A line passes through the point \((1, 8)\) and has a slope of -{2}. What is the x-intercept of the...

1. TRANSLATE the problem information

• Given information:
  • Point on line: \((1, 8)\)
  • Slope: \(\mathrm{m = -2}\)
  • Need to find: x-intercept (where line crosses x-axis)

• What this tells us: We need to find where \(\mathrm{y = 0}\)

2. INFER the approach

• To find the x-intercept, we need the line's equation first
• Then set \(\mathrm{y = 0}\) and solve for x
• Point-slope form works perfectly since we have a point and the slope

3. TRANSLATE into point-slope form

• Using \(\mathrm{y - y_1 = m(x - x_1)}\):
• \(\mathrm{y - 8 = -2(x - 1)}\)

4. INFER the next step to find x-intercept

• Set \(\mathrm{y = 0}\) (definition of x-intercept):
• \(\mathrm{0 - 8 = -2(x - 1)}\)

5. SIMPLIFY through algebraic steps

• \(\mathrm{-8 = -2(x - 1)}\)
• \(\mathrm{-8 = -2x + 2}\)
• \(\mathrm{-8 - 2 = -2x}\)
• \(\mathrm{-10 = -2x}\)
• \(\mathrm{x = 5}\)

6. TRANSLATE back to coordinate form

• The x-intercept is \((5, 0)\)

Answer: C \((5, 0)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse what "x-intercept" means and try to substitute \(\mathrm{x = 0}\) instead of \(\mathrm{y = 0}\).

This leads them to find: \(\mathrm{0 - 8 = -2(0 - 1)}\), which gives \(\mathrm{-8 = 2}\), resulting in confusion. They might then guess or incorrectly think the y-intercept \((0, 10)\) relates to the x-intercept somehow, possibly leading them to select Choice D \((10, 0)\).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{0 - 8 = -2(x - 1)}\) but make algebraic errors.

Common mistake: \(\mathrm{-8 = -2(x - 1)}\) becomes \(\mathrm{-8 = -2x - 2}\) (wrong sign on the 2), leading to \(\mathrm{-8 + 2 = -2x}\), so \(\mathrm{-6 = -2x}\), giving \(\mathrm{x = 3}\). Since \((3, 0)\) isn't an option, this causes confusion and guessing.

The Bottom Line:

This problem tests whether students truly understand what an x-intercept represents and can systematically work through the algebra without sign errors.