A line in the xy-plane has a slope of -{1/2} and passes through the point \((0, 3)\). Which equation represents...

1. TRANSLATE the problem information

• Given information:
  • Slope = \(-\frac{1}{2}\)
  • Line passes through point \((0, 3)\)

2. INFER the key insight about the point

• The point \((0, 3)\) is special because when \(\mathrm{x = 0}\), we're at the y-intercept
• This means the y-intercept \(\mathrm{b = 3}\)

3. INFER the approach

• Use slope-intercept form: \(\mathrm{y = mx + b}\)
• We have: \(\mathrm{m = -\frac{1}{2}}\) and \(\mathrm{b = 3}\)

4. SIMPLIFY by substituting values

• \(\mathrm{y = mx + b}\)
• \(\mathrm{y = (-\frac{1}{2})x + 3}\)
• \(\mathrm{y = -\frac{1}{2}x + 3}\)

Answer: B. \(\mathrm{y = -\frac{1}{2}x + 3}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students don't recognize that point \((0, 3)\) directly gives them the y-intercept. Instead, they might try to use point-slope form or get confused about what the coordinates mean.

This leads to confusion and potentially guessing among the answer choices, or attempting unnecessarily complex calculations.

Second Most Common Error:

SIMPLIFY execution error: Students understand the approach but make sign errors when writing the final equation. They might write \(\mathrm{y = -\frac{1}{2}x - 3}\) instead of \(\mathrm{y = -\frac{1}{2}x + 3}\), confusing the sign of the y-intercept.

This may lead them to select Choice A (\(\mathrm{y = -\frac{1}{2}x - 3}\)).

The Bottom Line:

The key insight is recognizing that when a line passes through \((0, \mathrm{y})\), that y-value is automatically your y-intercept in slope-intercept form. Missing this connection makes the problem much harder than it needs to be.