Which of the following systems of linear equations has no solution?

1. TRANSLATE each system to identify key characteristics

We need to analyze each system to determine slopes and y-intercepts:

• Choice A: \(\mathrm{x = 3}\), \(\mathrm{y = 5}\)
  • Vertical line and horizontal line
• Choice B: \(\mathrm{y = 6x + 6}\), \(\mathrm{y = 5x + 6}\)
  • \(\mathrm{Slope_1 = 6}\), \(\mathrm{y\text{-}intercept_1 = 6}\)
  • \(\mathrm{Slope_2 = 5}\), \(\mathrm{y\text{-}intercept_2 = 6}\)
• Choice C: \(\mathrm{y = 16x + 3}\), \(\mathrm{y = 16x + 19}\)
  • \(\mathrm{Slope_1 = 16}\), \(\mathrm{y\text{-}intercept_1 = 3}\)
  • \(\mathrm{Slope_2 = 16}\), \(\mathrm{y\text{-}intercept_2 = 19}\)
• Choice D: \(\mathrm{y = 5}\), \(\mathrm{y = 5x + 5}\)
  • Horizontal line and slanted line

2. INFER the key insight about systems with no solution

For a system to have no solution, the lines must be parallel but distinct:

• Same slope (parallel)
• Different y-intercepts (distinct/separate lines)

3. APPLY CONSTRAINTS to find systems with no solution

Checking each choice:

• Choice A: Vertical and horizontal lines intersect → Has solution
• Choice B: Different slopes (\(\mathrm{6 \neq 5}\)) → Lines intersect → Has solution
• Choice C: Same slopes (\(\mathrm{16 = 16}\)) but different y-intercepts (\(\mathrm{3 \neq 19}\)) → Parallel lines → No solution
• Choice D: Horizontal line intersects slanted line → Has solution

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't remember the relationship between parallel lines and systems with no solution.

They might think that having the same y-intercept (like Choice B) means no solution, or they might confuse "no solution" with "one solution." Without understanding that parallel lines (same slope, different y-intercepts) create systems with no solution, they end up guessing randomly.

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

Second Most Common Error:

Poor TRANSLATE reasoning: Students misidentify slopes and y-intercepts from the equations.

For example, they might not recognize that both equations in Choice C have slope 16, or they might get confused by the different formats (like the \(\mathrm{x = 3}\), \(\mathrm{y = 5}\) format in Choice A). This prevents them from properly comparing the characteristics needed to determine if lines are parallel.

This may lead them to select Choice B or get stuck and guess randomly.

The Bottom Line:

This problem tests whether students understand the geometric meaning behind "no solution" - that it occurs when you have two parallel lines that never meet. Success requires both translating equations correctly and connecting algebra to geometry.