In the xy-plane, consider the equations 3y - 6x = 12 and y = kx + 4, where k is...

1. TRANSLATE the first equation to standard form

• Given equations:
  • \(\mathrm{3y - 6x = 12}\)
  • \(\mathrm{y = kx + 4}\)

• TRANSLATE the first equation to slope-intercept form:
  \(\mathrm{3y - 6x = 12}\)
  \(\mathrm{3y = 6x + 12}\)
  \(\mathrm{y = 2x + 4}\)

2. INFER the comparison strategy

• Now we can compare both lines directly:
  • Line 1: \(\mathrm{y = 2x + 4}\) (slope = 2, y-intercept = 4)
  • Line 2: \(\mathrm{y = kx + 4}\) (slope = k, y-intercept = 4)

• Key insight: Both lines have the same y-intercept (4), so intersection behavior depends entirely on slope comparison.

3. INFER intersection conditions

• For two lines to intersect at exactly one point, they must have different slopes
• When slopes are equal but y-intercepts are equal: identical lines (infinitely many intersections)
• When slopes are equal but y-intercepts differ: parallel lines (no intersections)

4. APPLY CONSTRAINTS to find the answer

• If \(\mathrm{k = 2}\): Both lines become \(\mathrm{y = 2x + 4}\) (identical lines → infinitely many intersections)
• If \(\mathrm{k \neq 2}\): Different slopes → exactly one intersection point

• The question asks which value does NOT give exactly one intersection point
• Answer: \(\mathrm{k = 2}\) gives infinitely many intersections, not exactly one

Answer: C (2)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may incorrectly convert \(\mathrm{3y - 6x = 12}\) to slope-intercept form, perhaps getting \(\mathrm{y = -2x + 4}\) instead of \(\mathrm{y = 2x + 4}\) due to sign errors when rearranging terms.

With the wrong slope for the first line, they incorrectly conclude that \(\mathrm{k = -2}\) makes the lines identical, leading them to select Choice A (-2).

Second Most Common Error:

Poor INFER execution: Students understand the algebra correctly but misinterpret what the question is asking. They find that \(\mathrm{k = 2}\) creates identical lines but think this means the lines "intersect at exactly one point" rather than recognizing that identical lines have infinitely many intersection points.

This conceptual confusion about intersection types causes them to eliminate \(\mathrm{k = 2}\) as an answer and guess among the remaining choices.

The Bottom Line:

This problem requires careful attention to both algebraic manipulation and the precise meaning of "exactly one intersection point." Students must distinguish between one intersection point, infinitely many intersection points, and no intersection points.