In the xy-plane, a line representing a temperature change has a slope of 3/2 and passes through the point \((0,...

1. TRANSLATE the problem information

• Given information:
  • Slope \(\mathrm{m = \frac{3}{2}}\)
  • Line passes through point \(\mathrm{(0, 9)}\)
  • Need to find x-coordinate of x-intercept

• What this tells us: Since the line passes through \(\mathrm{(0, 9)}\), this point is the y-intercept \(\mathrm{(b = 9)}\)

2. INFER the equation setup

• Use slope-intercept form: \(\mathrm{y = mx + b}\)
• Substitute known values: \(\mathrm{y = \frac{3}{2}x + 9}\)

3. INFER what x-intercept means

• The x-intercept is where the line crosses the x-axis
• At this point, the y-coordinate equals 0
• Set \(\mathrm{y = 0}\) in our equation

4. SIMPLIFY to solve for x

• Start with: \(\mathrm{0 = \frac{3}{2}x + 9}\)
• Subtract 9 from both sides: \(\mathrm{-9 = \frac{3}{2}x}\)
• Multiply both sides by \(\mathrm{\frac{2}{3}}\) (reciprocal of \(\mathrm{\frac{3}{2}}\)): \(\mathrm{x = -9 \times \frac{2}{3}}\)
• Calculate: \(\mathrm{x = \frac{-18}{3} = -6}\)

Answer: -6

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion: Mixing up x-intercept and y-intercept definitions

Students might think "x-intercept means set x = 0" and conclude the answer is 9. This fundamental misunderstanding of intercept definitions leads to identifying the y-intercept instead of solving for the x-intercept. This leads to confusion and incorrect grid-in responses.

Second Most Common Error:

Weak SIMPLIFY execution: Algebraic manipulation errors when solving \(\mathrm{-9 = \frac{3}{2}x}\)

Students correctly set up the equation but make errors like:

• Dividing by \(\mathrm{\frac{3}{2}}\) instead of multiplying by \(\mathrm{\frac{2}{3}}\)
• Sign errors when moving terms
• Fraction arithmetic mistakes

This leads to answers like 6, -13.5, or other incorrect values being entered in the grid.

The Bottom Line:

This problem tests whether students truly understand what intercepts represent, not just how to manipulate equations. The key insight is recognizing that "x-intercept" means "where y equals zero," then executing clean algebraic steps to solve.