrm{= -3}\)

4. INFER the line equation using point-slope form

• Using point \(\mathrm{(m, 15)}\) and slope \(\mathrm{-3}\):
  \(\mathrm{y - 15 = -3(x - m)}\)
  \(\mathrm{y - 15 = -3x + 3m}\)
  \(\mathrm{y = -3x + 3m + 15}\)

5. SIMPLIFY to find the x-intercept

• Set \(\mathrm{y = 0}\) to find where line crosses x-axis:
  \(\mathrm{0 = -3x + 3m + 15}\)
  \(\mathrm{3x = 3m + 15}\)
  \(\mathrm{x = m + 5}\)
• So the x-intercept is \(\mathrm{(m + 5, 0)}\)

6. INFER how to use the given x-intercept form

• Given: x-intercept is \(\mathrm{(b, m - 25)}\)
• Since x-intercepts always have y-coordinate = 0: \(\mathrm{m - 25 = 0}\)
• Therefore: \(\mathrm{m = 25}\)

7. SIMPLIFY to find b

• Since x-intercept is \(\mathrm{(m + 5, 0)}\) and \(\mathrm{m = 25}\):
  \(\mathrm{b = m + 5}\)
  \(\mathrm{= 25 + 5}\)
  \(\mathrm{= 30}\)

Answer: 30

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that x-intercepts must have y-coordinate equal to 0, so they try to solve a more complex system thinking both coordinates of \(\mathrm{(b, m - 25)}\) are unknowns. They might attempt to substitute the x-intercept \(\mathrm{x = m + 5}\) directly into the original line equation without using the constraint that \(\mathrm{y = 0}\) at the x-intercept. This leads to confusion and often guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors when solving \(\mathrm{0 = -3x + 3m + 15}\), perhaps getting \(\mathrm{x = m - 5}\) instead of \(\mathrm{x = m + 5}\), or make sign errors in the calculation. This leads them to get \(\mathrm{b = 20}\) instead of 30, causing them to select an incorrect answer if such an option exists.

The Bottom Line:

This problem tests whether students understand the fundamental property that x-intercepts have y-coordinate 0, and whether they can work systematically through the multi-step process of finding a line equation and then its intercept.