Line r in the xy-plane has a slope of 4 and passes through the point \(\mathrm{(0, 6)}\). Which equation defines...

1. TRANSLATE the problem information

• Given information:
  • Slope = 4
  • Line passes through point \(\mathrm{(0, 6)}\)
• What this tells us: Since the point has x-coordinate 0, this is the y-intercept

2. INFER the appropriate approach

• Since we have both slope and y-intercept, we should use the slope-intercept form
• The slope-intercept form is: \(\mathrm{y = mx + b}\)
• We can directly substitute our known values

3. TRANSLATE our values into the formula

• \(\mathrm{m = 4}\) (the given slope)
• \(\mathrm{b = 6}\) (the y-intercept from point \(\mathrm{(0, 6)}\))
• Substitute: \(\mathrm{y = 4x + 6}\)

Answer: D. \(\mathrm{y = 4x + 6}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse which number represents the slope versus the y-intercept. They might see "slope of 4" and "point \(\mathrm{(0, 6)}\)" and incorrectly think the slope is 6 and y-intercept is 4, leading to \(\mathrm{y = 6x + 4}\).

This may lead them to select Choice B (\(\mathrm{y = 6x + 4}\))

Second Most Common Error:

Missing conceptual knowledge about y-intercepts: Students don't recognize that point \(\mathrm{(0, 6)}\) means the y-intercept is 6. They might try to use point-slope form unnecessarily or get confused about how to use the point information.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students can efficiently translate given information into the standard slope-intercept form. The key insight is recognizing that a point with x-coordinate 0 immediately gives you the y-intercept.