y = 2x + 1y = ax - 8In the system of equations above, a is a constant. If the...

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{y = 2x + 1}\)
  • \(\mathrm{y = ax - 8}\)
• We need to find the value of a that makes this system have no solution

2. INFER what "no solution" means for linear systems

• A system has no solution when the lines are parallel but not identical
• Parallel lines have the same slope but different y-intercepts
• If slopes are the same but y-intercepts differ, the lines never intersect

3. TRANSLATE slopes and y-intercepts from each equation

• First equation \(\mathrm{y = 2x + 1}\):
  • \(\mathrm{Slope = 2}\)
  • \(\mathrm{y\text{-}intercept = 1}\)
• Second equation \(\mathrm{y = ax - 8}\):
  • \(\mathrm{Slope = a}\)
  • \(\mathrm{y\text{-}intercept = -8}\)

4. INFER the required condition

• The y-intercepts are already different: \(\mathrm{1 ≠ -8}\) ✓
• For no solution, the slopes must be equal: \(\mathrm{a = 2}\)

Answer: D. 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Confusing "no solution" with "infinitely many solutions"

Students might think that for no solution, the equations must be completely different, or conversely, they might confuse the conditions and think both the slopes AND y-intercepts need to be the same. This conceptual confusion about when systems have no solution versus infinitely many solutions can lead to incorrect reasoning.

This may lead them to select Choice C (1) if they try to make the y-intercepts somehow related, or causes them to get stuck and guess.

Second Most Common Error:

Inadequate TRANSLATE reasoning: Not properly identifying slopes from slope-intercept form

Some students might misread the coefficient 'a' or not recognize that in \(\mathrm{y = ax - 8}\), the coefficient 'a' represents the slope. They might focus on the constants instead or get confused about which parts of the equations represent slopes versus y-intercepts.

This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem tests whether students truly understand the geometric meaning of "no solution" for linear systems - that it requires parallel lines that never meet.