In the xy-plane, line m passes through the points \((1, -6)\) and \((5, 2)\). What is the x-coordinate of the...

1. TRANSLATE the problem requirements

• Given information:
  • Line m passes through points (1, -6) and (5, 2)
  • Need to find x-coordinate of x-intercept

• What this tells us: We need to find where the line crosses the x-axis (where y = 0)

2. INFER the solution strategy

• To find the x-intercept, we need the equation of the line first
• Strategy: Find slope → Write line equation → Set y = 0 and solve for x

3. SIMPLIFY to find the slope

• Using slope formula with points (1, -6) and (5, 2):
  \(\mathrm{m = \frac{2 - (-6)}{5 - 1}}\)
  \(\mathrm{m = \frac{8}{4}}\)
  \(\mathrm{m = 2}\)

4. SIMPLIFY to find the line equation

• Using point-slope form with point (1, -6) and slope m = 2:
  \(\mathrm{y - (-6) = 2(x - 1)}\)
  \(\mathrm{y + 6 = 2x - 2}\)
  \(\mathrm{y = 2x - 8}\)

5. INFER and SIMPLIFY to find the x-intercept

• Set y = 0 (definition of x-intercept):
  \(\mathrm{0 = 2x - 8}\)
  \(\mathrm{2x = 8}\)
  \(\mathrm{x = 4}\)

Answer: D) 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misunderstand what "x-coordinate of the x-intercept" means and try to find where x = 0 instead of where y = 0. This leads them to substitute x = 0 into their line equation \(\mathrm{y = 2x - 8}\), getting y = -8, and then incorrectly thinking the answer is -8. Since -8 isn't an answer choice, this leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly find the slope and set up the equation, but make algebraic errors when solving \(\mathrm{0 = 2x - 8}\). Common mistakes include forgetting to add 8 to both sides or dividing incorrectly, leading to wrong values that might match incorrect answer choices.

The Bottom Line:

This problem requires clear understanding of intercept terminology and systematic algebraic manipulation. Students who rush through the algebra or misinterpret the x-intercept concept will struggle to reach the correct answer.