At how many points do the graphs of the equations y = x + 20 and y = 8x intersect...

Step-by-Step Solution

1. TRANSLATE the problem information

• Given information:
  • First equation: \(\mathrm{y = x + 20}\) (slope = 1, y-intercept = 20)
  • Second equation: \(\mathrm{y = 8x}\) (slope = 8, y-intercept = 0)
  • Need to find: number of intersection points

2. INFER the approach

• Intersection points occur where both equations give the same y-value for the same x-value
• Two solution paths available:
  • Algebraic: Set equations equal and solve
  • Conceptual: Compare slopes to determine intersection behavior

3. INFER the conceptual solution (fastest approach)

• Both equations represent straight lines
• Line 1 has \(\mathrm{slope = 1}\), Line 2 has \(\mathrm{slope = 8}\)
• Since the slopes are different \(\mathrm{(1 ≠ 8)}\), the lines will intersect at exactly one point
• Lines with the same slope are parallel (0 intersections) or identical (infinite intersections)

4. Alternative: SIMPLIFY through algebraic solution

• Set the equations equal: \(\mathrm{x + 20 = 8x}\)
• Solve for x: \(\mathrm{20 = 7x}\), so \(\mathrm{x = \frac{20}{7}}\)
• This gives us one solution, confirming exactly one intersection point

Answer: B. 1

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not understand what "intersection" means graphically, thinking it refers to something like the number of terms in the equations or confusing intersection with other mathematical concepts.

This leads to confusion and random guessing among the answer choices.

Second Most Common Error:

Conceptual confusion about linear functions: Students might not recognize that these are both linear equations representing straight lines, or they may not know the fundamental property that two non-parallel lines intersect at exactly one point.

This may lead them to select Choice C (2) thinking that since there are two equations, there must be two intersection points.

The Bottom Line:

This problem tests whether students understand the geometric meaning of "intersection" and can recognize basic properties of linear functions. The key insight is that two lines with different slopes must intersect exactly once - no algebraic work is actually required if this concept is solid.