Line k is defined by y = 1/4x + 1. Line j is parallel to line k in the xy-plane....

1. TRANSLATE the given equation into recognizable form

• Given information:
  • Line k: \(\mathrm{y = \frac{1}{4}x + 1}\)
  • Line j is parallel to line k

• This equation is already in slope-intercept form: \(\mathrm{y = mx + b}\)

2. TRANSLATE to identify the slope

• In \(\mathrm{y = mx + b}\) format:
  • \(\mathrm{m}\) (coefficient of x) = slope
  • \(\mathrm{b}\) (constant term) = y-intercept

• From \(\mathrm{y = \frac{1}{4}x + 1}\):
  • Slope = \(\mathrm{\frac{1}{4}}\)
  • Y-intercept = \(\mathrm{1}\)

3. INFER the relationship for parallel lines

• Key property: Parallel lines have identical slopes
• Since line j is parallel to line k, they must have the same slope
• Therefore: slope of line j = slope of line k = \(\mathrm{\frac{1}{4}}\)

Answer: 1/4 (equivalent to 0.25)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse slope with y-intercept in the slope-intercept form.

They might see \(\mathrm{y = \frac{1}{4}x + 1}\) and incorrectly identify the y-intercept (1) as the slope, thinking "the number at the end is the slope." This fundamental misunderstanding of slope-intercept form leads them to conclude that the slope is 1.

This may lead them to answer 1 instead of 1/4.

The Bottom Line:

This problem tests whether students truly understand slope-intercept form and the basic property of parallel lines. The key insight is recognizing that in \(\mathrm{y = mx + b}\), the coefficient of x (not the constant term) gives you the slope.