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

1. TRANSLATE the key concept

• For a system of linear equations to have no solution, the lines must be:
  • Parallel (same slope)
  • But distinct (different y-intercepts)

2. TRANSLATE each system to identify slopes and y-intercepts

Choice A: \(\mathrm{y = 6x + 3}\) and \(\mathrm{y = 6x + 9}\)

• \(\mathrm{Slope_1 = 6}\), \(\mathrm{y\text{-}intercept_1 = 3}\)
• \(\mathrm{Slope_2 = 6}\), \(\mathrm{y\text{-}intercept_2 = 9}\)

Choice B: \(\mathrm{y = 10}\) and \(\mathrm{y = 10x + 10}\)

• First equation: horizontal line (\(\mathrm{slope = 0}\), \(\mathrm{y\text{-}intercept = 10}\))
• Second equation: \(\mathrm{slope = 10}\), \(\mathrm{y\text{-}intercept = 10}\)

Choice C: \(\mathrm{y = 14x + 14}\) and \(\mathrm{y = 10x + 14}\)

• \(\mathrm{Slope_1 = 14}\), \(\mathrm{y\text{-}intercept_1 = 14}\)
• \(\mathrm{Slope_2 = 10}\), \(\mathrm{y\text{-}intercept_2 = 14}\)

Choice D: \(\mathrm{x = 3}\) and \(\mathrm{y = 10}\)

• First equation: vertical line
• Second equation: horizontal line

3. INFER which system has no solution

• Choice A: Same slopes (\(\mathrm{6 = 6}\)) ✓, Different y-intercepts (\(\mathrm{3 \neq 9}\)) ✓
  → Parallel but distinct lines → No solution
• Choice B: Different slopes (\(\mathrm{0 \neq 10}\)) → Lines intersect → Has solution
• Choice C: Different slopes (\(\mathrm{14 \neq 10}\)) → Lines intersect → Has solution
• Choice D: Vertical and horizontal lines → Always intersect → Has solution

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Confusing the conditions for no solution by thinking that when two equations have the same y-intercept but different slopes, there's no solution.

Looking at Choice C (\(\mathrm{y = 14x + 14}\) and \(\mathrm{y = 10x + 14}\)), students see both equations have \(\mathrm{y\text{-}intercept = 14}\) and think "they're the same line" or "they don't intersect." However, different slopes actually guarantee the lines will intersect at exactly one point (the y-intercept point in this case).

This may lead them to select Choice C (\(\mathrm{y = 14x + 14}\), \(\mathrm{y = 10x + 14}\)).

Second Most Common Error:

Inadequate TRANSLATE reasoning: Misidentifying what constitutes parallel lines by focusing only on coefficients rather than slopes.

Students might look at Choice A and think "\(\mathrm{6x}\) appears in both equations, but the constants are different, so they must intersect somewhere." They fail to recognize that identical slopes with different y-intercepts create parallel lines that never meet.

This leads to confusion and guessing among the other choices.

The Bottom Line:

This problem tests whether students truly understand the geometric meaning of slope and y-intercept in determining when lines are parallel versus intersecting. The key insight is that parallel lines (same slope) with different starting points (different y-intercepts) can never meet.