Line s in the xy-plane has a slope of 3 and an x-intercept of 2. Which equation defines line s?y...

1. TRANSLATE the problem information

• Given information:
  • Slope = 3
  • x-intercept = 2

• TRANSLATE what x-intercept means: The line crosses the x-axis at \(\mathrm{x = 2}\), so the point \(\mathrm{(2, 0)}\) is on our line

2. INFER the solution strategy

• We know one point \(\mathrm{(2, 0)}\) and the slope \(\mathrm{m = 3}\)
• Strategy: Use slope-intercept form \(\mathrm{y = mx + b}\) and substitute our known point to find \(\mathrm{b}\)

3. Set up the slope-intercept equation

• Start with: \(\mathrm{y = mx + b}\)
• Substitute slope: \(\mathrm{y = 3x + b}\)

4. SIMPLIFY to find the y-intercept

• Substitute point \(\mathrm{(2, 0)}\):
  \(\mathrm{0 = 3(2) + b}\)
  \(\mathrm{0 = 6 + b}\)
  \(\mathrm{b = -6}\)

5. Write the final equation

• \(\mathrm{y = 3x + (-6) = 3x - 6}\)

Answer: D. \(\mathrm{y = 3x - 6}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse x-intercept with y-intercept, thinking "x-intercept = 2" means the line passes through \(\mathrm{(0, 2)}\) instead of \(\mathrm{(2, 0)}\).

Following this misconception: \(\mathrm{0 = 3(0) + b}\) leads to \(\mathrm{b = 0}\), giving them \(\mathrm{y = 3x}\).
Since this isn't among the choices, this leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify point \(\mathrm{(2, 0)}\) but make algebraic errors when solving \(\mathrm{0 = 6 + b}\).

Common mistake: Thinking \(\mathrm{b = 6}\) instead of \(\mathrm{b = -6}\), leading them to select Choice A (\(\mathrm{y = 3x + 6}\)).

The Bottom Line:

This problem tests your ability to translate intercept terminology into coordinate points and then systematically apply the slope-intercept form. The key insight is recognizing that "x-intercept = 2" gives you a complete coordinate pair to work with.