y = 2x + 10y = 2x - 1At how many points do the graphs of the given equations intersect...

1. TRANSLATE the given equations

• Given information:
  • \(\mathrm{y = 2x + 10}\)
  • \(\mathrm{y = 2x - 1}\)
• Both equations are in slope-intercept form: \(\mathrm{y = mx + b}\)

2. INFER what the slope and intercept values tell us

• For \(\mathrm{y = 2x + 10}\): slope = 2, y-intercept = 10
• For \(\mathrm{y = 2x - 1}\): slope = 2, y-intercept = -1
• Key insight: Same slope but different y-intercepts

3. INFER the geometric relationship

• Lines with equal slopes are parallel
• Lines with different y-intercepts are distinct (not the same line)
• Therefore: These are two parallel lines that are distinct from each other

4. INFER the intersection behavior

• Parallel lines never meet
• Since these lines are parallel and distinct, they intersect at zero points

Answer: A. Zero

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students see that both equations have the same slope and conclude the lines are identical, leading them to think there are infinitely many intersection points.

They miss the crucial detail that the y-intercepts are different (10 vs -1), which means these are distinct parallel lines, not the same line. Without recognizing this distinction, they incorrectly reason that identical slopes mean identical lines.

This may lead them to select Choice D (Infinitely many)

Second Most Common Error:

Poor TRANSLATE reasoning: Students attempt to solve algebraically by setting the equations equal without first analyzing the structure.

When they get \(\mathrm{2x + 10 = 2x - 1}\), which simplifies to \(\mathrm{10 = -1}\), they may interpret this contradiction as meaning there's "exactly one" solution rather than recognizing it means no solution exists.

This may lead them to select Choice B (Exactly one)

The Bottom Line:

This problem tests whether students can connect algebraic properties (slope and intercept values) to geometric behavior (parallel vs intersecting lines). The key insight is distinguishing between "same slope" (which creates parallel lines) and "same line" (which requires both same slope AND same intercept).