In the xy-plane, line p has a slope of -{2/5} and a y-intercept of \((0, 3)\). What is the x-coordinate...

1. TRANSLATE the problem information

• Given information:
  • Slope: \(\mathrm{m = -\frac{2}{5}}\)
  • y-intercept: \(\mathrm{(0, 3)}\), so \(\mathrm{b = 3}\)
  • Need: x-coordinate of x-intercept

2. TRANSLATE into equation form

• Using slope-intercept form \(\mathrm{y = mx + b}\):
  \(\mathrm{y = -\frac{2}{5}x + 3}\)

3. INFER the approach for finding x-intercept

• The x-intercept occurs where the line crosses the x-axis
• At this point, \(\mathrm{y = 0}\)
• Strategy: Set \(\mathrm{y = 0}\) and solve for x

4. Set up and SIMPLIFY the equation

• Substitute \(\mathrm{y = 0}\):
  \(\mathrm{0 = -\frac{2}{5}x + 3}\)
• Add \(\mathrm{\frac{2}{5}x}\) to both sides:
  \(\mathrm{\frac{2}{5}x = 3}\)

5. SIMPLIFY to find x

• Divide both sides by \(\mathrm{\frac{2}{5}}\):
  \(\mathrm{x = 3 \div \frac{2}{5}}\)
• Convert division to multiplication:
  \(\mathrm{x = 3 \times \frac{5}{2} = \frac{15}{2}}\)

Answer: \(\mathrm{\frac{15}{2}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students confuse x-intercept with y-intercept terminology and try to use the given y-intercept \(\mathrm{(0, 3)}\) as their answer, focusing on the x-coordinate of the y-intercept.

This leads them to answer 0 instead of recognizing they need to find where \(\mathrm{y = 0}\).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{0 = -\frac{2}{5}x + 3}\) but make arithmetic errors when solving for x, particularly with the fraction division step.

Common mistakes include:

• Getting \(\mathrm{\frac{2}{5}x = 3}\) but then calculating \(\mathrm{x = 3 \times \frac{2}{5} = \frac{6}{5}}\) (using wrong reciprocal)
• Sign errors when rearranging the equation

This causes calculation confusion and leads to incorrect numerical answers.

The Bottom Line:

This problem tests whether students can distinguish between different intercept types and execute fraction arithmetic accurately. The conceptual understanding is straightforward, but execution precision is critical.