In the xy-plane, line ell has y-intercept -{8}. The point \((3, 7)\) lies on ell. What is the x-intercept of...

1. TRANSLATE the problem information

• Given information:
  • y-intercept = \(-8\) (line passes through point \((0, -8)\))
  • Line passes through point \((3, 7)\)
  • Need to find: x-intercept

2. INFER the approach

• Since we know the y-intercept and one other point, we can find the slope and write the complete equation
• Once we have the equation, set \(\mathrm{y = 0}\) to find where the line crosses the x-axis

3. SIMPLIFY to find the slope

• Using slope-intercept form \(\mathrm{y = mx + b}\) with \(\mathrm{b = -8}\):
• Substitute point \((3, 7)\): \(\mathrm{7 = m(3) - 8}\)
• Solve:
  \(\mathrm{7 = 3m - 8}\)
  \(\mathrm{15 = 3m}\)
  \(\mathrm{m = 5}\)

4. INFER the complete equation

• With slope \(\mathrm{m = 5}\) and y-intercept \(\mathrm{b = -8}\):
• Line equation: \(\mathrm{y = 5x - 8}\)

5. SIMPLIFY to find the x-intercept

• Set \(\mathrm{y = 0}\): \(\mathrm{0 = 5x - 8}\)
• Solve:
  \(\mathrm{5x = 8}\)
  \(\mathrm{x = \frac{8}{5}}\)
• X-intercept: \(\mathrm{(\frac{8}{5}, 0)}\)

Answer: C. \(\mathrm{(\frac{8}{5}, 0)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may confuse which intercept they're given versus which they need to find, or incorrectly interpret "y-intercept -8" as the point \((-8, 0)\) instead of \((0, -8)\).

This fundamental misunderstanding of intercept definitions derails the entire solution process. They might try to use \((-8, 0)\) and \((3, 7)\) to find the slope, getting \(\mathrm{m = \frac{7}{11}}\), leading to a completely wrong equation and ultimately selecting an incorrect answer choice or becoming confused.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the equations but make algebraic errors when solving \(\mathrm{7 = 3m - 8}\) or \(\mathrm{0 = 5x - 8}\).

For example, they might forget to add 8 to both sides in the first step, getting \(\mathrm{m = \frac{7}{3}}\) instead of \(\mathrm{m = 5}\). This leads to the wrong equation \(\mathrm{y = \frac{7}{3}x - 8}\), and subsequently the wrong x-intercept. Such calculation errors often lead to confusion when their answer doesn't match any of the given choices.

The Bottom Line:

This problem requires solid understanding of intercept definitions and careful algebraic manipulation. Students who rush through the setup or make sign errors in their algebra will struggle to reach the correct answer.