In the xy-plane, line m has a y-intercept of \((0, 12)\) and a slope of 4/3. What is the x-coordinate...

1. TRANSLATE the problem information

• Given information:
  • y-intercept: \((0, 12)\)
  • Slope: \(\frac{4}{3}\)
  • Need: x-coordinate of x-intercept

2. INFER the approach

• Since we have slope and y-intercept, we can write the line equation in slope-intercept form
• To find x-intercept, we need the point where the line crosses the x-axis (where \(\mathrm{y} = 0\))

3. TRANSLATE given information into equation form

Write the equation: \(\mathrm{y} = \frac{4}{3}\mathrm{x} + 12\)

4. INFER the method to find x-intercept

• At the x-intercept, \(\mathrm{y} = 0\)
• Substitute \(\mathrm{y} = 0\) into our equation and solve for x

5. SIMPLIFY to solve for x

• Set up: \(0 = \frac{4}{3}\mathrm{x} + 12\)
• Subtract 12: \(-12 = \frac{4}{3}\mathrm{x}\)
• Divide by \(\frac{4}{3}\): \(\mathrm{x} = -12 \div \frac{4}{3}\)
• Convert division to multiplication: \(\mathrm{x} = -12 \times \frac{3}{4} = -9\)

Answer: -9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not understanding what "x-intercept" means
Students might think the x-intercept is related to the y-intercept value or try to use the slope directly. They may confuse x-intercept with y-intercept and incorrectly think the answer should be 12 or try to manipulate the slope value \(\frac{4}{3}\).
This leads to confusion and random guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making errors when dividing by a fraction
Students correctly set up \(-12 = \frac{4}{3}\mathrm{x}\) but then incorrectly calculate \(\mathrm{x} = -12 \times \frac{4}{3} = -16\) instead of \(\mathrm{x} = -12 \times \frac{3}{4} = -9\). They multiply by the fraction instead of its reciprocal.
This leads them to select an incorrect positive or negative value.

The Bottom Line:

This problem tests whether students truly understand what intercepts mean geometrically and can correctly manipulate equations involving fractions. The key insight is recognizing that x-intercept means "y equals zero."